Analysis of Chimera Behavior in Coupled Logistic Maps on Different Connection Topologies

Transcription

1 Analysis of Chimera Behavior in Coupled Logistic Maps on Different Connection Topologies Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science by Research in Computational Natural Science by Pranneetha Bellamkonda International Institute of Information Technology (Deemed to be University) Hyderabad , INDIA August, 2015

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3 Copyright Pranneetha Bellamkonda, 2015 All Rights Reserved ii

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9 ACKNOWLEDGMENTS I would like to express my deepest gratitude to my supervisor Dr. Nita Parekh for her constant support and thorough guidance. She gave me the opportunity to pursue research and motivated me from time to time. The regular detailed discussions helped me in building concepts and improving the understanding of the subject. I would also like to thank her for encouraging and helping me to shape my interest and ideas. I want to thank the Center for Computational Natural Sciences and Bioinformatics at the International Institute of Information Technology, Hyderabad for giving me support in various forms throughout my research. I would also like to thank the faculty of the centre, Dr. Harjinder Singh, Dr. U. Deva Priyakumar, Dr. Prabhakar, Dr. Krishnan, Dr. Abhijit Mitra, Dr. Gopalakrishnan and Dr. Rameshwar for encouraging and helping me to enhance my knowledge in various fields. I would also like to thank Dr. Kamal Karlapalem and Dr. Kavita Vemuri for their valuable guidance in my CS projects. Thanks to my dear friends Rashmi tonge, Avni Verma and Mounica Maddela for creating a healthy environment in the workspace and giving their honest inputs whenever needed. Special thanks to my fellow CND students Jaya and Supriya for being there with me all the time. Lastly, I would like to take this opportunity to thank my parents for everything they have done for me. They made selfless sacrifices throughout their life so that I can get the best possible opportunity for growth. This work is a result of their blessings and good wishes. viii

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11 ABSTRACT In the last decade there has been considerable interest in a novel dynamical phenomenon, chimera behavior in which regions of coherence and incoherence coexist. In this study we analyze the emergence of chimera states in different connection topologies. Firstly we analyze chimera behavior in non-locally coupled logistic maps for varying range and strength of coupling. The dependence on initial conditions is shown by considering different initial conditions viz., uniform, random and hump shaped. To capture the dependence of the strength of coupling on the spatial distance between the nodes, we considered two decaying functions for the coupling strength and analyzed the chimera behavior in non-locally coupled networks. The robustness of the chimera behavior in the presence of noise in the system parameters and connection topology is also studied. To capture different connection topologies and their effect on the dynamical behavior of the whole system, we also analyzed nearest-neighbor diffusely coupled, globally coupled, randomly connected networks. Apart from these dynamical behavior on small-world and scale-free networks is also discussed. The objective of this analysis is to introduce heterogeneity in the connection topology and analyze the dynamical behavior of the whole system. In various physical and biological systems, viz., power grids and excitable tissues, synchronous behavior of all the components is generally desired. In such situations there is a real need to control the undesired asynchronous behavior of a few nodes for proper functioning of the system. With this objective we study the control of asynchronous regions in the chimera behavior. The analysis has also been extended to two-dimensional systems. x

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13 CONTENTS CHAPTER 1 INTRODUCTION DYNAMICAL BEHAVIOR IN SPATIAL SYSTEMS CHIMERA BEHAVIOR CHIMERA STATES ON OTHER NETWORKS CONDITIONS FOR OBSERVATION OF CHIMERA STATES Initial Condition Robustness CHIMERAS IN EXPERIMENTAL SYSTEMS ORGANIZATION OF THESIS 31 CHAPTER 2 MATERIALS AND METHODS GRAPHS AND THEIR PROPERTIES Graphs Degree Degree Distribution Average Path Length Average Clustering Coefficient NETWORK TOPOLOGIES Regular Network Random Network Small-World Network Scale-Free Network NON-LINEAR DYNAMICS AND LOGISTIC MAPS Non-linear Dynamics Logistic Map Lyapunov Exponent Quantification of Chaos on a Network Effect of Connection Topologies on Dynamics of Coupled Logistic Map Dynamics CHIMERA STATES IN LOGISTIC MAP MODELS Model and Simulations 56 CHAPTER 3 RESULTS AND DISCUSSION 58 xii

14 3.1 INTRODUCTION ANALYSIS OF REGULAR NETWORKS: SPATIAL EXTENT OF COUPLING SENSITIVITY TO INITIAL CONDITIONS ANALYSIS OF PARAMETER SPACE Type I Initial Conditions and r = Type II Initial Conditions and r = Type I Initial Conditions and r = NON-UNIFORM COUPLING ROBUSTNESS Noise in System Parameters and Initial Conditions Noise in Topological Connections EFFECT OF EXTERNAL PERTURBATIONS CHIMERA STATES IN TWO-DIMENSIONAL LATTICES Initial Conditions Control of Incoherence CHIMERA STATES ON COMPLEX NETWORKS Scale-Free Networks Random Networks Small-world Networks 87 CHAPTER 4 CONCLUSION 90 xiii

15 LIST OF FIGURES Figure 1.1: The temporal behavior of a node in a locally coupled regular network where each node follows logistic map dynamics, x n+1 = rx n (1-x n ) a) Stable fixed point at r = 2.5 b) Periodic behavior at r = 3.1 and c) chaotic temporal behavior for r = 4.0. For plots 1b and 1c all the other parameters remain the same. Here N = 400 and ԑ = 0.2. The initial conditions were randomly chosen in the range [0, 1] and dynamics is shown after eliminating transients Figure 1.2: Chimera state observed for Ginzburg Landau oscillators for the parameters α = and r = 0.3. This figure has been reproduced from Kuramoto and Battogtokh(2002) s paper Figure 1.3: X axes of all the four plots correspond to time whereas the Y axes correspond to mean intensity of the oscillators. The inset in each of the subplots shows the spatial phase distribution of 40 oscillators. The color blue corresponds to the synchronized group A and red corresponds to group B. This figure has been reproduced from Tinsley et al. (2012) Figure 2.1: G 1 is a directed graph with 4 vertices and 4 edges and G 2 is an undirected graph with 5 nodes and 7 edges Figure 2.2: A graph with 7 nodes. The degree of node 3 is 5 and the average degree of the graph is 24/7 = Figure 2.3: A locally coupled regular network with N = 20 nodes in which each node is connected to two immediate neighbors on either side is shown with r c = 2/20 = Figure 2.4: A globally coupled regular network with N = 15 nodes is shown in which each node is connected to all the remaining nodes of the network. Here r c = 7/15 = Figure 2.5: A non-locally coupled regular network with N = 20 nodes in which each node is connected to five neighbors on each side i.e. rc = Figure 2.6: The degree distribution of a regular network of size 500 where a) locally coupled (r c = 0.004) b) non-locally coupled (r c = 0.4) and c) globally coupled (r c = 0.498) Figure 2.7: A random network of size, N = 20 with average degree k = 4 is shown Figure 2.8: The degree distribution of a random network of size N = 500 with average degree, k = Figure 2.9: Construction of a small-world network. As p changes, the network transforms from regular to small-world to random. This figure has been reproduced from Watts and Strogatz, (1998) Figure 2.10: The behavior of L p /L 0 (in blue) and C p /C 0 (in green) with respect to log(p). This plot has been obtained for 50 different configurations (50 values of p) Figure 2.11: Degree distribution of a small-world network of size N = 500 with an average degree of 6 is shown. It may be noted that k = 5, 6 and 7 have non-zero probabilities xiv

16 Figure 2.12: Construction of a scale-free network through preferential attachment, starting with a seed of size 2. This figure has been reproduced from Barabási and Bonabeau (2003) Figure 2.13: (a)the degree distribution of a 500 node scale-free network generated using Albert Barabasi s algorithm is shown. (b) The degree distribution of the same on a negative log scale shows a linear fit with γ = Figure 2.14: The dynamical behavior of a logistic map with respect to the bifurcation parameter r exhibiting period doubling route to chaos is shown Figure 2.15: a, b, c, d depict the temporal dynamics of the logistic map at r = 2.9, 3.2, 3.8 and 4 respectively. e, f, g, h show the corresponding phase plane plot of x t+1 vs x t. Dynamics for 100 time steps have been plotted after eliminating the transients Figure 2.16: Three dimensional non-linear system showing trajectories starting from three initial points Figure 2.17: The spatio-temporal dynamics and x n+1 vs x n behavior of a regular-locally coupled logistic map lattice of size 100 for r = 2.5, 3.2, 3.6 and 3.9 are depicted in a-b, c-d, e-f and g-h respectively transients have been eliminated and ԑ = 0.3. Random initial conditions have been used Figure 2.18: The dynamics of (a) locally coupled, (b) non-locally coupled, (c) random, (d) small-world and (d) scale-free network all of size N = 500. The coupling strength ԑ = 0.3 and r = 3.6 (weak chaos) transients have been eliminated and randomly distributed initial conditions have been used. The λ values for (a) (e) are , , 0.031, and respectively Figure 2.19: a) Spatial dynamics of x given by equation 2.17 b) spatial dynamics of R. The system size N = 400, and the system parameters are r = 3.8, r c = 0.32, ԑ = Regions shaded in blue are the regions of incoherence transients have been eliminated. Type I initial conditions have been used Figure 3.1: Spatial dynamics of R for a) local b) non-local c) global coupling on a regular network for type I initial conditions is shown if. The parameters (r c, ε) for the plots a, b and c are (0.0025, 0.24), b, e are (0.32, 0.25) and c, f are (0.498, 0.24) and r = Figure 3.2: Spatio-temporal dynamics of the non-locally coupled logistic maps with (a) uniform, (b) type I, (c) random initial conditions in the interval (0, 1) for r = 3.8, r c = 0.32 and ԑ = Figure 3.3: a c The three types of initial conditions considered here, viz., type I, II and III shown. d f spatial dynamics of non-locally coupled logistic maps shown after eliminating transients for the corresponding initial conditions in a - c. g i depict the spatial behavior of R corresponding to the dynamics in d f respectively. The parameters (r c, ε) for the plots d, g are (0.32, 0.24), e, h are (0.25, 0.25) and f, i are (0.24, 0.24) and r = xv

17 Figure 3.4: r c - ԑ parameter-space plot shown for non-locally coupled logistic maps with N = 400 on considering type I initial conditions. Here r = 3.8. The chimera states emerge in regions shown in blue, purple and yellow, while red and green regions correspond correspond to temporal periodic dynamics with period-4 and period-2 respectively. The yellow region indicates intersection of blue and green regions and purple region, the intersection of blue and red regions. K denotes the wave number of the spatial dynamics Figure 3.5: a - e The spatial dynamics of x i and f j the spatial behavior of Ri shown for the points a, b, c, d and e in Figure 3.2 respectively. The parameter values (r c, ε) for the points a, b, c, d and e in Figure 3.2 correspond to (0.32, 0.08), (0.32, 0.2), (0.32, 0.24) and (0.32, 0.42), (0.15, 0.26) respectively. The spatio-temporal dynamics for the corresponding points is shown in the third column. MLE values corresponding to dynamical behavior in a e are 0.290, , , and respectively Figure 3.6: Dependence of degree of incoherence on the radius of coupling, r c and coupling strength, ε is shown and for (a) spatial behavior of R for r c : 0.2 (red), 0.32 (green) and 0.4 (blue) for a constant coupling strength, ε = 0.24 and (b)spatial behavior of R for r c : 0.2 (red), 0.32 (green) and 0.4 (blue) for a constant coupling strength, ε = 0.24 and (b)spatial behavior of R for ε : 0.24 (red), 0.36 (green) and 0.42 (blue) for a radius of coupling, r c = Type I initial conditions have been used Figure 3.7: r c - ԑ parameter-space plot shown for non-locally coupled logistic maps with N = 400, r = 3.8 and type II initial conditions Figure 3.8: (a) - (e) depicts the spatial dynamics (xi vs i) at points a, b, c, d and e respectively in the r c - ԑ parameter-space plot (Figure 3.7). (e) (h) spatial plots of R i corresponding to the plots (a) (e). The parameter values (rc, ε) for the points a b, c, d and e in Figure 3.7 correspond to (0.25, 0.1), (0.25, 0.25) and (0.25, 0.3), (0.25, 0.36), (0.2, 0.25) respectively. The points a, b, c and d correspond to wave number K = 1, while point e to wave number K = Figure 3.9: r c - ԑ parameter-space plot for r = 3.6. The chimera states emerge in regions shown in purple, green and red. The purple region corresponds to a temporal dynamics with period greater than 4. The green and red regions correspond to temporal periodic dynamics with period-4 and period-2 respectively Figure 3.10: Figure 3.8: (a) - (e) depicts the spatial dynamics (x i vs i) and (f) (j) depict spatial plots of R i vs i corresponding to dynamics in (a) (e). The (r c, ε) values corresponding to the points a, b, c, d and e in Figure 3.9 are (0.05,0.08), (0.125, 0.08), (0.125, 0.2), (0.125, 0.35) and (0.325,0.2) respectively. The wave number K = 1, r = 3.6 and N = 400. The spatiotemporal dynamics is shown xvi

18 in the third column. MLE values for spatio-temporal dynamics corresponding to a e are 0.001, , , and respectively. Type I initial conditions have been used Figure 3.11: To linearly decaying coupling function, ԑ ij as a function of distance d ij between nodes i and j is shown Figure 3.12: a - d depicts the spatial dynamics (x i vs i) and (e) (h) depict spatial plots of R i vs i corresponding to the plots a d. The values of corresponding to a, b, c and d are 0.24, 0.3, 0.36 and 0.4 respectively for a constant r c = Type I initial conditions have been used Figure 3.13: Exponentially decaying coupling function. Here k = 2.0 and μ = Figure 3.14: a - d the spatial dynamics (x i vs i) and (e) (h) spatial plots of R i vs i corresponding to the plots a d. The values of μ corresponding to a, b, c and d are 0.35, 0.4, 0.45 and 0.5 respectively for a constant r c = 0.32 and k = 2. Type I initial conditions have been used Figure 3.15: In (a) (d) is shown the spatial dynamics of x and their corresponding spatio-temporal plots (in the 2nd column) for different coupling strengths, ԑ = 0.08, 0.2, 0.24 and 0.42 respectively ± and r c = The parameters values of the lattice are r = 3.8 ± 0.04, and the random variations to the type I initial conditions are x i (0) ± transients were eliminated Figure 3.16: The maximum percentage of edges that can be rewired for chimera states to emerge for ε = 0.2. The initial conditions are of type I and N = Figure 3.17: The spatial dynamics of x for a constant r c = 0.32 and percentage of nodes rewired being 15% and 30% in a and b respectively and the corresponding spatial plots of R are shown in c and d. The initial conditions are of type I and r = Figure 3.18: (a) Spatial behavior of R for the parameters N = 400, r c = 0.32, ε = 0.24, shown on selectively pinning the incoherent regions with varying strengths of (a) negative pinning: (green) and (red); (b) positive pinning: 0.02 (green), 0.05 (red). The plots in blue correspond to no pinning in both (a) and (b). It is clear that negative pinning suppresses while positive pinning enhances incoherence. Type I initial conditions have been used Figure 3.19: Spatial behavior of R: i) no pinning (blue), ii) selective pinning of (green) after eliminating transients and iii) eliminating transients after removing the selective pinning (red). Here, r = 3.8, N = 400, r c = 0.32, ԑ = Type I initial conditions have been used.. 78 Figure 3.20: Initial conditions on a 2 dimensional lattice of size 100x100. The nodes in the outer rectangle (blue in color) have a constant value of 0.45 while those in the inner rectangle (red in color) have a constant value of 0.9. The patch in between has random values from the set (0.4, 0.5) (0.85, 0.95) given by equations xvii

19 Figure 3.21: In (a) (d) is shown the spatial dynamics of x and (e) (h), the spatial dynamics of R on a 2- d lattice of dimensions 100x100 for the parameters, r = 3.8, δ = 2. The value of r c is 0.16 for all the plots and the values of ε for a, b, c and d are 0.15, 0.2, 0.25 and 0.3 respectively Figure 3.22: : In (a) (d) is shown the spatial dynamics of x and (e) (h), the spatial dynamics of R on a 2-d lattice of dimensions 100x100 for the parameters r = 3.8, δ = 2. The value of ε is 0.2 for all the plots and the values of r c for a, b, c and d are 0.08, 0.12, 0.16 and 0.2 respectively Figure 3.23: In (a) (c) is shown the spatial dynamics of R on a 2-d lattice of dimensions 100x100 for the parameters r = 3.8, δ = 2, r c = 0.2, ε = 0.2. (a) corresponds to dynamics with no pinning, (b) with a pinning strength of -0.2 and (c) dynamics, steps after removal of pinning Figure 3.24: The degree distributions of scale free networks of size 1000 and mean degree 2 (points in blue) and 6 (points in green) Figure 3.25: In a, b is shown the spatial dynamics of x and c, d, the spatial dynamics of R on a scale-free network for the parameters N = 1000, r = 3.6, ε = 0.11 with mean degree 2 and 6 respectively. Type I initial conditions have been used Figure 3.26: Degree distributions of random networks of size 400. Plot in blue and green indicate the degree distributions with average degrees 10 and 40 respectively Figure 3.27: In a, b is shown the spatial dynamics of x and c, d, the spatial dynamics of R on a random network for the parameters N = 400, r = 3.6, ε = The average degree for the random networks corresponding to a and b are 10 and 60 respectively. Type I initial conditions have been used Figure 3.28: In a and b, the behavior of L p /L 0 (in blue) and C p /C 0 (in green) with respect to log(p) for small-world networks of size 500 and mean degrees 6 and 40 respectively are shown. This plot has been obtained for 50 different configurations (50 values of p) Figure 3.29: In a and b is shown the spatial dynamics of x and c and d, the spatial dynamics of R on a small-world networks of mean degrees 6 and 40 respectively for the parameters N = 400, r = 3.6, ε = 0.2 and p = Type I initial conditions have been used xviii

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21 LIST OF TABLES Table 2.1: The C and L values for a regular network of size 500 for increasing r c Table 2.2: The variation in C and L values of a network of size 500 with p Table 2.3: Dependence of linear and non-linear functions on initial conditions Table 2.4: Sensitive Dependence on initial Conditions for the logistic map when r = Table 3.1 Expected and actual values of C and L for a random network of size xx

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23 Chapter 1 INTRODUCTION 1.1 DYNAMICAL BEHAVIOR IN SPATIAL SYSTEMS Dynamical systems are those whose states evolve over the course of time. At any given time a dynamical system has a state which can be represented as a point in its state space or phase space. Every dynamical system has an evolution rule that is deterministic, which is mathematically represented by one or more functions. When any of these functions is non-linear, the system shows chaotic dynamics and sensitive dependence to initial conditions. One of the most widely studied non-linear model that exhibits a wide variety of dynamical behavior including chaos is the logistic map. The logistic map is a non-linear difference equation proposed by Robert May (1976) to study population dynamics. Mathematically it defines the rule for the change in the population density at a given time as a function of growth rate and the population at the previous time step. A number of non-linear models have been proposed to study real systems such as Hodgkin Huxley (Hodgkin and Huxley, 1952), FitzHugh-Nagumo (Sherwood, 2014), Hindmarsh Rose models for neuronal dynamics (Hindmarsh and Rose, 1984) ; Rossler system (Rössler, 1976) and Belousov Zhabotinsky reaction for modeling chemical systems (Tomita and Tsuda, 1979); Lorentz system (Lorenz, 1963)for modeling atmospheric dynamics, etc. Since real systems are not isolated points but are multi-particle systems, the dynamical behavior is constructed by considering multiple systems coupled in space. Such a coupled system is called a network. There has been considerable interest in the role of connection topology on the dynamics of a network. Spatio-temporal dynamics of various non-linear systems have been widely studied in the past. Labyrinthine patterns, spiral waves and spots are some of the complex spatiotemporal patterns that have been observed in non-linear systems. While spiral and labyrinthine patterns have been observed mainly in chemical reaction-diffusion systems and some neuronal systems (Hagberg and Meron, 1994); (Lindsay and Schulthess, 1958),

24 spot patterns have been widely observed in a number of non-linear systems. In a spatiotemporal system, dynamical behaviors such as fixed point, periodic, quasi-periodic and chaotic behavior have also been observed and can be characterized by the maximal Lyapunov exponent. In Figure 1.1 is shown fixed point, periodic and chaotic for a particular node in a locally coupled logistic lattice of size 500. Similar to the temporal behavior of any node, the spatial behavior may exhibit a wide variety of dynamical behavior ranging from periodic to chaotic. Furthermore the spatial dynamics may be characterized as synchronous (all nodes exhibiting the same value at a given time step) or asynchrounous. Earlier partly synchronous and partly asynchronous spatial behavior was observed only in systems in which all nodes were not identical (e.g. operating in different parameter regime). Kuramoto and Battogtokh (2002) for the first time observed this novel dynamical behavior, chimera behavior in a non-locally coupled system of identical Ginzburg Landau oscillators wherein a group of oscillators are coherent while the others are incoherent, in spite of the oscillators being identical. Since then much attention has focused over the last decade on the analysis of chimera behavior in various systems. In this thesis we studied the generation, characteristics and stability or robustness of the chimera dynamical behavior on different network topologies. Dynamics on different types of connection topologies is studied to understand the role of connection topology in obtaining chimera states, both in one- and two-dimensional lattices. The topology of a network determines how different nodes in the network are connected to each other. It plays an important role in the resultant spatial dynamics of the system. For instance, neuro-plasticity which amounts to local changes in the underlying topology of neuronal networks leads to improved memory and intelligence (Vicente et al., 2008). Different network topologies varying from simple regular lattice in which every node is connected to a fixed number of its nearest neighbors on either side; to the more complex small-world and scale-free network topologies with heterogeneous connectivity have been studied. A regular lattice can be further classified into locally coupled, non-locally coupled and globally coupled lattices depending on the spatial extent of coupling. In a locally coupled lattice, each node is connected to only its nearest neighbors whereas in a globally coupled lattice each node is connected to every other 23

25 node in the lattice. In a non-locally coupled lattice each node is connected to a fixed number of neighbors, k (1 < k < N). The effect of non-local coupling on the dynamics has been rigorously studied to generate chimera patterns (Kuramoto, 1995). Here we investigate the role of non-local coupling to observe chimera states in spatially coupled logistic maps for varying extent of spatial coupling. In this study we also considered random, small-world and scale free networks to observe chimera behavior. Two variants of a random graph model, G(N, p ) and G(N, M) were proposed by Edgar Gilbert (1959) and Erdős and Rényi (1964) respectively. According to the G(N, M) model, a graph of size N is chosen uniformly at random from all possible M-edge graphs. And according to the G(N, p ) model, every possible edge in a graph is included independently of the other edges with a probability p. Both the variants generate random graphs with Poisson degree distribution. Small path length and clustering coefficient and Poisson degree distribution distinguish random networks from other kinds of networks. Watts and Strogatz (1998) proposed small world networks, wherein majority of the nodes have nearest neighbor interactions (similar to a regular network) and very few nodes have long range interactions. This results in high clustering coefficient (similar to regular networks) and small path lengths (similar to random networks) and a node can be reached by any other node in a few hops. Small-world networks are characterized by small path length and high clustering coefficient. Watts in his work Six Degrees describes the small-world phenomenon which is, any two people in the world are connected through a chain of acquaintances that has no more than five intermediates (Watts, 2004). Scale-free networks, proposed by Barabási and Albert (1999), are constructed by preferential attachment algorithm. According to this algorithm, a new node is connected to an already existing node with a probability proportional to the degree of the existing node. Therefore scale-free networks have a hub structure, where a small fraction of nodes (hubs) have a very high degree when compared to the degrees of the remaining nodes. Because of the huge difference in the degrees of the hubs and normal nodes, the network has no scale and hence the name scale-free. As a result of preferential attachment, scale-free networks exhibit power law behavior i.e. P(k) α k -γ, which distinguishes scale-free networks from other complex networks. Real world 24

26 networks such as social networks, World Wide Web, protein-protein interaction networks, airline networks, etc. have been shown to exhibit the scale-free property. Figure 1.1: The temporal behavior of a node in a locally coupled regular network where each node follows logistic map dynamics, x n+1 = rx n (1-x n ) a) Stable fixed point at r = 2.5 b) Periodic behavior at r = 3.1 and c) chaotic temporal behavior for r = 4.0. For plots 1b and 1c all the other parameters remain the same. Here N = 400 and ԑ = 0.2. The initial conditions were randomly chosen in the range [0, 1] and dynamics is shown after eliminating transients. 1.2 CHIMERA BEHAVIOR Kuramoto and Battogtokh (2002) showed that a ring of identical non-locally coupled Ginzburg Landau oscillators given by the equation 1.1 showed a peculiar behavior for certain chosen initial conditions. They observed that some of the oscillators in the ring synchronized while the others were asynchronous. Chimera states were then defined as spatiotemporal patterns of synchrony and disorder in homogeneous, nonlocally coupled excitable systems. Such a state was given the name chimera by Steve Strogatz for its similarity with the Greek mythological creature which is a hybrid of a lion, a goat and a snake. What makes the chimera behavior interesting is the coexistence of distinct spatial regions of synchronized behavior and irregular incoherent behavior, in networks of identical and symmetrically coupled units. Until recently much attention has not been focused on networks with non-local coupling in spite of applications in wide areas, viz., chemical oscillators (Mikhailov and 25

27 Showalter, 2008), excitable systems e.g., neural tissue (Vicente et al., 2008), Josephson junctions (Wiesenfeld et al., 1996) etc. There is now renewed interest in non-local networks with the recent discovery of chimera states. In various numerical studies, it has been shown that non-local coupling is a necessary condition for the occurrence of chimera states (Abrams and Strogatz, 2006); with global or local coupling, identical oscillators either synchronize or oscillate incoherently. In addition, the emergence of chimera states is extremely sensitive to the initial conditions and is observed only for carefully chosen initial conditions in most systems (Omelchenko et al., 2011). In this system, the variation in the intensity of oscillators causes multi-stability by increasing the number of fixed points, thereby causing chimera states. In all these networks the emergence and stability of chimera states typically depends on connection topology, coupling strength and time lag if any. Depending on these parameters, in some systems, chimera states with multiple coherent and incoherent regions were also observed (Maistrenko et al., 2014). 1.1 where ф i denotes the phase of the oscillator i, ω the natural frequency of the system. K is the fixed number of neighbors of each node and α denotes the phase coupling parameter. G ij is the coupling strength between oscillators i and j which decays linearly with the distance between two nodes. Figure 1.2: Chimera state observed for Ginzburg Landau oscillators for the parameters α = and r = 0.3. This figure has been reproduced from Kuramoto and Battogtokh(2002) s paper. 26

28 Figure 1.2 shows a spatially asynchronous state at the centre of the lattice with synchronized oscillators on either sides. Stable chimera states were observed only for specially prepared initial conditions. The initial state was prepared such that a group of oscillators were phase-synchronized with each other, while others were asynchronous. Kuramoto and Battogtokh also showed that the appearance of chimera states depends on parameters such as time lag α and the strength of coupling between the oscillators given by G ij. Abrams and Strogatz (2004) performed a numerical analysis of Kuramoto and Battogtokh s paper and predicted that chimera states can be observed in experimental systems with non-local coupling such as laser array systems, superconducting arrays of Josephson junctions and biochemical oscillators (such as oscillators in the Brusselator reaction diffusion system). Chimera behavior is observed in nature in Uni-hemispheric sleep in neuronal systems of birds and dolphins which sleep with half of their brain (synchronous state) while the other half remains awake (asynchronous state) (Mathews et al., 2006). Uni-hemispheric sleep helps the dolphins to look out for danger while they rest, stop them from drowning and maintain body heat needed to survive in frigid oceans. Therefore such a chimera state is very much needed for the survival of dolphins. Successful functioning of power grid depends on the synchronous working of the generators in it. A small group of generators falling out of synchrony forms a chimera state will lead to power failure. Hence chimera dynamics in a power grid is undesirable and hence there is a need to take the power grid back to the state where all the generators work synchronously. Spiral chimera states have been observed in the heart tissue during cardiac fibrillation when an area in tissue contracts asynchronously when compared to the other areas. This condition may lead to sudden cardiac death. A pacemaker is used to treat this condition by forcing the whole tissue to contract synchronously. Hence there is a need to study the conditions suitable for the emergence of chimera states, conditions under which they seize to exist and the role that connection topology plays in the chimera behavior. 27

29 1.3 CHIMERA STATES ON OTHER NETWORKS In earlier studies it was assumed that non-local coupling was a necessary condition for the emergence of chimera states. However, recently chimera states have been observed in lattices with global coupling as well as in scale-free and random networks. Yeldesbay et al. (2014) observed the emergence of chimera states in a network of globally coupled identical oscillators. They showed that the emergence of chimera states in this case was caused by the bi-stability which appears dynamically due to internal delayed feedback in individual oscillators. They further showed that even without delayed feedback, chimera behaviour can emerge in globally coupled lattices if the individual units are naturally bi-stable. Wang et al. (2009) observed chimera states on scale-free networks with transmission delays between neurons. The neuronal dynamics was simulated using the Rulkov map and it was found that the chimeras were not only robust to variations in the size of the network and the scaling exponent of scale-free network but also to the intensity of external noise induced. Recently, Zhu et al. (2014) observed chimera states in both scale-free and random networks. They analyzed the dependence of chimera states on the mean degree of the network. They observed that for random networks the degree of oscillators in the coherent region were almost the same. However in the scale-free network, the degree distribution of oscillators in the coherent regions exhibited power law. 1.4 CONDITIONS FOR OBSERVATION OF CHIMERA STATES Initial Condition An important factor for the emergence of chimera states apart from the underlying topology and dynamical parameters is the initial state the system is in. Kuramoto and Battogtokh (2002) placed the system in a partially synchronous and partially asynchronous initial state which was observed to be crucial for the emergence of chimera states in their study. Specially prepared initial conditions have also been used in systems of coupled 28

30 phase oscillators (Omelchenko et al., 2011), oscillators exhibiting bubble dynamics (Hsiao and Chahine, 2001), neuronal oscillators (Sheeba et al., 2009) for the emergence of chimera states. These special initial conditions are generally hump shaped in order to induce bistability and give the system a jump start (Abrams and Strogatz, 2006). However emergence of chimera states have been shown for random initial conditions also. Omelchenko et al. (2011) observed spiral chimera states in a two-dimensional lattice for random initial conditions. They used a phase lag parameter to induce chimera behavior. Similarly Sethia et al. (2008) observed chimera behavior in a system of Kuramoto oscillators coupled in a non-local and time-delayed fashion with a random distribution of initial phases (between 0 and 2π), which are spatially organized in a mirror-symmetric fashion. These chimera states have a number of spatially disconnected regions of coherence with intervening regions of incoherence. To see the dependence of chimera behavior on initial conditions, three different specially prepared initial conditions are considered in this study. In our analysis we were unable to observe chimera states for random initial conditions Robustness Perturbations or small variations are intrinsic to any real system. A number of studies have been carried out to check the robustness of chimera states to structural perturbations. Recently chimera states were shown to persist in non-locally coupled lattices with small perturbations in their spatial structures. Yao et al. (2013) observed that chimera states in a system of non-locally coupled Ginzburg Landau oscillators are robust to removal of random links upto a certain threshold. Laing et al. (2012) analyzed chimera states in a two-cluster network of phase oscillators with strong coupling within the subnetworks and weak coupling between them. These chimera states were robust to random removal of a few links, but the range of parameters for which the chimera states exist was reduced. In this thesis we analyze the robustness of chimera states to changes in connection topology by rewiring edges in the network with certain probability. We have also analyzed the robustness of chimera states to small perturbations in the initial 29

31 conditions and system parameters such as bifurcation parameter of the logistic map and coupling strength. 1.5 CHIMERAS IN EXPERIMENTAL SYSTEMS The initial studies on chimera states were only based on computational results. In 2012, two different groups showed the occurrence of chimera states in experimental systems. The first group Tinsley et al. (2012) divided photosensitive oscillators into two groups (A and B) and used light to provide feedback for the reactions. Chimera states were observed when oscillators were weakly coupled with those of the other group and strongly coupled with the oscillators within the same group with a time delay. The distinctive temporal behavior of groups A and B for a certain set of parameters are shown in Figure 1.3. In addition to chimera states, clusters (shown in Figure 1.3a) were also observed. In another study Hagerstrom et al. (2012) experimentally constructed optical coupled map lattices (CMLs) with periodic boundary conditions and used a liquid-crystal spatial light modulator (SLM) to control the polarization properties of an optical wavefront. They showed that for both one-dimensional and two-dimensional CMLs, the emergence of chimera behavior depends on coupling strength and radius of coupling. They also observed that chimera states emerged when there is chaos in the local dynamics. Martens et al. (2013) placed groups of metronomes on two swings, which were coupled with the help of springs. They observed that the emergence of chimera states depends on the spring constant and the metronome frequency. They showed that for intermediate values of spring constant, there exists a wide range of metronome frequency for which chimera states emerge. The chimera states were observed when initially, one of the populations of metronomes was completely synchronized and the other was desynchronized. Larger et al. (2013) constructed an experimental realization of a time delayed system of Ikeda oscillators. They observed that the emergence of chimera states depends on the system parameters, time delay and characteristic response time. In the one-dimensional experimental setup, it was observed that random initial conditions 30

32 gave rise to chimera states whereas specially prepared initial conditions are necessary for the emergence of chimera states in two and three dimensional systems. Figure 1.3: X axes of all the four plots correspond to time whereas the Y axes correspond to mean intensity of the oscillators. The inset in each of the subplots shows the spatial phase distribution of 40 oscillators. The color blue corresponds to the synchronized group A and red corresponds to group B. This figure has been reproduced from Tinsley et al. (2012). 1.6 ORGANIZATION OF THESIS In this thesis, we have analyzed different topological networks to observe chimera behavior using logistic map as the dynamical node on the networks. In Chapter 2 the dynamical behavior of a single logistic map and that of networks of coupled logistic maps is discussed. Characterization of different spatiotemporal dynamical behavior is also discussed. In chapter 3 we discuss the conditions for observing chimera states on different topological networks of logistic maps, including two-dimensional lattices; their robustness to perturbations in system parameter well as structural parameters and control of chimera states. In chapter 4 we discuss the conclusions of our study. 31

33 Chapter 2 MATERIALS AND METHODS In this chapter we introduce the concept of a graph and the various centrality measures of graphs that will be used in the study. In section 2.2, various topologies such as regular lattices with local, non-local and global coupling, random, small world and scale-free networks are defined and their construction is discussed. In section 2.3 we logistic map as a dynamical system is introduced to model different dynamical behavior. Finally in section 2.4, a novel dynamical behavior, chimera behavior is introduced; we give an introduction to chimera behavior; a parameter to quantify chimera behavior and the initial conditions which lead to the emergence of chimera states in networks of logistic maps are discussed. 2.1 GRAPHS AND THEIR PROPERTIES Graphs A graph is a collection of vertices and edges. Every edge in a graph connects two vertices. An edge between two nodes i, j is denoted by e i,j. A Graph is represented mathematically by an adjacency matrix [a ij ] where a ij = 1 only when there exists an edge between the nodes i and j and is 0 otherwise. In this thesis we use the term nodes to refer to vertices and the term network to refer to a graph. Figure 2.1 shows two graphs G 1 (directed) and G 2 (undirected). In the graph G 1 vertex 1 is a neighbor of vertex 2 but not vice versa as each edge in G 1 is a directed edge. Since graph G 2 is an undirected graph, vertex 5 is a neighbor of 6 and vice versa. In this thesis we will consider undirected networks. Graph theory is the mathematical study of graphs and their properties. It can help in identifying important nodes using centrality measures of a graph such as degree, betweenness, closeness, principal eigenvector etc. For example, in a social network, nodes with high degrees are considered to be the public figures in the network, whereas 32

34 the nodes with high closeness (harmonic mean of the distance to all vertices) may not be public figures but are influential within their local community network. Apart from social networks, graph theory has also been useful in the study of wide variety of systems such as complex molecules, computer networks, protein-protein networks, the internet, citation networks, food webs, metabolomic networks, etc. Figure 2.1: G 1 is a directed graph with 4 vertices and 4 edges and G 2 is an undirected graph with 5 nodes and 7 edges. Graphs can also be classified depending upon whether the topology is fixed or changes with time. Networks with fixed topologies have a constant set of nodes and edges. Networks with changing topologies can either have a constant set of nodes and variable set of edges or have varying sets of both nodes and edges. Most real world networks such as communication networks, social networks, neuronal networks, web, etc. have varying sets of nodes and edges, but the time scale of dynamics on each node is far less than the time scale of change in the connection topology. Earlier studies on non-linear dynamical systems were mainly on nearest neighbor coupled lattices. In recent studies, non-homogeneous systems with non-local and heterogeneous connection topology is being studied since most real systems do not exhibit regular connectivity between its individual entities. Studies on molecular (Mathews et al., 2006), neuronal (Apolloni et al., 2009) and communication networks (Midgley et al., 1992) revealed that alterations in the connection topology may have a significant effect on the spatio-temporal dynamics of the system. In this thesis we study 33

35 the role of connection topology on coupled logistic map dynamics, in particular, on a novel dynamical behavior, chimera behavior Degree Degree of a node i in a network is the number of its neighbors and is defined as 2.1 where a ij is the ij th term of the adjacency matrix and takes a value 1 if nodes i and j are connected and 0 otherwise. Average degree k avg of a network can be defined as the average of the degree of all the nodes in the network. In Error! Reference source not ound. 2.2 is shown a network where N = 7, k 1 = 3, k 3 = 5 and k avg = 3. Figure 2.2: A graph with 7 nodes. The degree of node 3 is 5 and the average degree of the graph is 24/7 = The importance of a node in a network may be determined by its degree. Nodes with high degree, called hubs have the potential to influence the network globally. Hubs also act as easy targets to collapse the entire network by external attack Degree Distribution Degree distribution is the probability distribution of the degree of all the nodes in a network and is given by P(k) = N k /N

36 where N k is the number of nodes with degree k. The degree distribution of a network is useful in characterizing different topological networks as discussed in Section Average Path Length Average path length gives a rough estimate of how quickly can information flow across the network can spread in it. It is defined as the arithmetic mean of the shortest distance between every pair of nodes in the network. In order to define average path length we need to first get familiar with the notion of the shortest path between two nodes. The shortest path between two nodes i and j is the path with least number of edges between. To calculate the average path length of a network, we calculate the sum of the shortest path lengths between every pair of nodes using Dijkstra's algorithm. Consider the nodes 1 and 6 in the graph shown in Figure 2.2. There exist a number of paths between nodes 1 and 6 of varying lengths of which the paths 1 2 6, are shortest with length 2. Therefore the shortest path length between the nodes 1 and 6 is 2. Average path length of a network, L is 2.3 where d ij is the distance between nodes i and j and N is the size of the network. The average path length (L) for the graph shown in Figure 2.2 is Average Clustering Coefficient The clustering coefficient of a node shows how well connected the neighborhood Nbr i of a node i is. The neighborhood Nbr i of a node i is defined as the set of nodes that are directly connected to node i. The local clustering coefficient of a node and the global average clustering coefficient (C) of a network is defined as 35

37 Here k i denotes the degree of node i and e jl is an edge connecting nodes j, l belong to Nbr i. In Figure 2.2, node 3 has 5 neighbors, 0, 1, 2, 4 and 6. There are only 4 connections, ((0 1), (1 2), (2 6), (6 4)) between these 5 neighbors. So C 3 = 2 4/5(5-1) = 0.4. The average clustering coefficient of the graph in Figure 2.2 is Since the clustering coefficient gives us how well connected the neighborhood of a node is, it is a local property of a network. Average path length on the other hand is the global property of a network as it gives an estimate of the spread of information through the network. Average path length and clustering coefficient are two important topological properties used to characterize networks with different connection topologies. 2.2 NETWORK TOPOLOGIES The topology of a network schematically represents how the nodes in a network are connected to each other. It has been observed that topology has a significant effect on the dynamics of the network (Plaut, 2003; Wang, 2002). Below we discuss various network topologies considered in this study Regular Network A regular network with periodic boundary conditions has equal number of connections on either side of each node in the network, i.e. the degree of each node is a constant, say k. Regular networks can be classified as local, non-local and globally coupled networks depending on the extent of coupling to the neighboring nodes. Here, the extent of coupling on one side of node i is defined as radius of coupling r c = k/2n. 36

38 Locally Coupled Network A regular network in which each node is connected to a small number of its neighbors on either side is called a locally coupled network, i.e. r c ~ k/2n, where k << N. Figure 2.3 shows a locally coupled regular network of size 20 with every node connected to 2 nodes on each side. 2 Figure 2.3: A locally coupled regular network with N = 20 nodes in which each node is connected to two immediate neighbors on either side is shown with r c = 2/20 = Globally Coupled Network A regular network in which each node is coupled to all the remaining nodes in the network is called a globally coupled network i.e. r c ~ ~ 0.5. In this case k ~ N-1. Figure 2.4 shows a globally coupled regular network with r c =

39 Figure 2.4: A globally coupled regular network with N = 15 nodes is shown in which each node is connected to all the remaining nodes of the network. Here r c = 7/15 = Non-locally Coupled Network In a non-locally coupled network, the radius of coupling is neither as small as in a locally coupled network nor is a large as (N-1)/2 as in a globally coupled network i.e. 1/N < r c < (N-1)/2N. Figure 2.5 shows a non-locally coupled regular network with r c = Figure 2.5: A non-locally coupled regular network with N = 20 nodes in which each node is connected to five neighbors on each side i.e. rc =

40 Average Clustering Coefficient and Average Path Length of Regular Networks Let us examine how the average clustering coefficient (C) and average path length (L) vary as the radius of coupling, r c in a regular network. In Table 2.1 is listed the C and L values corresponding to increasing r c values for a network of size N = 500. Table 2.1: The C and L values for a regular network of size 500 for increasing r c S S. No r c C L In Table 2.1, the first row corresponds to a locally coupled network with only one edge on either side of each node i, with r c = 1/500 = while the fifth row corresponds to a globally coupled network where a node i is connected all other nodes in the network i.e. r c = 248/500 = The second, third and fourth rows correspond to non-locally coupled networks with 10, 50 and 100 nodes on each side respectively. From Table 2.1 we observe that Average clustering coefficient increases with an increase in the radius of coupling in a regular network. Average path length decreases with an increase in the radius of coupling in a regular network. 39

41 As the radius of coupling in a regular network increases each node is connected to an increasing number of its neighbors, which in turn are connected to an increasing number of their neighbors. We have already seen how clustering coefficient is a measure of how well connected the neighborhood of each node is. Hence non-locally coupled lattices have a higher clustering coefficient and lower path length when compared to that of locally coupled lattices which is evident from Table Degree Distribution of Regular Networks As discussed in section 2.1.3, degree distribution is the probability distribution of the degree of all the nodes in a network. In a regular network, all its nodes have the same degree. In Figure 2.6 is shown the degree distributions for locally, non-locally and globally coupled lattices corresponding to 2 (r c = 0.004), 100 (r c = 0.2) and 249 (r c = 0.498) connections on each side of every node. Figure 2.6: The degree distribution of a regular network of size 500 where a) locally coupled (r c = 0.004) b) non-locally coupled (r c = 0.4) and c) globally coupled (r c = 0.498). We observe degree distributions for the three networks is similar with the peak at k = Nr c Random Network One of the first definitions of random networks was given by Erdős and Rényi (1964). They defined two random-graph models named G(N, M) and G(N, p') where N 40

42 and M denote the number of vertices and edges of the graph and p denotes the probability of rewiring the edges. 1. The G(N,M) model chooses a graph uniformly at random from a collection of graphs with N vertices and M edges. Hence every graph in the model is chosen with a probability of 2. In the G(N, p') model, each edge in a graph of N vertices is included independently of other edges with a probability of p'. In this model, each graph with N vertices and M edges is chosen with a probability of and the expected number of edges is. Asymptotically, both the models behave similarly because the expected number of edges according to G(N, p') model, = M as N. The degree distribution of a G(N, p ) random graph is in the form of a binomial distribution and is a Poisson distribution for a large values of N. Later Watts and Strogatz proposed another approach to construct a random network. In this approach, starting with a regular network of degree k, rewire every edge in the network with a rewiring probability p. When the rewiring probability is 1, then a random network of average degree k is obtained. For a random network thus obtained Watts and Strogatz observed that C ~ k/n and L ~ ln(n)/ln(k) and the degree distribution is a Poisson distribution. Figure 2.7 is a random network of size 20 and average degree 4 generated using the Watts and Strogatz approach. In this study we use Watts and Strogatz approach to construct the random networks. 41

43 \ Figure 2.7: A random network of size, N = 20 with average degree k = 4 is shown. The degree distribution of a random network with N = 500 nodes and average degree 6, constructed by Watts and Strogatz approach is shown in Figure 2.8. It exhibits Poisson distribution with mean equal to 6, the average degree of the network. Figure 2.8: The degree distribution of a random network of size N = 500 with average degree, k = 6. Table 2.2: The variation in C and L values of a network of size 500 with rewiring probability, p P C p L p C Reg L Reg

44 C Ran L Ran The values of C and L for networks with N = 500, average degree 6 and increasing p values are depicted in Table 2.2. The values of C and L for the random network whose degree distribution is shown in Figure 2.8 are and (corresponding to p = 1.0 in Table 2.2). It may be noted that the clustering coefficient and path length of this random network are both very small when compared to that of a regular network with the same average degree (corresponding to p = 0.0 in Table 2.2). The decrease in average path length is because as long range connections are introduced, nodes that were farther from each other get connected thus decreasing the average path length Small-World Network Small-world networks lie between the two extremes, regular networks and random networks. Social networks, internet and gene networks exhibit small world characteristics. Watts and Strogatz (1998) proposed a small world network model in which we start with a regular network and rewire each of its edges with a certain probability. By rewiring each edge of a regular network with probability p, long range connections are being introduced in the network as shown in Figure 2.9. As a result of the long range connections, it can be seen, from Figure 2.9, that only 2 hops are needed to reach node 2 from node 1 in the small-world network (Figure 2.9b) when compared to 4 hops in the regular network (Figure 2.9a). Before we define a small-world network let us denote average path length (clustering coefficient) of a network with a rewiring probability of p as L p (C p ). In Figure 2.10 is plotted the values of C p /C 0 and L p /L 0 on varying rewiring probability from 0 to 1. We observe a narrow window of p values (shown by the shaded portion in Figure 2.10) for which clustering coefficient C p ~ C 0 and the path length L p ~ L 1. The networks thus obtained with p in this window are called small-world networks. That is, a small-world network is well-clustered, like a regular network and has a small average path length, like a random network. 43

45 Figure 2.9: Construction of a small-world network. As p changes, the network transforms from regular to small-world to random. This figure has been reproduced from Watts and Strogatz, (1998) Figure 2.10: The behavior of L p /L 0 (in blue) and C p /C 0 (in green) with respect to log(p). This plot has been obtained for 50 different configurations (50 values of p) For small world networks the rewiring probability is that value between 0 and 1 for which the average path length of the network L p is nearly equal to L 1 (average path length of a random network with the same average degree) and the average clustering coefficient, C p is nearly equal to C 0 (average clustering coefficient of a regular lattice with the same average degree). Hence in the shaded region in Figure 2.10 where p lies between and 0.1, we observe the small-world property i.e. Lp/L 1 ~ 1 and Cp/C 0 ~ 1. That is, a small-world network is well-clustered, like a regular network and has a small average path length, like a random network. 44

46 Figure 2.11 shows the degree distribution of a small world network with N = 500 and average degree 6. We can see that since the probability of rewiring is small, there is only a small fraction of nodes with degree 5 and 7 and the degree distribution is very similar to that of a regular network with mean degree 6. Figure 2.11: Degree distribution of a small-world network of size N = 500 with an average degree of 6 is shown. It may be noted that k = 5, 6 and 7 have non-zero probabilities Scale-Free Network A scale-free network is defined as a network which exhibits degree distribution follows a power law, P(k) = k -γ. The value of γ for most scale-free networks is a real number that lies between 2 and 3. The topologies of a gamit of real world networks like the network of web-pages, the collaborative network of Hollywood actors, citation networks have been observed to exhibit scale-free properties. To construct a scale-free network we start with a very small network known as the seed and keep adding nodes iteratively. The new nodes are connected to the rest of the network through a process called preferential attachment, where a new node gets attached to an existing node with a probability proportional to its degree. In Figure 2.12 is shown the process of preferential attachment, starting with a seed network of size 2 and at each step a new node (in green) is attached preferentially to an existing node (in red) with a higher degree. This preferential attachment algorithm has been proposed by Albert and Barabasi (1999) to construct a scale-free network. This gives rise to a hub-structure, where very few nodes having a very high degree in comparison to a large number of 45

47 nodes having a very small degree, which is not seen in any other type of network. Hence scale-free networks are robust to random failures and vulnerable to non-random attacks. In Figure 2.13a is shown the degree distribution plot P(k) vs k and in Figure 2.13b log(p(k)) vs log(k) for a scale-free network of size 500. The degree distribution of a scale-free network, though looks like an exponential function (Figure 2.13a), is not so, because an exponential degree distribution would account for a considerable number of nodes with intermediate degrees. However, due to the hub-structure, scale-free networks do not have nodes with intermediate degrees between the two extremes, which is in accordance with power law behavior. A linear plot is observed in the log-log plot with the slope of the straight line, γ = -2.4 giving the degree exponent. Figure 2.12: Construction of a scale-free network through preferential attachment, starting with a seed of size 2. This figure has been reproduced from Barabási and Bonabeau (2003). Figure 2.13: (a)the degree distribution of a 500 node scale-free network generated using Albert Barabasi s algorithm is shown. (b) The degree distribution of the same on a negative log scale shows a linear fit with γ =

48 The values of clustering coefficient and path length for this network are 0.1 and 3.0 respectively. Comparing these values with the C and L values of Table 2.2 we observe that the average path length of the above scale-free network (with average degree 6) is close to that of a random network (3.69) with the same average degree. The clustering coefficient of the scale-free network however lies between that of a regular (0.6) and random network (0.0097). Hence we can say that scale-free networks like small world networks have short path lengths comparable to those of random networks. However unlike small world networks the scale-free networks do not have a high clustering coefficient. 2.3 NON-LINEAR DYNAMICS AND LOGISTIC MAPS Non-linear Dynamics Most real systems such as neuronal systems, weather dynamics and population dynamics are nonlinear in nature. These systems show a non-intuitive behavior where simple changes in one part of the system may produce complex changes throughout the system. A representative example of linear and non-linear functions is given in Equations 2.6 and 2.7. x t+1 = f(x t ) = 3x t x t+1 = g(x t ) = x t 2.7 In linear systems a small error in the initial conditions translates into a proportionally small error in the final state. On the other hand in a non-linear system, as said by Henri Poincare, small differences in the initial conditions produce very great ones in the final phenomenon. A small error in the former will produce an enormous one in the latter. This is called sensitive dependence on initial conditions which is typical of non-linear systems, demonstrated in Table 2.3 for the equations 2.6 and

49 Table 2.3: Dependence of linear and non-linear functions on initial conditions Time Linear function f Non-linear function g step t x t x t δx t x t x t δx t In Table 2.3 when x is governed by a linear function we observe that a small difference in the initial value of x only translates proportionally. On the other hand when x is governed by a non-linear function we observe that a small difference in the initial value of x translates into a large difference in the value of x at the end of time step 4. This difference continues to increase exponentially with time Logistic Map Let us discuss the origin of the logistic map, a logistic model created by Pierre Francois Verhulst in 1838 that represents population growth. Although the logistic model depicts the trend in population growth it is far from reality as we assume a constant birth and death rate in this model as shown in equation 2.8. n t+1 = (b_rate d_rate) (n t n 2 t /M) 2.8 where n t denotes the population at time step t, b_rate and d_rate denote the birth and death rates respectively and M denotes the maximum possible population. Dividing equation 2.8 by M we get n t+1 /M = (b_rate d_rate) (n t /M n 2 t /M 2 ) 2.9 Defining growth rate r = b_rate d_rate and population density at time t, x t = n t /M we obtain the difference equation given by equation

50 x t+1 = r(x t x t 2 ) 2.10 Equation 2.10 represents a logistic map (popularized by Robert May (1976)) where x t ranges from 0 to 1. From Figure 2.12 (bifurcation plot of logistic map) we observe a fixed point at r = 2.9 (point a), i.e. whatever be the initial x 0 after a few time steps x t converges to a fixed point. At r = 3.2 (point b), the dynamics is periodic with period 2. From Figure 2.12 we also observe that as the value of r increases the period increases by multiples of 2. This is known as the period-doubling behavior of the logistic map. At r = 3.8 and 4.0 in Figure 2.12 it becomes hard to predict what the period is. This state represents chaos.. Figure 2.14: The dynamical behavior of a logistic map with respect to the bifurcation parameter r exhibiting period doubling route to chaos is shown. 49

51 Figure 2.15: a, b, c, d depict the temporal dynamics of the logistic map at r = 2.9, 3.2, 3.8 and 4 respectively. e, f, g, h show the corresponding phase plane plot of x t+1 vs x t. Dynamics for 100 time steps have been plotted after eliminating the transients. Figures 2.15 a, b c and d show the temporal dynamics of a single logistic map corresponding to points a, b, c and d respectively in Figure From Figure 2.15d we observe that there is no finite period at r = 4.0 and hence it is a chaotic state. Here we demonstrate sensitive dependence on initial conditions in Table 2.4 by observing the behavior of x for two different initial conditions. As is clearly seen in Table 2.4 value of δx t diverges very quickly exhibiting sensitive dependence to initial conditions when the system is in the chaotic state (r = 4). Thus we observe that though the dynamics is governed by a well defined equation, long time behavior is unpredictable as a consequence of sensitivity to initial conditions. Table 2.4: Sensitive Dependence on initial Conditions for the logistic map when r = 4 Time step t x t (x 0 = 0.15) x t (x 0 = ) δx t

52 Lyapunov Exponent Lyapunov exponent (LE) is a measure of chaos in a dynamical system, i.e. it is a quantitative measure of the sensitive dependence on initial conditions. It quantifies the rate of separation of two nearby trajectories of a system and is given by where λ is the rate of separation between two trajectories of the system. Shown in Figure 2.16 is a three dimensional system where the distance between the black and blue trajectories is very small initially (δx(0)) and keeps growing with time (δx(t)). From Figure 2.16 we also observe that the sphere that contains the initial set of points in the neighborhood of x transforms into an ellipsoid after time t. At t, λ in equation 2.12 is the Lyapunov exponent for the system shown in Figure Given the equations of motion of a system, where x i and x j are two variables, δx i (t)/δx j (0) is given by the Jacobian matrix as shown in equation where J ij is the term in the Jacobian matrix corresponding to the i th row and j th column. The sign of λ quantifies different dynamical behavior. λ < 0 indicates a fixed point or a periodic orbit 51

53 λ = 0 indicates a marginally stable orbit λ > 0 indicates chaos The Lyapunov exponents corresponding to the r values, 2.5, 3.2, 3.8 and 4.0 in Figures 2.15a, b, c and d are -1.2, -0.8, 0.42 and 0.69 respectively. Clearly, we observe negative λ values for periodic dynamics and positive λ values for chaotic dynamics. Figure 2.16: Three dimensional non-linear system showing trajectories starting from three initial points Quantification of Chaos on a Network We have already seen that mathematically Lyapunov exponent can be defined as a quantity that characterizes the rate of separation of infinitesimally close trajectories. In a spatial system of n dimensions, this rate of separation can be different in different dimensions. Hence for an n dimensional system we have a vector of n LEs called the Lyapunov spectrum. When we have a system of m logistic maps then the Jacobian matrix is defined as 2.14 Let us define the Lyapunov spectrum with the help of the Jacobian matrix. The LE corresponding to the i th dimension is 52

54 At the t th iteration, The size of the Lyapunov exponent spectrum is equal to the number of dimensions in the system, but it suffices to compute the maximum lyapunov exponent (MLE) which is an indicator of the extent of chaos in the system. The MLE is computed for the dynamics of locally coupled logistic map network depicted in Figure 2.17 for different values of the bifurcation parameter r. Figure 2.17: The spatio-temporal dynamics and x n+1 vs x n behavior of a regular-locally coupled logistic map lattice of size 100 for r = 2.5, 3.2, 3.6 and 3.9 are depicted in a-b, c-d, e-f and g-h respectively transients have been eliminated and ԑ = 0.3. Randomly chosen values in the range [0, 1] have been used as initial conditions. The maximum Lyapunov exponent for coupled logistic map dynamics is computed by using the algorithm proposed by Wolf et al. (1985). In order to calculate the Lyapunov spectrum for a set of non-linear equations, the non-linear equations are linearized using the Jacobian matrix for these equations. These linearized equations are applied on a set of N orthonormal vectors for a certain number of time steps which results in the divergence of these vectors. The resulting vectors are reorthonormalized using Gram Schmidt 53

55 Reorthonormalization. The MLEs thus obtained corresponding to dynamics in Figures 2.17a, c, e and g are , , and respectively. It is clear that the MLE is positive for chaotic dynamics (Figure 2.17e, g) and negative for stable and periodic dynamics (Figure 2.17 a, c) Effect of Connection Topologies on Dynamics of Coupled Logistic Map Dynamics Until now we have seen the variety of dynamical behavior exhibited by a single logistic map ranging from fixed point, periodic, to chaotic behavior. However in reality systems are spatially extended. We see that individual systems interact with each other and as a result influence each others dynamical behavior. In this thesis we consider the logistic map on different network topologies representing different connectivity patterns (discussed in section 2.2) and study their effect on the dynamics of the coupled system. Mathematically the coupled logistic map on a network is represented as 2.15 f(x) = rx(1-x), k i denotes the degree of node i, and j ~ i indicates that there exists a connection between i and j and ԑ is the coupling strength. In Figure 2.18 is shown the spatio-temporal dynamics of the logistic map on five different topologies viz., (a) regular local coupling, r c = (b) regular non-local coupling, r c = 0.4 (c) random, (d) smallworld and (e) scale-free networks for r = 3.6 and ԑ = 0.3. From the spatio-temporal dynamics in Figure 2.18a - e we observe that for the locally coupled regular network (λ = ), non-locally coupled regular network (λ = ) and small world network (λ = ), spatio-temporal dynamics shows periodic behavior whereas for the random (λ = 0.031) and scale-free networks (λ = 0.186) spatio-temporal behavior show chaotic behavior. 54

56 Figure 2.18: The dynamics of (a) locally coupled, (b) non-locally coupled, (c) random, (d) small-world and (d) scale-free network all of size N = 500. The coupling strength ԑ = 0.3 and r = 3.6 (weak chaos) transients have been eliminated and randomly distributed initial conditions have been used. The λ values for (a) (e) are , , 0.031, and respectively. 2.4 CHIMERA STATES IN LOGISTIC MAP MODELS In spatially extended non-linear systems different spatio-temporal patterns such as spiral, spot and labyrinthine patterns have been observed in reaction-diffusion and 55

57 ecological systems. The spatial behavior alone has also been studied extensively for nonlinear systems and can be classified broadly into coherence and incoherence. The coexistence of these two kinds of spatial behavior has been shown in a system of identical coupled oscillators by Kuramoto and Battogtokh (2002). Since then chimera states have been observed in numerous computational and experimental studies. In this thesis we characterize the emergence of chimera states on different connection topologies and study the sensitivity to initial conditions in the emergence of chimera states Model and Simulations In our model, we consider identical logistic maps at every node of the lattice which are coupled to P neighbors on either side on the spatial lattice. The spatio-temporal dynamical system considered here is given by the equation 2.16 where i = 1, 2,, N, t denotes the time step, ε the coupling strength and each node is coupled to P number of nodes in either direction, i.e., a total of 2P connections. The local function considered here is a logistic map given by f(x) = rx(1 - x), r being the bifurcation parameter. The radius of coupling r c = P/N is a constant for all the nodes in the lattice. For a locally coupled network (nearest-neighbor coupling), r c = 1/N while for a globally coupled lattice, r c = (N-1)/2N ~ 1/2. Here we consider non-local coupling defined as 1/N << r c << 1/2. To mathematically quantify incoherence, the parameter R i which gives the degree of incoherence in a local region surrounding the node i is defined as 2.17 where i = 1, 2,, N, δ denotes the neighborhood of a node i on its either side over which the extent of incoherence is measured. The value of R i is directly proportional to the level of incoherence in the region around the node i of size δ on each side with R i = 0 56

58 in a perfectly coherent region and R i > 0 in a region of incoherence. In Figure 2.19a is shown the spatial dynamics of a non-locally coupled logistic map lattice at a given time t, after eliminating the transients. In Figure 2.19b is shown the corresponding spatial behavior of R for δ = 10. It may be noted that the value of R in the coherent region is close to 0, while R > 0 in the incoherent regions. Thus, analyzing the behavior of R i helps in detecting the presence of chimera states in the lattice. (Note that δ = 10 is used in the calculation of R calculations when N = 400 in the rest of the study). Figure 2.19: a) Spatial dynamics of x given by equation 2.17 b) spatial dynamics of R. The system size N = 400, and the system parameters are r = 3.8, r c = 0.32, ԑ = Regions shaded in blue are the regions of incoherence transients have been eliminated. Type I initial conditions have been used. 57

59 Chapter 3 RESULTS AND DISCUSSION 3.1 INTRODUCTION In this study we analyze the chimera states in coupled 1 dimensional and 2 dimensional logistic maps. First we present our analysis of chimera behavior in nonlocally coupled regular networks for three different initial conditions and three different types of coupling functions. To analyze the sensitivity to initial conditions, we introduce noise in the initial conditions and study the emergence of chimera behavior. Since it is unlikely that in any physical system, all the units are exactly identical, the effect of heterogeneity on their collective behavior is of interest. With this objective we analyze the emergence and stability of chimera states in the presence of noise in the system parameters, viz., the bifurcation parameter and the coupling strength. In addition to this we also study the robustness of chimera states to changes in the connectivity of nonlocally coupled networks by rewiring the edges. In the event of the system exhibiting undesirable dynamical behavior in localized regions, viz., cardiac arrhythmia, epileptic neural activity, desynchronization in coupled chemical reactors, etc there is a need to enhance or curb the incoherent dynamics in certain regions. With this objective we analyze the effect of external perturbation or pinning given selectively to regions of incoherence/coherence on the spatiotemporal dynamics of the whole system. The emergence of chimera states in two-dimensional non-locally coupled networks is also analyzed and control of the local region of coherence/incoherence shown by external pinning. Finally we show the existence of chimera dynamical behavior on complex networks such as small-world, scale-free and random networks. 3.2 ANALYSIS OF REGULAR NETWORKS: SPATIAL EXTENT OF COUPLING To assess the effect of extent of coupling in the emergence of chimera states, we next consider logistic maps on a regular nearest neighbor coupled lattice, non-locally 58

60 coupled lattice and globally coupled lattice with specially prepared, Type I initial conditions (Section 3.3). On a nearest neighbor coupled lattice, we observe that R i > 0 for all i, suggesting asynchronous behavior in the lattice as shown in Figure 3.1a. Similarly on a globally coupled lattice, R i = 0 for all i suggesting synchronous behavior in the lattice as shown in Figure 3.1c. However on a non-locally coupled lattice, chimera states emerge with incoherent regions interspersed between regions of coherence while for nonlocally coupled network, the lattice exhibits regions of coherence interspersed between incoherent regions (Figure 3.1c) Figure 3.1: Spatial dynamics of R for a) local b) non-local c) global coupling on a regular network for type I initial conditions is shown if. The parameters (r c, ε) for the plots a, b and c are (0.0025, 0.24), b, e are (0.32, 0.25) and c, f are (0.498, 0.24) and r =

61 3.3 SENSITIVITY TO INITIAL CONDITIONS We analyze the spatio-temporal dynamics for uniform, type I and random initial conditions. The spatio-temporal dynamics of the lattice was analyzed for 50 different random initial configurations and chimera behaviour was not observed in any of these cases. On using uniform initial conditions, we observe that the lattice is completely synchronized as depicted in Figure 3.2a. In Figures 3.2b the spatiotemporal dynamics clearly exhibits chimera states on using type I initial conditions. On choosing random initial conditions x i (0) ϵ (0,1), the spatio-temporal dynamics is completely asynchronous as depicted in Figures 3.2c. Figure 3.2: Spatio-temporal dynamics of the non-locally coupled logistic maps with (a) uniform, (b) type I, (c) random initial conditions in the interval (0, 1) for r = 3.8, r c = 0.32 and ԑ = In numerous studies it has been shown that hump-shaped initial conditions give rise to chimera states. Here we consider three different hump-shaped initial distributions of x to induce chimera states in the 1d spatially coupled lattice. TYPE I: In this case, the initial distribution of x is composed of two distinct coherent regions separated by regions of incoherence. In the incoherent regions x randomly takes either of the two coherent values as defined below and depicted in Figure 3.3a: x i (0) = 0.45, i (0, N/8) and (7N/8, N) 3.1 x i (0) = 0.9, i (N/4, 3N/4) 3.2 x i (0) I, I = (0.4, 0.5) (0.85, 0.95), i (N/8, N/4) and (3N/4, 7N/8)

62 After eliminating the transients we observe that the system exhibits chimera behavior with the spatial lattice compartmentalized into alternate coherent and incoherent regions as shown in Figure 3.3d. The spatial behavior of R in Figure 3.3g further confirms the chimera state. TYPE II: This is defined as a double tanh function on the whole lattice as shown in Figure 3.3b and given by, i [0, N/2) 3.4, i [N/2, N) 3.5 The chimera behavior is observed in this case also as shown in Figure 3.3e and 3.3h, very similar to that observed with type I initial conditions. TYPE III: In this case the initial distribution of x is defined by a smooth sine wave function defined over the lattice given by i [0, N), init x i = sin(i /N) 3.6 as shown in Figure 3.3c. The spatial behavior of x in Figure 3.3f and R i in Figure 3.3i clearly exhibits the emergence of chimera behavior in this case also. Unlike type I and type II conditions, in this case there is neither a sharp discontinuity in the x value along the lattice, nor a flat region with nodes having the same value. In the case of type II or type III initial conditions, the chimera states are observed only when the local dynamics is chaotic, while the type I initial conditions give rise to chimera states even when the local dynamics is periodic. Thus we observe that specially prepared initial conditions give rise to chimera states as long as the coupling is non-local. 61

63 Figure 3.3: a c The three types of initial conditions considered here, viz., type I, II and III shown. d f spatial dynamics of non-locally coupled logistic maps shown after eliminating transients for the corresponding initial conditions in a - c. g i depict the spatial behavior of R corresponding to the dynamics in d f respectively. The parameters (r c, ε) for the plots d, g are (0.32, 0.24), e, h are (0.25, 0.25) and f, i are (0.24, 0.24) and r = ANALYSIS OF PARAMETER SPACE We observe that apart from initial conditions, the emergence of chimera states is observed to be dependent on two parameters, the range of coupling, r c, and the strength of coupling, ε. Next we analyzed the (r c, ε) parameter space for observing chimera dynamics in non-locally coupled logistic map lattices when the local dynamics exhibits high chaos (r = 3.8) and weak chaos (r = 3.6). We also compared the (r c, ε) parameter spaces for both type I and type II initial conditions when the local dynamics exhibits high chaos Type I Initial Conditions and r = 3.8 Considering the individual logistic maps in high chaotic regime (r = 3.8) and type I initial conditions, we analyzed (r c, ε) parameter space for non-locally coupled maps. 62

64 The dynamical behavior exhibited is mainly spatially and temporally periodic as shown in Figure 3.4. The chimera behavior is observed over a small coupling strength ranging from 0.15 to K denotes the wave number of the spatial dynamics. It is observed that for constant coupling strength, chimera states with higher wave numbers (K = 2) are observed at lower radii of coupling than with wave number (K = 1). Also, as the range of coupling r c is increased, the chimera behavior is observed for a very narrow range of coupling strengths (0.05, 0.1). Though r = 3.8 belongs to the chaotic regime for an individual logistic map, the chimera states that emerge for the three types of initial conditions (defined in Section 3.3) are temporally and spatially periodic even in the incoherent regions. In Figure 3.4, the red and green regions correspond to temporally periodic dynamics (in the incoherent regions) with period = 4 and 2 respectively. The yellow region corresponds to chimera dynamics with temporal period = 4, while the purple region indicates chimera dynamics with temporal period = 2 and the blue region corresponds to chimera dynamics with periods other than 2 and 4. Figure 3.4: r c - ԑ parameter-space plot shown for non-locally coupled logistic maps with N = 400 on considering type I initial conditions. Here r = 3.8. The chimera states emerge in regions shown in blue, purple and yellow, while red and green regions correspond correspond to temporal periodic dynamics with period-4 and period-2 respectively. The yellow region indicates intersection of blue and green regions and purple region, the intersection of blue and red regions. K denotes the wave number of the spatial dynamics. 63

65 In Figure 3.5 the spatiotemporal dynamical behavior of the system at points marked a, b, c, d and e in Figure 3.4 is depicted in the third column. The coupling strength corresponding to point a in Figure 3.4 is very low (ε = 0.08) and we observe incoherent dynamics in Figure 3.5a, which is quantified by R in plot 3.5b. It may be noted from the spatial dynamics of R corresponding to point b (ε = 0.2) and c (ε = 0.24) in Figures 3.5g and h, that the degree of incoherence decreases with increase in coupling strength ε (for fixed P = 128). On further increasing the coupling strength (ε > 0.35), the regions of incoherence disappear and the lattice is seen to exhibit coherent dynamics, depicted in Figures 3.5d and i. The spatio-temporal dynamics at larger coupling strength, ε = 0.42 (P = 128) corresponding to d is coherent. That is, at higher coupling strengths for fixed radius of coupling, the incoherent regions in the spatiotemporal lattice are reduced and the lattice can exhibit synchronous dynamics above a certain threshold, ε. The spatio-temporal dynamics corresponding to point e where P = 60 and ε = 0.26 exhibits higher wave number (K = 2) and period = 4. The incoherent regions in the chimera states corresponding to Figures 3.5b, c and e exhibit spatial period, 2. The dynamical behavior shown in Figure 3.5 is quantified by computing the MLE and except at very low coupling strength (ε = 0.08) in Figure 3.5a, all other cases, the spatiotemporal dynamical behavior is periodic. This is also observed in the R plots in Figures 3.5f - i, i.e., as the coupling strength, ε increases, the degree of incoherence R i is reduced and the lattice exhibits synchronous behavior at ε = In Figure 3.6a is depicted the spatial behavior of degree of coherence, R i, for different values of radius of coupling r c, corresponding to 80, 128 and 160 connections on either side for a constant ԑ = In Figure 3.6b is depicted the the spatial plot of R i for the ԑ values 0.24, 0.36 and 0.42 and a constant P = 128. Thus, as the coupling strength is increased for a fixed radius of coupling, the degree of incoherence is reduced. We observe that an increase in coupling strength ԑ, and radius of coupling r c, results in an overall reduction in the degree of incoherence. 64

66 Figure 3.5: a - e The spatial dynamics of x i and f j the spatial behavior of Ri shown for the points a, b, c, d and e in Figure 3.2 respectively. The parameter values (r c, ε) for the points a, b, c, d and e in Figure 3.2 correspond to (0.32, 0.08), (0.32, 0.2), (0.32, 0.24) and (0.32, 0.42), (0.15, 0.26) respectively. The spatio-temporal dynamics for the corresponding points is shown in the third column. MLE values corresponding to dynamical behavior in a e are 0.290, , , and respectively. 65

67 Figure 3.6: Dependence of degree of incoherence on the radius of coupling, r c and coupling strength, ε is shown and for (a) spatial behavior of R for r c : 0.2 (red), 0.32 (green) and 0.4 (blue) for a constant coupling strength, ε = 0.24 and (b)spatial behavior of R for r c : 0.2 (red), 0.32 (green) and 0.4 (blue) for a constant coupling strength, ε = 0.24 and (b)spatial behavior of R for ε : 0.24 (red), 0.36 (green) and 0.42 (blue) for a radius of coupling, r c = Type I initial conditions have been used Type II Initial Conditions and r = 3.8 Next we examine the parameter space of the dynamics obtained with type II initial conditions. In Figure 3.5, the red and green colored region correspond to the (r c, ε) parameter space for which chimera states are observed. In the region in red chimera states are observed with wave number K = 2, while the region in green corresponds to the chimera states with K = 1. Figure 3.7: r c - ԑ parameter-space plot shown for non-locally coupled logistic maps with N = 400, r = 3.8 and type II initial conditions. Comparing Figures 3.4 and 3.7 we clearly observe the dependence of initial conditions on (r c, ε) parameter space for observing chimera behavior. A clear reduction in the parameter space is observed for type II initial conditions compared to type I. While 66

68 the range of r c is the same in both the cases, the range of ԑ is much narrower. This may be because type I initial state is a chimera state in itself, whereas type II initial state is a continuous curve. The spatial plots of x and R corresponding to the points a, b, c, d and e in Figure 3.7 are shown in Figures 3.8a e. In this case also we observe that for a fixed radius of coupling r c, as the strength of coupling increases, the extent of incoherence in the chimera states decrease (clear from Figures 3.8a d) and above a certain value of ε (0.32), the spatial dynamics is nearly synchronous. Also by comparing the spatial dynamics of R in Figures 3.8f - i where r c is a constant (0.25), we observe that the degree of incoherence decreases with increase in ε (from 0.1 to 0.36). As in the earlier case we also observe is that dynamics with wave number K = 2, occur at lower radius of coupling compared to the dynamics with wave number K = 1. Figure 3.8: (a) - (e) depicts the spatial dynamics (xi vs i) at points a, b, c, d and e respectively in the r c - ԑ parameter-space plot (Figure 3.7). (e) (h) spatial plots of R i corresponding to the plots (a) (e). The parameter values (rc, ε) for the points a b, c, d and e in Figure 3.7 correspond to (0.25, 0.1), (0.25, 0.25) and (0.25, 0.3), (0.25, 0.36), (0.2, 0.25) respectively. The points a, b, c and d correspond to wave number K = 1, while point e to wave number K = 2. 67

69 3.4.3 Type I Initial Conditions and r = 3.6 To see if the complexity of the local dynamics affects the emergence of chimera states in the network, we next consider weak chaotic dynamics at each node corresponding to r = 3.6. Figure 3.9: r c - ԑ parameter-space plot for r = 3.6. The chimera states emerge in regions shown in purple, green and red. The purple region corresponds to a temporal dynamics with period greater than 4. The green and red regions correspond to temporal periodic dynamics with period-4 and period-2 respectively. From the r c - ԑ parameter space plot shown in Figure 3.9 we observe that there are no chimera states with wave number, K = 2 for r = 3.6. In this case the chimera states are observed even at very low coupling strengths such as ε = 0.08, while the range of r c is the same as in the case of r = 3.8. Further we observe that temporal period in the incoherent regions reduces from period > 4, in the purple region to period = 4 in the green region to period = 2 in the red region. 68

70 Figure 3.10: Figure 3.8: (a) - (e) depicts the spatial dynamics (x i vs i) and (f) (j) depict spatial plots of R i vs i corresponding to dynamics in (a) (e). The (r c, ε) values corresponding to the points a, b, c, d and e in Figure 3.9 are (0.05,0.08), (0.125, 0.08), (0.125, 0.2), (0.125, 0.35) and (0.325,0.2) respectively. The wave number K = 1, r = 3.6 and N = 400. The spatiotemporal dynamics is shown in the third column. MLE values for spatio-temporal dynamics corresponding to a e are 0.001, , , and respectively. Type I initial conditions have been used. As expected, in this case also we observe that for a constant value of radius of coupling where P = 50, r c = 0.125, as the coupling strength increases (from ε = 0.08 in Figure 3.10b to ε = 0.35 in Figure 3.10d), the degree of incoherence decreases. Similarly the degree of incoherence decreases significantly with increase in radius of coupling (corresponding to Figure 3.10a and b) for low coupling strength (ε = 0.08). However no such reduction in the degree of incoherence is observed with increase in r c at higher 69

71 coupling strength (ε = 0.2), as seen in Figures 3.10h and j. Weak spatio-temporal chaos is observed ( λ max = 0.001) when ε and r c are low and becomes periodic at large radius of coupling. This is contrast to that observed in the case of high chaotic dynamics, wherein reduction in the degree of incoherence was observed even at higher coupling strengths (ε = 0.24) with increase in r c. The complexity of the dynamics is quantified by the MLE in Figure 3.10c - e. 3.5 NON-UNIFORM COUPLING So far, uniform coupling strength has been used irrespective of the distance between nodes. To mimic a realistic situation wherein the coupling strength decreases with increase in the distance between two nodes, we consider two different coupling functions as decaying functions of distance. Linearly Decaying Coupling Function: In this case, the coupling strength is defined by a linearly decreasing function of distance where d ij is the distance between the nodes i and j in the lattice and P is the number of nodes connected to a node on either side i.e. 2P is the degree of each node. 3.7 Figure 3.11: To linearly decaying coupling function, ԑ ij as a function of distance d ij between nodes i and j is shown. 70

72 The function defined in equation 3.8 is depicted in Figure 3.11, decreasing linearly on either side of the i th node as the distance from its neighboring nodes increases. In this case also chimera states are observed and are shown in Figures 3.12a - d. As in the case of uniform coupling, in this case also the degree of incoherence decreases with increase in coupling strength or radius of coupling. However the (r c, ε) parameter space for observing chimera states is different compared to the case with uniform coupling and the chimera states are observed even at higher coupling strengths (ε > 0.35). Figure 3.12: a - d depicts the spatial dynamics (x i vs i) and (e) (h) depict spatial plots of R i vs i corresponding to the plots a d. The values of corresponding to a, b, c and d are 0.24, 0.3, 0.36 and 0.4 respectively for a constant r c = Type I initial conditions have been used. Exponentially Decaying Coupling Function: given by The other non-uniform function considered is the double exponential function,

73 where λ ij = 3d ij /N and k is a constant which determines the extent of decay in coupling and μ is the scaling constant. The decaying exponential function on either of node i is depicted in Figure Figure 3.13: Exponentially decaying coupling function. Here k = 2.0 and μ = 0.5. Figure 3.14: a - d the spatial dynamics (x i vs i) and (e) (h) spatial plots of R i vs i corresponding to the plots a d. The values of μ corresponding to a, b, c and d are 0.35, 0.4, 0.45 and 0.5 respectively for a constant r c = 0.32 and k = 2. Type I initial conditions have been used. We observe that chimera states emerge even for the double exponentially decaying coupling function as shown in Figure 3.12c - d. As expected we observe that as the value of μ increases the degree of incoherence in the spatial dynamics decreases. 72

74 3.6 ROBUSTNESS Noise in System Parameters and Initial Conditions In real practical situations, it is unlikely to have exactly the same system parameters over the entire spatial domain, e.g., the junctional coupling strengths, ε may vary between cells in a neural tissue, or the growth parameter r may not be same in all subpopulation patches, etc. To mimic such a scenario, we introduce small random variations in the system parameters r and ε, i.e., r ± δr i and ε ± δε i. Since the occurrence of chimera states is sensitive to the initial conditions, we also introduce noise in the initial conditions, i.e., x i (0) ± δx i (0) (x i (0) refers to type I initial conditions), and analyze the emergence and stability of the chimera states in such a heterogeneous coupled logistic map lattice. In Figure 3.15a-d is shown the spatial dynamics of x and in the adjacent plots (2 nd panel) the spatio-temporal dynamics for varying coupling strengths. The maximum permissible values of δr i, δε i and δx i (0) are 0.04, and 0.05 respectively. For larger perturbations, the chimera states are observed to break down. in the presence of noise, chimera states are observed for a narrower range of (r c, ε) parameter space as can seen in Figures 3.15b and c. For low ε, the network exhibits spatially coherent dynamics as can For low ε, the lattice exhibits asynchronous dynamics (Figure 3.15c) and for ε > 0.35, the lattice exhibits spatially coherent dynamics as can be seen in Figure 3.15d. Further, the regions of coherence are also noisy as can be seen in Figures 3.15 b and c. Each neuron in neuronal network differs from the other in terms of its excitability and synaptic input. The study of robustness of chimera states to noise in system parameters is relevant in neuronal networks (where each neuron slightly differs from the other in terms of excitability), which exhibit uni-hemispheric sleep. 73

75 Figure 3.15: In (a) (d) is shown the spatial dynamics of x and their corresponding spatiotemporal plots (in the 2nd column) for different coupling strengths, ԑ = 0.08, 0.2, 0.24 and 0.42 respectively ± and r c = The parameters values of the lattice are r = 3.8 ± 0.04, and the random variations to the type I initial conditions are x i (0) ± transients were eliminated Noise in Topological Connections We next analyze the stability of chimera states on a logistic map lattice introducing noise in the connection topology by rewiring the edges. Let p be the probability of rewiring, i.e. the probability with which each node is rewired to a node to which it is not already connected. This rewiring also introduces long range connections. As is clear from Figure 3.16, the maximum percentage of connections that can be rewired decreases with increase in r c. 74

76 Figure 3.16: The maximum percentage of edges that can be rewired for chimera states to emerge for ε = 0.2. The initial conditions are of type I and N = 400. From Figure 3.17 we observe that even when nodes are rewired to the maximum permissible extent, chimera states do emerge suggesting their robustness to perturbations in the topology. In Figure 3.17a and b is shown the variation in the spatial behavior of x and R for 15% and 30% rewiring of the edges for P = 128 (r c = 0.32). Figure 3.17: The spatial dynamics of x for a constant r c = 0.32 and percentage of nodes rewired being 15% and 30% in a and b respectively and the corresponding spatial plots of R are shown in c and d. The initial conditions are of type I and r =

77 From Figure 3.17a and b, we observe that there is small noise in the values of x even in the coherent intervals, similar to the dynamics observed when noise was introduced in r and ԑ values and initial conditions which decreases as the number of rewired nodes is increased. The fact that chimera states are robust to small noise in topological connections has important applications in the study of chimera behavior in power grids (power failures), neuronal networks of some animals (uni-hemispheric sleep) and human heart (cardiac fibrillation), which are prone to minor changes in the topological connections in them. 3.7 EFFECT OF EXTERNAL PERTURBATIONS In various physical and biological systems such as power grids or excitable tissues (e.g., cardiac or neuronal tissues), the synchronous movement of all their parts is extremely crucial for their proper functioning (Rohden et al., 2012). Localized regions of incoherence (asynchronous beat of a small number of tissues in the heart) in such systems may cause hindrance to their performance (cardiac fibrillation) and in extreme situations may even completely destabilize the system (Martens et al., 2013). In such situations, there is clearly a need to address local disturbances/incoherent dynamics and bring the system back to its original synchronous or asynchronous dynamical state. Here we attempt to analyze the effect of applying localized external perturbation or pinning to the incoherent regions and see if the system is driven to spatially synchronous state or the whole lattice exhibits incoherent dynamics. The application of pinning is analogous to using pacemakers used during cardiac fibrillation to restore the normal functioning of the heart making all the heart tissues beat synchronously. The objective is to manipulate the dynamics in the event of the system exhibiting undesired local dynamics. For example, extended periods of synchronization in the brain, results in epileptic seizures and there exists need for external intervention. In diffusively coupled logistic maps, it has been shown by Parekh et al., (1998) that negative pinning suppresses chaos while positive pinning induces/enhances chaos in logistic coupled map systems. In Figure 3.18a, we observe that on applying negative pinning to the incoherent regions, the degree of 76

78 incoherence is reduced and can be completely suppressed, while applying positive pinning the degree of incoherence is enhanced as shown in Figure 3.18b. However, it may be noted that the coherent dynamics attained by selectively pinning the regions of incoherence exists only as long as the pinning is being given. In Figure 3.19 is shown the effect of removing the external pinning after having applied for a certain period of time. Initially the system is considered to be exhibiting chimera behavior when no pinning is applied (blue). The region of incoherence is decreased on applying negative pinning selectively to the nodes in the incoherent region (green). On switching off the external pinning and eliminating transients, we observe that the system goes back to the original dynamical state and exhibits spatial incoherence regions (red) which are much narrower than the initial case, i.e., on switching off the external perturbation, the extent of spread of the incoherent dynamics is reduced, but not completely eliminated. Figure 3.18: (a) Spatial behavior of R for the parameters N = 400, r c = 0.32, ε = 0.24, shown on selectively pinning the incoherent regions with varying strengths of (a) negative pinning: (green) and (red); (b) positive pinning: 0.02 (green), 0.05 (red). The plots in blue correspond to no pinning in both (a) and (b). It is clear that negative pinning suppresses while positive pinning enhances incoherence. Type I initial conditions have been used. 77

79 Figure 3.19: Spatial behavior of R: i) no pinning (blue), ii) selective pinning of (green) after eliminating transients and iii) eliminating transients after removing the selective pinning (red). Here, r = 3.8, N = 400, r c = 0.32, ԑ = Type I initial conditions have been used. 3.8 CHIMERA STATES IN TWO-DIMENSIONAL LATTICES Most real world networks such as neuronal networks, wireless networks, population systems are two or three dimensional systems. Hence we extend our study of chimera behavior to two dimensional logistic maps. Shima and Kuramoto (2004) for the first time observed two dimensional spiral chimera states in a non-locally coupled reaction diffusion system. They argued that the spiral dynamics appeared because of the non-locality in coupling whose effect was stronger at lower coupling strengths. Later Martens et al. (2013) analyzed that such spiral chimera states may also occur in networks of neurons. Apart from two dimensional lattices, chimera states have also been observed on three dimensional structures such as torus and sphere (Panaggio and Abrams, 2013, 2014). We consider a non-locally coupled two-dimensional lattice given by 3.9 where the function f is the logistic map. Every node in the lattice is coupled to its neighbors in a square of size 2P+1 centered around node i,j with radius of coupling defined as P/N. The nodes in the lattice are said to be non-locally coupled when the value 78

80 of r c lies between 1/N and (N-1)/2N in either dimension. The measure of degree of incoherence R i,j is defined in this case as 3.10 δ denotes the size of neighborhood around each node, consisting of all nodes within a square of size (2δ+1) centered around the node i,j Initial Conditions The pattern of initial conditions used for the two-dimensional lattice to observe chimera states is similar to that of type I initial conditions in the one-dimensional case. Mathematically the initial conditions for the two dimensional lattice are given by following equations. x ij = 0.45, i ϵ [0, N/8) and [7N/8, N); and j ϵ [0, N) 3.11 x ij = 0.45, i ϵ [N/8, 7N/8) and j ϵ [0, N/8) and [7N/8, N) 3.12 x ij = 0.90, i ϵ [N/4, 3N/4) and j ϵ [N/4, 3N/4) 3.13 x ij ϵ I, i ϵ [N/8, N/4) and [3N/4, 7N/8) and j ϵ [N/8, 7N/8) 3.14 x ij ϵ I, i ϵ [N/4, 3N/4) and j ϵ [N/8, N/4) and [3N/4, 7N/8) 3.15 where I = (0.4, 0.5) (0.85, 0.95) Figure 3.20: Initial conditions on a 2 dimensional lattice of size 100x100. The nodes in the outer rectangle (blue in color) have a constant value of 0.45 while those in the inner rectangle (red in color) have a constant value of 0.9. The patch in between has random values from the set (0.4, 0.5) (0.85, 0.95) given by equations

81 Starting with the initial condition shown in Figure 3.19, the two dimensional network is allowed to evolve for different values of r c and ε and summarized in Figures 3.21 and Figure 3.21: In (a) (d) is shown the spatial dynamics of x and (e) (h), the spatial dynamics of R on a 2-d lattice of dimensions 100x100 for the parameters, r = 3.8, δ = 2. The value of r c is 0.16 for all the plots and the values of ε for a, b, c and d are 0.15, 0.2, 0.25 and 0.3 respectively. 80

82 Figure 3.22: : In (a) (d) is shown the spatial dynamics of x and (e) (h), the spatial dynamics of R on a 2-d lattice of dimensions 100x100 for the parameters r = 3.8, δ = 2. The value of ε is 0.2 for all the plots and the values of r c for a, b, c and d are 0.08, 0.12, 0.16 and 0.2 respectively. In Figures 3.21a - d, we observe that the region of incoherence (shown by yellowish red color) in the second panel is decreasing with increase in the coupling strength ε (for constant r c ), and in Figure 3.19d most of the lattice exhibits synchronous behavior. Similarly in Figure 3.22, for constant coupling strength, with increase in radius if coupling, the regions of incoherence (reddish yellow regions) reduce. This is in 81

83 accordance with the observations on one dimensional non-locally coupled networks. From this analysis it is clear that the strength of coupling has a stronger effect on reducing the degree of incoherence compared to the radius of coupling Control of Incoherence In non-locally coupled logistic maps we observed that the incoherence in chimera states can be curbed temporarily by giving negative external perturbation selectively to the incoherent regions. We now analyze the effect of external perturbation in two dimensional logistic maps. Figure 3.21a and b show the dynamics of R with no external perturbation and a negative external perturbation (-0.2) respectively. Figure 3.23: In (a) (c) is shown the spatial dynamics of R on a 2-d lattice of dimensions 100x100 for the parameters r = 3.8, δ = 2, r c = 0.2, ε = 0.2. (a) corresponds to dynamics with no pinning, (b) with a pinning strength of -0.2 and (c) dynamics, steps after removal of pinning. 82

84 In Figure 3.23a is shown the spatial dynamics of R in the absence of any pinning, and in Figure 3.23b after applying negative pinning of strength -0.2 to the incoherent regions only. Clearly the whole two dimensional lattice now exhibits synchronous behavior. On removing the pinning and eliminating transients we observe that chimera state re-emerges, but the degree of incoherence is reduced. Thus as in the one dimensional case, the effect of pinning is observed as long as the external perturbation is applied. The proper functioning of a power grid system depends on total synchronization of generators to avoid breaks in power transmissions. Fibrillations in the heart are caused due to asynchronous movement of some of the heart tissues, a chimera state. Thus the stabilization of undesirable asynchronous dynamics in a two dimensional lattice can have important applications in power grid systems and fibrillations of the cardiac tissue which are known to exhibit chimera dynamics. 3.9 CHIMERA STATES ON COMPLEX NETWORKS Until now we have studied the emergence of chimera states in non-locally coupled lattices where the nodes are homogeneously coupled in their neighborhood. We now study the emergence of chimera behavior on networks with heterogeneity in degree or in spatial extent viz., scale-free, random and small-world networks. The major difference between random, scale-free and non-locally coupled regular lattices is in their degree distribution. While in non-locally coupled lattices each node has same degree and a homogenous neighborhood, random networks have a Poisson degree distribution where every node has a non-homogenous neighborhood. Scale-free networks exhibit a power law degree distribution with a very few nodes, called hubs, having a very large degree compared to other nodes in the network as discussed in Section 1.1. In a power law degree distribution, hubs shape the way the entire network operates. Small world networks differ from regular lattices in their average path length (very low path lengths) owing to a few long range connections in small-world networks. Most real world networks exhibit heterogeneous connections and so, studying the dynamics on these complex networks is important. Moreover, chimera behavior has been 83

85 observed in the neuronal networks of some animals which undergo uni-hemispheric sleep (Rattenborg et al., 2000). Also, bump states (where some neurons are quiescent while the others are firing) in neuronal dynamics are very similar to chimera states. In a recent study, González-Avella et al. (2014) observed chimera states in a model of two interacting populations of social agents with heterogeneous interactions. Hence we extend our analysis of chimera states to complex networks Scale-Free Networks A number of real world networks such as World Wide Web, protein-protein networks and neuronal networks etc. exhibit scale-free properties and hence it will be interesting to study the existence of chimera states on scale-free topologies. In earlier studies, chimera states have been shown to occur in scale-free networks by Wang et al. (2009) and Zhu et al. (2014). In this section we analyze the emergence of chimera dynamics on a scale-free network of logistic maps of size 1000 with average degrees of 2 and 6 constructed using Barabasi s algorithm. The degree distribution of these networks on the log-log scale is shown in Figure The scaling exponent of the degree distribution for the network with mean degree 2 is 2.03 and that with mean degree 6 is In Figures 3.25 is shown chimera states on scale-free network for r = 3.6 using type I initial conditions. Figure 3.24: The degree distributions of scale free networks of size 1000 and mean degree 2 (points in blue) and 6 (points in green). 84

86 Figure 3.25: In a, b is shown the spatial dynamics of x and c, d, the spatial dynamics of R on a scale-free network for the parameters N = 1000, r = 3.6, ε = 0.11 with mean degree 2 and 6 respectively. Type I initial conditions have been used. Figure 3.25a and b correspond to dynamics on the network with mean degree 2 and Figure 3.25c and d for the network with mean degree, 6. We observe that the noise in the coherent regions is higher for scale free networks with smaller mean degree. On increasing chaos in the local dynamics (r = 3.8), no chimera states were observed. In the scale-free network chimera states are observed only for low coupling strengths ( ), probably because the hub nodes drive the system to synchronous or asynchronous states for stronger coupling strengths. These observations may have important applications in the study of neuronal networks showing chimera dynamics (unihemispheric sleep). However chimera states emerge in scale-free configurations for smaller mean degrees (mean degree of 2 for N = 1000) as compared to non-locally coupled logistic maps (mean degree of 64 for N = 400). This may also be a consequence due to the power law degree distribution. 85

87 3.9.2 Random Networks In this thesis, we use the Watts and Strogatz (1998) model, discussed in Section to generate two random networks of size 400 and average degrees 10 and 60. The results in this section have been averaged over 50 random network configurations. In Figure 3.26 is shown the degree distribution of the random networks of average degree 10 (points in blue) and 60 (points in green). As discussed in Section the average clustering coefficient and path length for a random network must be k/n and ln(n)/ln(k) respectively. Shown in Table 3.1 are the clustering coefficient and path length of the two random networks which match the expected values, confirming that the networks generated are indeed random networks. Figures 3.27 shows chimera dynamics on the random network for r = 3.6. We do not observe chimera states at r = 3.8 in the case of strong chaos in the local dynamics Table 3.1 Expected and actual values of C and L for a random network of size 400 Average degree C k/n L ln(n)/ln(k) Figure 3.26: Degree distributions of random networks of size 400. Plot in blue and green indicate the degree distributions with average degrees 10 and 40 respectively. 86

88 Figure 3.27: In a, b is shown the spatial dynamics of x and c, d, the spatial dynamics of R on a random network for the parameters N = 400, r = 3.6, ε = The average degree for the random networks corresponding to a and b are 10 and 60 respectively. Type I initial conditions have been used. From Figure 3.27a and b we observe that there is noise in the coherent regions in the spatial dynamics, which reduces for the random network with a higher average degree (60). The range of ε for which chimera states are observed for a random network with average degree 10 is [0.05, 0.15] and for the random network with average degree 60 is [0.08, 0.17], much lower compared to that in non-locally coupled lattices (as can be seen from Figure 3.9). However similar to that in scale-free networks, in this case also, chimera states are observed for smaller average degrees compared to non-locally coupled lattices. These differences may be due to the heterogeneous connections in random networks and scale-free networks Small-world Networks Previously chimera states were observed in small world networks of pulsecoupled oscillators which mimic neuronal dynamics (Rothkegel and Lehnertz, 2014). Here we use the Watts and Strogatz (1998) model to construct a small world network of size 500 and mean degree 6, as discussed in Section We consider two configurations of small-world networks with mean degrees, 6 and 40 and the behavior of C p /C 0 and L p /L 0 for these two networks is shown in Figure In the shaded regions in 87

89 Figure 3.26a and b, we observe the small world property (C p /C 0 ~ 1 and L p /L 1 ~ 1). In Figure 3.29a and b is shown the spatial dynamics of x and R for small world networks of size 500 and mean degree 6 and 40 respectively with p = Figure 3.28: In a and b, the behavior of L p /L 0 (in blue) and C p /C 0 (in green) with respect to log(p) for small-world networks of size 500 and mean degrees 6 and 40 respectively are shown. This plot has been obtained for 50 different configurations (50 values of p). Figure 3.29: In a and b is shown the spatial dynamics of x and c and d, the spatial dynamics of R on a small-world networks of mean degrees 6 and 40 respectively for the parameters N = 400, r = 3.6, ε = 0.2 and p = Type I initial conditions have been used. For the small-world configurations in Figures 3.29a and b, we observe that as the mean degree increases from 6 to 40, the noise present in the coherent regions and the degree of incoherence in the asynchronous regions of chimera states reduces. The range 88