Synchronization in delaycoupled bipartite networks
|
|
- Christopher Webb
- 5 years ago
- Views:
Transcription
1 Synchronization in delaycoupled bipartite networks Ram Ramaswamy School of Physical Sciences Jawaharlal Nehru University, New Delhi February 20, 2015
2 Outline Ø Bipartite networks and delay-coupled phase oscillators Ø Globally synchronized solutions (GS) Ø Mismatch between the partitions Ø Time-varying frequencies and Remote Synchronization (RS) Ø Chimera states (CS) and Individual Synchronization (IS) Ø Summary
3 Networks Most natural systems are nonlinear and rarely isolated, functioning through a collective behavior of many interacting subsystems. These are networks, which have been extensively studied to describe wide variety of phenomena in physics, biology, chemistry, ecology, sociology etc. Examples -- networks Biological networks, social
4 Bipartite topology The bipartite topology describes a connection structure where the nodes of a network can be divided into two groups such that any two nodes that are coupled must belong to different groups. Different Immune cells on a bipartite network Biological systems (Immune system)
5 Bipartite Topology Different birds and plants relationship. Ecological systems (Birds and plants system)
6 File sharing protocols Computer network (Bit torrent) Bipartite network reduces search traffic cost and query response time.
7 Metabolic networks are bipartite
8 Other cases The bipartite network also covers some of the well studied topologies- Ring with nearest neighbor coupling and even number of oscillators. Star topology
9 Collective synchronization Ensembles of coupled oscillators typically show emergent properties, namely, the features that are not shown by the individual units. An example of such emergent behavior is collective synchronization which is observed in the systems ranging from simple mechanical oscillator to pacemaker heart cells, circadian rhythms, flashing-fireflies, clapping of the large audience, chemical, biological oscillators, laser arrays and Josephson junctions.
10 The Kuramoto Model The Kuramoto model describes a large population of coupled limit cycle oscillators, used to understand collective behavior of populations in several disciplines. The model equations are given by θ i (t) = ω i + K N N a ij sin(θ j (t) θ i (t)), i =1,...,N j =1 Here θ i is the phase of ith oscillator, K is the coupling strength. ω s are the natural frequencies drawn from a distribution g(ω). For simplicity, the distribution g(ω) is unimodal and symmetric around the mean frequency Ω. The factor (1/N) ensures that the model is well behaved as N à.
11 The Kuramoto order parameter The model is known to exhibit a spontaneous second order phase transition: with increasing coupling K, the system abruptly goes from the incoherent state to a synchronized one at the critical coupling strength K c. To visualize the dynamics of the phases, a complex order parameter is defined as: re iψ = 1 N N j =1 This value can be interpreted as the collective macroscopic rhythm produced by the whole population. r(t) measures the phase coherence and ψ(t) is the average phase. e iθ j
12 θ i = ω i + Krsin(ψ θ i ), i =1,...,N Phase transition Rewriting the system equation in terms of complex order parameter This indicates that the interactions among the oscillators are through the mean field quantities (ψ and r). The effective coupling strength (Kr) in the equation works as a positive feedback between coupling and coherence. The oscillators are pulled towards the mean phase ψ, the ones merging in the synchronized set are termed as phase locked while others are drifting oscillators. The competition between the two makes the population selflimiting.
13 Time Delay Coupling Time-delay is inherent in the couplings due to finite speed of signal propagation. Depending upon the time-scale of the system, the time delay coupling may modify the resulting dynamics of the system, including aspects of synchronization, AD, PF, multistability etc.
14 Time-delay coupling In order to investigate the effect of time delay on synchronization of nonlinear oscillators, Schuster and Wagner took two limit cycle oscillators coupled with time delay θ 1 = ω 1 + K sin(θ 2 (t τ) θ 1 (t)) θ 2 = ω 2 + K sin(θ 1 (t τ) θ 2 (t)) Here θ 1 and θ 2 are the phases of oscillators with individual frequencies ω 1 and ω 2. K is the coupling strength and τ is time delay. The interaction term tend to synchronize both oscillators.
15 Synchronized solutions The most general synchronized solution contains a common time dependent phase part ( Ψ(t) ) and a part corresponding to the constant phase difference between the oscillators ( α ) i.e. θ 1 (t) =ψ(t) + α 2 θ 2 (t) =ψ(t) α 2 Putting these θ s in the system equations, leads to a condition- cos( ψ(t) ψ(t τ) ) = ω 1 ω 2 2K sin(φ) = const.
16 Synchronized solutions The condition cos( ψ(t) ψ(t τ) ) = const. is satisfied if, Ψ(t) = Ωt, so the only possible synchronized solutions for the given set of parameters are those for which both oscillators have same common frequency Ω and they differ only by a constant phase shift α. i.e. θ 1,2 (t) = Ωt ± α 2
17 Synchronized solutions: Collective frequency and Synchronized solutions phase difference In contrast to the case with delay time = 0 where there exists only one synchronization frequency (Ω=ϖ), here we see from the transcendental equation that there can be multiple solutions for a given set of parameters. The number of solutions increases with coupling strength and time delay. ω =1, Δω = 0.4, K = 2
18 Synchronized solutions The frequencies and phase differences between the solutions vary with coupling strength. The number of solutions increases with coupling strength and the system jumps to new stable frequency- ω =1, Δω = 0.4, τ =1.0
19 Multiple frequency solutions: basins of attraction In the time interval [ -τ, 0 ], starting with the function θ 1,2 (t) = Ωt ± α 2 With different values of Ω and α as starting point, the basin of attraction: ω =1, Δω = K = 2.0, τ = 2.0 For the set of parameter values, there are total five possible frequencies out of which three are stable as can be seen in the figure.
20 Relay Synchronization with delay Network of oscillators with time delayed coupling are studied theoretically and experimentally because of their applications to neurobiology, laser arrays, microwave devices, electronic circuits and also because of their inherent mathematical interest. If there is time delay in the interaction, oscillators 1 and 3 become phase synchronized and 1, 2 and 2, 3 are lag synchronized. [ ε, [ ε, 1 τ ] 2 τ ] 3
21 Relay Synchronization with delay This is a simple bipartite network. How general are such solutions for other bipartite networks?
22 A Bipartite network of Delay coupled oscillators Consider a system of N coupled phase oscillators on a bipartite topology with distributed delay coupling, θ i = ω i + ε N ( + a k ij g* θ j (t s) f (s)ds θ i (t)-, i ), j =1 0 i =1,...,N Here θ s are the corresponding phases, ω the natural frequencies, ε denotes the coupling strength, τ is the delay. k i is the number of inputs received by the i th oscillator and a ij is the adjacency matrix which reflects the connection topology.
23 Synchronization For synchronized solutions, θ A,B (t) = Ωt ± φ /2 where Ω is the common frequency and ϕ is the phase difference between the partitions. The necessary condition for the existence of phase-locked solutions of this form is Ω = ω +εg( Ωτ φ) = ω +εg( Ωτ + φ). The function g is periodic with period 2π, the two solutions are Ω 0 = ω +εg( Ω 0 τ) Ω π = ω +εg( Ω π τ π) = ω +εg( Ω π τ + π)
24 For stability With stability conditions, ε g #( Ω 0 τ) > 0 for in phase solutions ε g #( Ω π τ + π) > 0 for anti phase solutions
25 Bipartite Kuramoto Consider delay coupled phase oscillators on a bipartite network, with standard sinusoidal coupling function. θ i = ) ω + ε + k i * + ω + ε, + k i sin( θ j (t τ) θ i (t)), i A, j B sin( θ j (t τ) θ i (t)), j A i B
26 Bipartite Kuramoto The collective frequencies and phase differences between the partitions is given by And the stability conditions are Ω = ω ε sin(ωτ) ) 0 if cos(ωτ) > 0 Δφ = * + π otherwise cos(ωτ) > 0 for cos(ωτ) < 0 for in phase solutions anti phase solutions
27 Bipartite Kuramoto For different number of oscillators in two partitions, if the number of connections for every oscillator is same, the results for common frequencies and phase differences remain unaltered. A π Complete bipartite: different number of oscillators in each partition. B Each partition works as a big oscillator. 0 The phase difference between the partition flips between zero and π
28 Collective frequencies Stable collective frequencies and corresponding phase difference. From the transcendental equation (ω = 1): Ω =1 ε sin(ωτ ) The blue lines are solutions corresponding to in-phase and red lines are anti-phase solutions. The green boxes are the regions where more than one frequencies are stable. ε = 0.05
29 Numerical results: collective frequencies Collective frequencies As the value of delay increases, the number of solution frequencies also increases. However, irrespective to the number of solutions, we can have only two phase solutions (Φ = 0, π). ε = 0.05 Green boxes are the regions where same phase solutions (in- or antiphase) can be obtained from different collective frequency solutions.
30 Numerical results: collective frequencies Solutions for different coupling strengths. The overlapping region (multistable region) increases with increasing value of coupling strength. ε = 0.05, 0.125, 0.25, 0.5
31 Phase-diagram for N=6 Numerical frequencies for 6 oscillators N A = N B = 3, plotted with stable analytical solutions from the transcendental equation: in-phase anti-phase multistable region Ω =1 ε sin(ωτ )
32 Hysterisis and Multistability Numerical frequencies plotted with stable analytical solutions.
33 Solution Branches Analytic forms of solutions branches The width of multistable regions increase linearly with time-delay: Stable solution branches are indicated by B n, where even and odd values of n indicate in-phase and anti-phase branches, respectively. The dashed boxes are the region of multistability and the dotted boxes, indicated by the arrows, show highest number of overlaps three overlapping branches in this case.
34 Stable Solutions The number of stable frequencies increases approximately linearly with parameters. Ω =1 ε sin(ωτ) Ω max =1+ε Ω min =1 ε
35 Stable Solutions As the time delay determines the actual value of collective frequency, it effectively acts as a frequency selector in a range that is determined by the coupling strength. Ω =1 ε sin(ωτ ) Ω max =1+ε Ω min =1 ε ε = 0.5
36 Other oscillators on a bipartite network: Other oscillators on bipartite networks Landau-Stuart (L-S) oscillators Rössler oscillators ( 2 ( A + iω A Z i )Z ii + ε a k ij Z j (t τ) Z i (t) Z i ( ), i A, * j B i = ) 2 *( A + iω B Z i )Z ii + ε a k ij Z j (t τ) Z i (t) i ( ), i B + * j A With some approximations, we can reduce the phase dynamics of coupled L-S oscillators and Rössler oscillators on the bipartite networks to that of Kuramoto phase oscillators. ( ω A y i z i + ε a k ij i ( x j (t τ) x i (t)), i A, * j B x i = ) * ω B y i z i + ε a k ij x j (t τ) x i (t) i ( ), i B + * j A ( y i = ω A x i + ay i i A, ) + ω B x i + ay i i B z i = f + z i (x i c) ( ω A + K * θ k i i = ) * ω B + K + * k i a ij sin( θ j (t τ) θ i (t)), i A, j B a ij sin( θ j (t τ) θ i (t)), j A i B K = ε K = ε 2 For L-S oscillators For Rössler oscillators
37 Landau Stuart Frequencies and time series: Parameter values: A =1,ω A = ω B =1.0 ε =1.0 K = ε =1
38 Rössler Oscillators Parameter values: a = b = 0.2,c =1.0 ω A = ω B =1.0 ε = 2.0 K = ε 2 =1.0 Frequencies and time series.
39 Mismatched Partitions
40 We study a system of delay-coupled phase oscillators on a bipartite topology, when there is a mismatch between the partitions - Mismatched oscillators θ i = $ & & % & & ' ω A + ε k i ω B + ε k i a ij sin( θ j (t τ ) θ i (t)), i A, j B a ij sin( θ j (t τ ) θ i (t)), j A i B ω A,B are natural frequencies of the two partitions.
41 Global Synchrony $ & θ i = % & '& Ωt + φ 2 Ωt φ 2 if if i A i B Ω is the collective frequency and ϕ is the constant phase difference between the partitions.
42 Collective frequencies and phases Making the ansatz that there is a common oscillation for the oscillators in the two partitions, one can get a transcendental equation for the collective frequency and for the corresponding phase difference. F (Ω) = ω Ω ε tan(ωτ ) cos 2 (Ωτ ) (Δω)2 4ε 2 $ Δω ' φ = arcsin& ); if cos(ωτ ) > 0 % 2ε cos(ωτ )( $ Δω ' φ = π arcsin& ); otherwise % 2ε cos(ωτ )(
43 GS solutions and asynchrony Frequencies and phase differences between the partitions (N = 64) Ω = ω ε tan(ω τ ) cos 2 (Ω τ ) (Δω)2 4ε 2 Condition for globally synchronized solutions to exist $ cos 2 (Ωτ ) Δω ' & ) % 2ε ( 2 We observe windows (W 1, W 2 ) where globally synchronized solutions do not exist.
44 Global behaviour Phase behavior in the parameter space: Δω = 0.08 F (Ω) = 0 F + (Ω) = 0 multistable region desynchronized region Δω = 0.2 F (Ω) = 0 F + (Ω) = 0 multistable region desynchronized region
45 Global frequencies Time-averaged frequencies for the two partitions (N = 64) In regions (I) and (V): globally synchronized solutions exist Ω A = Ω B = Ω In regions (II), (III) and (IV): globally synchronized solutions do not exist Ω A Ω B $ & Ω i = Ω A if i A % ' & Ω i = Ω B if i B Time-averaged frequencies of the oscillators from the same partitions are equal.
46 Partition frequencies Time-averaged frequencies for the two partitions for different coupling strength and frequency mismatch θ A,B = Ωt ± Δφ 2 θ i = G i (t)+ c i Ω i (t) = G i (t) Ω i = Ω A if i A Ω i = Ω B if i B
47 Time-dependent frequencies and RS Remote synchronization due to locking of time-dependent frequencies (N = 64) Ω j A = Ω j B = Ω + Ω j A = f (t) Ω j B = g(t)+ c j Ω j A = f (t) Ω j B = g(t) Ω j A = f (t)+ c j Ω j B = g(t) Ω j A = Ω j B = Ω
48 Remote Synchronization (RS) In a system of coupled oscillator networks, RS is said to occur when indirectly coupled oscillators are phase synchronized, while not being in synchrony with the relaying unit(s), Remote synchronization: Chimera States (CS), and Individual Synchronization (IS) Remote synchronization has not been shown in a network of phase oscillators without time-delay
49 Phase evolution of oscillators Time evolution of oscillator phases at different values of time delay showing global synchronization (GS), chimera states (CS) and individual synchronization (IS).
50 Partition order-parameters A complex order parameter can be defined for each partition: z A,B = r A,B e iψ = 1 N j A,B e iθ j chimera states: One of r A and r B is 1 and other is 0 individual synchronization: r A = r B = 1 incoherent behavior: r A = r B = 0 Variation of the real parts r A and r B of the partition order parameters as a function of time delay.
51 Order parameter regimes Level sets of the average order parameter, r = (r A +r B )/2 as a function of ε and τ Chimera states Incoherent regions Individual synchronization Global Sync. solutions
52 Order parameter regimes Figure showing Arnold t o n g u e o f g l o b a l l y synchronized solutions and scenarios of RS in! Δω ε space. Chimera states Incoherent regions Individual synchronization Global Sync. solutions
53 Time-dependence W h e n t h e i n t r i n s i c frequencies of oscillators in each partition are drawn from two Gaussian distributions with mean values µ 1,2 = ϖ ± Δω and variance σ = Δω/( ).
54 The order parameter Chimera states Incoherent regions Individual synchronization Global Sync. solutions
55 Summary Ø In complete bipartite case, identical oscillators in the same partitions are always in-phase synchronized and are either in inphase or anti-phase synchronization state with the oscillators in the other partition. Ø The system flips between in-phase and anti-phase states when it jumps from one collective frequency state to another. The number of possible stable frequency solutions increase linearly when delay or coupling is varied. Ø Stability for in as well as anti phase states are obtained. There can be several stable collective frequencies indicating multistability. The system can switch from in-phase state to anti-phase state depending upon the initial conditions.
56 Summary Ø Reducing the phase dynamics of Landau-Stuart oscillators and Rössler oscillators to simple phase oscillators, we can predict the phase behavior of these rather complex systems in parameter space. Ø The nonlinearity and the periodic nature of the coupling is responsible for the stabilization of the anti-phase states Ø Delay coupled phase oscillators on a bipartite network with frequency mismatch show interesting and novel collective states. In addition to global synchrony when all the oscillators in the network lock onto a common frequency, there can also be remote synchronization, with different groups of indirectly connected oscillators displaying distinct patterns of phase coherence.
57 Summary Ø There can be several stable collective frequencies indicating multi-stability. The system can switch from in-phase state to antiphase state depending upon the initial conditions. Ø For a range of parameters, where global synchrony exist, we find solutions such that systems within a partition are in complete synchrony, while there is lag synchronization between the partitions. Ø Outside this range, one observes scenarios of remote synchronization namely, chimeras and individual synchronization, where either one or both of the partitions are synchronized independently. In the absence of time-delay such states are not observed in phase oscillators.
58 References: Ø H. G. Schuster and P. Wagner, Prog. Theor. Phys. 81, 939 (1989). M. G. Earl and S. H. Strogatz, Phys. Rev. E 67, (2003). Ø D. V. R. Reddy, A. Sen and G. L. Johnston, Phys. Rev. Lett. 80, 5109 (1998). D. V. R. Reddy, A. Sen, G. L. Johnston, Physica D 129, 15 (1999); Ø Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, Berlin, 1984). Ø F. M. Atay, Phil. Trans. Roy. Soc. A 371, (2013). N. Punetha, R. Ramaswamy, and F. M. Atay, submitted. N. Punetha, S. R. Ujjwal, F. M. Atay, and R. Ramaswamy, in press.
59 Thank you
arxiv:nlin/ v1 [nlin.cd] 4 Oct 2005
Synchronization of Coupled Chaotic Dynamics on Networks R. E. Amritkar and Sarika Jalan Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India. arxiv:nlin/0510008v1 [nlin.cd] 4 Oct 2005 Abstract
More informationPhase Synchronization
Phase Synchronization Lecture by: Zhibin Guo Notes by: Xiang Fan May 10, 2016 1 Introduction For any mode or fluctuation, we always have where S(x, t) is phase. If a mode amplitude satisfies ϕ k = ϕ k
More informationChimera states in networks of biological neurons and coupled damped pendulums
in neural models in networks of pendulum-like elements in networks of biological neurons and coupled damped pendulums J. Hizanidis 1, V. Kanas 2, A. Bezerianos 3, and T. Bountis 4 1 National Center for
More informationDynamics of delayed-coupled chaotic logistic maps: Influence of network topology, connectivity and delay times
PRAMANA c Indian Academy of Sciences Vol. 7, No. 6 journal of June 28 physics pp. 1 9 Dynamics of delayed-coupled chaotic logistic maps: Influence of network topology, connectivity and delay times ARTURO
More informationUniversity of Colorado. The Kuramoto Model. A presentation in partial satisfaction of the requirements for the degree of MSc in Applied Mathematics
University of Colorado The Kuramoto Model A presentation in partial satisfaction of the requirements for the degree of MSc in Applied Mathematics Jeff Marsh 2008 April 24 1 The Kuramoto Model Motivation:
More informationSurvey of Synchronization Part I: Kuramoto Oscillators
Survey of Synchronization Part I: Kuramoto Oscillators Tatsuya Ibuki FL11-5-2 20 th, May, 2011 Outline of My Research in This Semester Survey of Synchronization - Kuramoto oscillator : This Seminar - Synchronization
More informationPHASE-LOCKED SOLUTIONS IN A HUB CONNECTED OSCILLATOR RING NETWORK
Copyright c 29 by ABCM PHASE-LOCKED SOLUTIONS IN A HUB CONNECTED OSCILLATOR RING NETWORK Jacqueline Bridge, Jacqueline.Bridge@sta.uwi.edu Department of Mechanical Engineering, The University of the West
More informationSynchronization and Phase Oscillators
1 Synchronization and Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 Synchronization
More informationPhase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265 269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
More informationNETWORKS Lecture 2: Synchronization Fundamentals
Short-course: Complex Systems Beyond the Metaphor UNSW, February 2007 NETWORKS Lecture 2: Synchronization Fundamentals David J Hill Research School of Information Sciences and Engineering The ANU 8/2/2007
More informationHysteretic Transitions in the Kuramoto Model with Inertia
Rostock 4 p. Hysteretic Transitions in the uramoto Model with Inertia A. Torcini, S. Olmi, A. Navas, S. Boccaletti http://neuro.fi.isc.cnr.it/ Istituto dei Sistemi Complessi - CNR - Firenze, Italy Istituto
More informationHSND-2015, IPR. Department of Physics, University of Burdwan, Burdwan, West Bengal.
New kind of deaths: Oscillation Death and Chimera Death HSND-2015, IPR Dr. Tanmoy Banerjee Department of Physics, University of Burdwan, Burdwan, West Bengal. Points to be discussed Oscillation suppression
More informationSynchronization Transitions in Complex Networks
Synchronization Transitions in Complex Networks Y. Moreno 1,2,3 1 Institute for Biocomputation and Physics of Complex Systems (BIFI) University of Zaragoza, Zaragoza 50018, Spain 2 Department of Theoretical
More informationEntrainment of coupled oscillators on regular networks by pacemakers
PHYSICAL REVIEW E 73, 036218 2006 Entrainment of coupled oscillators on regular networks by pacemakers Filippo Radicchi* and Hildegard Meyer-Ortmanns School of Engineering and Science, International University
More informationSpontaneous Synchronization in Complex Networks
B.P. Zeegers Spontaneous Synchronization in Complex Networks Bachelor s thesis Supervisors: dr. D. Garlaschelli (LION) prof. dr. W.Th.F. den Hollander (MI) August 2, 25 Leiden Institute of Physics (LION)
More informationApplicable Analysis and Discrete Mathematics available online at FRUSTRATED KURAMOTO MODEL GENERALISE EQUITABLE PARTITIONS
Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. x (xxxx), xxx xxx. doi:10.98/aadmxxxxxxxx α-kuramoto PARTITIONS FROM THE FRUSTRATED KURAMOTO
More informationDevelopment of novel deep brain stimulation techniques: Coordinated reset and Nonlinear delayed feedback
Development of novel deep brain stimulation techniques: Coordinated reset and Nonlinear delayed feedback Oleksandr Popovych 1, Christian Hauptmann 1, Peter A. Tass 1,2 1 Institute of Medicine and Virtual
More informationCommon noise vs Coupling in Oscillator Populations
Common noise vs Coupling in Oscillator Populations A. Pimenova, D. Goldobin, M. Rosenblum, and A. Pikovsky Institute of Continuous Media Mechanics UB RAS, Perm, Russia Institut for Physics and Astronomy,
More informationSaturation of Information Exchange in Locally Connected Pulse-Coupled Oscillators
Saturation of Information Exchange in Locally Connected Pulse-Coupled Oscillators Will Wagstaff School of Computer Science, Georgia Institute of Technology, Atlanta, Georgia 30332, USA (Dated: 13 December
More informationFirefly Synchronization. Morris Huang, Mark Kingsbury, Ben McInroe, Will Wagstaff
Firefly Synchronization Morris Huang, Mark Kingsbury, Ben McInroe, Will Wagstaff Biological Inspiration Why do fireflies flash? Mating purposes Males flash to attract the attention of nearby females Why
More informationPhase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusion
Phase synchronization of an ensemble of weakly coupled oscillators: A paradigm of sensor fusion Ariën J. van der Wal Netherlands Defence Academy (NLDA) 15 th ICCRTS Santa Monica, CA, June 22-24, 2010 Copyright
More informationarxiv: v1 [q-fin.st] 31 Oct 2011
1 Coupled Oscillator Model of the Business Cycle with Fluctuating Goods Markets Y. Ikeda 1, H. Aoyama, Y. Fujiwara 3, H. Iyetomi 4, K. Ogimoto 1, W. Souma 5, and H. Yoshikawa 6 arxiv:1110.6679v1 [qfin.st]
More informationThe Kuramoto Model. Gerald Cooray. U.U.D.M. Project Report 2008:23. Department of Mathematics Uppsala University
U.U.D.M. Project Report 008:3 The Kuramoto Model Gerald Cooray Examensarbete i matematik, 30 hp Handledare och examinator: David Sumpter September 008 Department of Mathematics Uppsala University THE
More informationPhase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos
Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 682 688 c International Academic Publishers Vol. 35, No. 6, June 15, 2001 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional
More informationBifurcations and global stability of synchronized stationary states in the Kuramoto model for oscillator populations
PHYSICAL REVIEW E, VOLUME 64, 16218 Bifurcations and global stability of synchronized stationary states in the Kuramoto model for oscillator populations J. A. Acebrón, 1,2, * A. Perales, 3 and R. Spigler
More informationα-kuramoto partitions: graph partitions from the frustrated Kuramoto model generalise equitable partitions
α-kuramoto partitions: graph partitions from the frustrated Kuramoto model generalise equitable partitions Stephen Kirkland 1 and Simone Severini 1 Hamilton Institute, National University of Ireland Maynooth,
More informationSpiral Wave Chimeras
Spiral Wave Chimeras Carlo R. Laing IIMS, Massey University, Auckland, New Zealand Erik Martens MPI of Dynamics and Self-Organization, Göttingen, Germany Steve Strogatz Dept of Maths, Cornell University,
More informationSynchronization of coupled stochastic oscillators: The effect of topology
PRMN c Indian cademy of Sciences Vol. 70, No. 6 journal of June 2008 physics pp. 1165 1174 Synchronization of coupled stochastic oscillators: The effect of topology MITBH NNDI 1, and RM RMSWMY 1,2 1 School
More informationDynamics of slow and fast systems on complex networks
Indian Academy of Sciences Conference Series (2017) 1:1 DOI: 10.29195/iascs.01.01.0003 Indian Academy of Sciences Dynamics of slow and fast systems on complex networks KAJARI GUPTA and G. AMBIKA * Indian
More informationarxiv: v1 [nlin.cd] 4 Dec 2017
Chimera at the phase-flip transition of an ensemble of identical nonlinear oscillators R. Gopal a, V. K. Chandrasekar a,, D. V. Senthilkumar c,, A. Venkatesan d, M. Lakshmanan b a Centre for Nonlinear
More informationarxiv: v2 [nlin.ps] 22 Jan 2019
Pattern selection in a ring of Kuramoto oscillators Károly Dénes a, Bulcsú Sándor a,b, Zoltán Néda a, a Babeş-Bolyai University, Department of Physics, 1 Kogălniceanu str., 400084 Cluj, Romania b Goethe
More informationEntrainment Alex Bowie April 7, 2004
Entrainment Alex Bowie April 7, 2004 Abstract The driven Van der Pol oscillator displays entrainment, quasiperiodicity, and chaos. The characteristics of these different modes are discussed as well as
More informationAn analysis of how coupling parameters influence nonlinear oscillator synchronization
An analysis of how coupling parameters influence nonlinear oscillator synchronization Morris Huang, 1 Ben McInroe, 2 Mark Kingsbury, 2 and Will Wagstaff 3 1) School of Mechanical Engineering, Georgia Institute
More informationarxiv: v1 [math.oc] 28 Mar 2013
Synchronization of Weakly Coupled Oscillators: Coupling, Delay and Topology arxiv:133.7248v1 [math.oc] 28 Mar 213 Enrique Mallada and Ao Tang Cornell University, Ithaca, NY 14853 Abstract There are three
More informationExternal Periodic Driving of Large Systems of Globally Coupled Phase Oscillators
External Periodic Driving of Large Systems of Globally Coupled Phase Oscillators T. M. Antonsen Jr., R. T. Faghih, M. Girvan, E. Ott and J. Platig Institute for Research in Electronics and Applied Physics
More informationDESYNCHRONIZATION TRANSITIONS IN RINGS OF COUPLED CHAOTIC OSCILLATORS
Letters International Journal of Bifurcation and Chaos, Vol. 8, No. 8 (1998) 1733 1738 c World Scientific Publishing Company DESYNCHRONIZATION TRANSITIONS IN RINGS OF COUPLED CHAOTIC OSCILLATORS I. P.
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationarxiv: v1 [nlin.ao] 8 Dec 2017
Nonlinearity 31(1):R1-R23 (2018) Identical synchronization of nonidentical oscillators: when only birds of different feathers flock together arxiv:1712.03245v1 [nlin.ao] 8 Dec 2017 Yuanzhao Zhang 1 and
More informationCoherence of Noisy Oscillators with Delayed Feedback Inducing Multistability
Journal of Physics: Conference Series PAPER OPEN ACCESS Coherence of Noisy Oscillators with Delayed Feedback Inducing Multistability To cite this article: Anastasiya V Pimenova and Denis S Goldobin 2016
More informationThe Hamiltonian Mean Field Model: Effect of Network Structure on Synchronization Dynamics. Yogesh Virkar University of Colorado, Boulder.
The Hamiltonian Mean Field Model: Effect of Network Structure on Synchronization Dynamics Yogesh Virkar University of Colorado, Boulder. 1 Collaborators Prof. Juan G. Restrepo Prof. James D. Meiss Department
More informationSynchronizationinanarray of globally coupled maps with delayed interactions
Available online at www.sciencedirect.com Physica A 325 (2003) 186 191 www.elsevier.com/locate/physa Synchronizationinanarray of globally coupled maps with delayed interactions Cristina Masoller a;, Arturo
More informationOTHER SYNCHRONIZATION EXAMPLES IN NATURE 1667 Christiaan Huygens: synchronization of pendulum clocks hanging on a wall networks of coupled Josephson j
WHAT IS RHYTHMIC APPLAUSE? a positive manifestation of the spectators after an exceptional performance spectators begin to clap in phase (synchronization of the clapping) appears after the initial thunderous
More informationOscillator synchronization in complex networks with non-uniform time delays
Oscillator synchronization in complex networks with non-uniform time delays Jens Wilting 12 and Tim S. Evans 13 1 Networks and Complexity Programme, Imperial College London, London SW7 2AZ, United Kingdom
More informationarxiv: v1 [nlin.ao] 23 Sep 2015
One node driving synchronisation Chengwei Wang 1,*, Celso Grebogi 1, and Murilo S. Baptista 1 1 Institute for Comple Systems and Mathematical Biology, King s College, University of Aberdeen, Aberdeen,
More informationUsing controlling chaos technique to suppress self-modulation in a delayed feedback traveling wave tube oscillator
Using controlling chaos technique to suppress self-modulation in a delayed feedback traveling wave tube oscillator N.M. Ryskin, O.S. Khavroshin and V.V. Emelyanov Dept. of Nonlinear Physics Saratov State
More informationCHAOTIC ATTRACTOR IN THE KURAMOTO MODEL
International Journal of Bifurcation and Chaos, Vol., No. () 7 66 c World Scientific Publishing Company CHAOTIC ATTRACTOR IN THE URAMOTO MODEL YURI L. MAISTRENO, OLESANDR V. POPOVYCH and PETER A. TASS
More informationSTUDY OF SYNCHRONIZED MOTIONS IN A ONE-DIMENSIONAL ARRAY OF COUPLED CHAOTIC CIRCUITS
International Journal of Bifurcation and Chaos, Vol 9, No 11 (1999) 19 4 c World Scientific Publishing Company STUDY OF SYNCHRONIZED MOTIONS IN A ONE-DIMENSIONAL ARRAY OF COUPLED CHAOTIC CIRCUITS ZBIGNIEW
More informationChimera State Realization in Chaotic Systems. The Role of Hyperbolicity
Chimera State Realization in Chaotic Systems. The Role of Hyperbolicity Vadim S. Anishchenko Saratov State University, Saratov, Russia Nizhny Novgorod, July 20, 2015 My co-authors Nadezhda Semenova, PhD
More informationWeakly Pulse-Coupled Oscillators: Heterogeneous Delays Lead to Homogeneous Phase
49th IEEE Conference on Decision and Control December 5-7, 2 Hilton Atlanta Hotel, Atlanta, GA, USA Weakly Pulse-Coupled Oscillators: Heterogeneous Delays Lead to Homogeneous Phase Enrique Mallada, Student
More informationPhase Oscillators. and at r, Hence, the limit cycle at r = r is stable if and only if Λ (r ) < 0.
1 Phase Oscillators Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 Phase Oscillators Oscillations
More informationDynamical Systems and Chaos Part II: Biology Applications. Lecture 10: Coupled Systems. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part II: Biology Applications Lecture 10: Coupled Systems. Ilya Potapov Mathematics Department, TUT Room TD325 Foreword In order to model populations of physical/biological
More information9 Atomic Coherence in Three-Level Atoms
9 Atomic Coherence in Three-Level Atoms 9.1 Coherent trapping - dark states In multi-level systems coherent superpositions between different states (atomic coherence) may lead to dramatic changes of light
More informationHow fast elements can affect slow dynamics
Physica D 180 (2003) 1 16 How fast elements can affect slow dynamics Koichi Fujimoto, Kunihiko Kaneko Department of Pure and Applied Sciences, Graduate school of Arts and Sciences, University of Tokyo,
More informationarxiv: v1 [cond-mat.stat-mech] 2 Apr 2013
Link-disorder fluctuation effects on synchronization in random networks Hyunsuk Hong, Jaegon Um, 2 and Hyunggyu Park 2 Department of Physics and Research Institute of Physics and Chemistry, Chonbuk National
More informationSynchronization of Weakly Coupled Oscillators: Coupling, Delay and Topology
Synchronization of Weakly Coupled Oscillators: Coupling, Delay and Topology Enrique Mallada and A. Kevin Tang Cornell University, Ithaca, NY 4853 Abstract There are three key factors of a system of coupled
More informationReentrant synchronization and pattern formation in pacemaker-entrained Kuramoto oscillators
Reentrant synchronization and pattern formation in pacemaker-entrained uramoto oscillators Filippo Radicchi* and Hildegard Meyer-Ortmanns School of Engineering and Science, International University Bremen,
More informationInfluence of noise on the synchronization of the stochastic Kuramoto model
Influence of noise on the synchronization of the stochastic Kuramoto model Bidhan Chandra Bag* Institute of Physics, Academia Sinica, ankang, Taipei 11529, Taiwan and Department of Chemistry, Visva-Bharati
More informationPhase Model for the relaxed van der Pol oscillator and its application to synchronization analysis
Phase Model for the relaxed van der Pol oscillator and its application to synchronization analysis Mimila Prost O. Collado J. Automatic Control Department, CINVESTAV IPN, A.P. 4 74 Mexico, D.F. ( e mail:
More informationNonlinear systems, chaos and control in Engineering
Nonlinear systems, chaos and control in Engineering Module 1 block 3 One-dimensional nonlinear systems Cristina Masoller Cristina.masoller@upc.edu http://www.fisica.edu.uy/~cris/ Schedule Flows on the
More informationDipartimento di Ingegneria dell Informazione, Università di Padova, Via Gradenigo, 6/B, Padova, Italy
The Kuramoto model Juan A. Acebrón Dipartimento di Ingegneria dell Informazione, Università di Padova, Via Gradenigo, 6/B, 35131 Padova, Italy L. L. Bonilla Departamento de Matemáticas, Universidad Carlos
More informationarxiv: v3 [physics.soc-ph] 12 Dec 2008
Synchronization in complex networks Alex Arenas Departament d Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain Institute for Biocomputation and Physics of Complex
More informationNonlinear and Collective Effects in Mesoscopic Mechanical Oscillators
Dynamics Days Asia-Pacific: Singapore, 2004 1 Nonlinear and Collective Effects in Mesoscopic Mechanical Oscillators Alexander Zumdieck (Max Planck, Dresden), Ron Lifshitz (Tel Aviv), Jeff Rogers (HRL,
More informationCollective and Stochastic Effects in Arrays of Submicron Oscillators
DYNAMICS DAYS: Long Beach, 2005 1 Collective and Stochastic Effects in Arrays of Submicron Oscillators Ron Lifshitz (Tel Aviv), Jeff Rogers (HRL, Malibu), Oleg Kogan (Caltech), Yaron Bromberg (Tel Aviv),
More informationarxiv: v1 [nlin.cd] 31 Oct 2015
The new explanation of cluster synchronization in the generalized Kuramoto system Guihua Tian,2, Songhua Hu,2,3, Shuquan Zhong 2, School of Science, Beijing University of Posts And Telecommunications.
More informationTeam Metronome. Quinn Chrzan, Jason Kulpe, Nick Shiver
Team Metronome Quinn Chrzan, Jason Kulpe, Nick Shiver Synchronization Fundamental in nonlinear phenomena Commonly observed to occur between oscillators Synchronization of periodic cicada emergences Synchronization
More informationDesign of Oscillator Networks for Generating Signal with Prescribed Statistical Property
Journal of Physics: Conference Series PAPER OPEN ACCESS Design of Oscillator Networks for Generating Signal with Prescribed Statistical Property To cite this article: Tatsuo Yanagita 2017 J. Phys.: Conf.
More informationCONTROLLING CHAOS. Sudeshna Sinha. The Institute of Mathematical Sciences Chennai
CONTROLLING CHAOS Sudeshna Sinha The Institute of Mathematical Sciences Chennai Sinha, Ramswamy and Subba Rao: Physica D, vol. 43, p. 118 Sinha, Physics Letts. A vol. 156, p. 475 Ramswamy, Sinha, Gupte:
More informationPhase-locked patterns and amplitude death in a ring of delay-coupled limit cycle oscillators
PHYSICAL REVIEW E 69, 056217 (2004) Phase-locked patterns and amplitude death in a ring of delay-coupled limit cycle oscillators Ramana Dodla,* Abhijit Sen, and George L. Johnston Institute for Plasma
More informationarxiv:chao-dyn/ v1 5 Mar 1996
Turbulence in Globally Coupled Maps M. G. Cosenza and A. Parravano Centro de Astrofísica Teórica, Facultad de Ciencias, Universidad de Los Andes, A. Postal 26 La Hechicera, Mérida 5251, Venezuela (To appear,
More informationSYNCHRONIZATION IN CHAINS OF VAN DER POL OSCILLATORS
SYNCHRONIZATION IN CHAINS OF VAN DER POL OSCILLATORS Andreas Henrici ZHAW School of Engineering Technikumstrasse 9 CH-8401 Winterthur, Switzerland andreas.henrici@zhaw.ch Martin Neukom ZHdK ICST Toni-Areal,
More informationChimeras in networks with purely local coupling
Chimeras in networks with purely local coupling Carlo R. Laing Institute of Natural and Mathematical Sciences, Massey University, Private Bag -94 NSMC, Auckland, New Zealand. phone: +64-9-44 8 extn. 435
More informationProceedings of Neural, Parallel, and Scientific Computations 4 (2010) xx-xx PHASE OSCILLATOR NETWORK WITH PIECEWISE-LINEAR DYNAMICS
Proceedings of Neural, Parallel, and Scientific Computations 4 (2010) xx-xx PHASE OSCILLATOR NETWORK WITH PIECEWISE-LINEAR DYNAMICS WALTER GALL, YING ZHOU, AND JOSEPH SALISBURY Department of Mathematics
More informationSynchronization and Swarming: Clocks and Flocks
Synchronization and Swarming: Clocks and Flocks Andrew J. Bernoff, Harvey Mudd College Thanks to Chad Topaz, Macalester College Andrew J. Bernoff Harvey Mudd College ajb@hmc.edu Chad M. Topaz Macalester
More informationarxiv: v1 [physics.data-an] 28 Sep 2009
Synchronization in Networks of Coupled Harmonic Oscillators with Stochastic Perturbation and Time Delays arxiv:99.9v [physics.data-an] Sep 9 Yilun Shang Abstract In this paper, we investigate synchronization
More informationConsensus seeking on moving neighborhood model of random sector graphs
Consensus seeking on moving neighborhood model of random sector graphs Mitra Ganguly School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India Email: gangulyma@rediffmail.com Timothy Eller
More informationDetecting Synchronisation of Biological Oscillators by Model Checking
Detecting Synchronisation of Biological Oscillators by Model Checking Ezio Bartocci, Flavio Corradini, Emanuela Merelli, Luca Tesei School of Sciences and Technology, University of Camerino, Via Madonna
More informationarxiv: v1 [nlin.ao] 21 Feb 2018
Global synchronization of partially forced Kuramoto oscillators on Networks Carolina A. Moreira and Marcus A.M. de Aguiar Instituto de Física Física Gleb Wataghin, Universidade Estadual de Campinas, Unicamp
More informationExperimental observation of direct current voltage-induced phase synchronization
PRAMANA c Indian Academy of Sciences Vol. 67, No. 3 journal of September 2006 physics pp. 441 447 Experimental observation of direct current voltage-induced phase synchronization HAIHONG LI 1, WEIQING
More informationGeneralized Chimera States in Two Interacting Populations of Kuramoto Oscillators
Generalized Chimera States in Two Interacting Populations of Kuramoto Oscillators A RESEARCH PROJECT SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, OF THE UNIVERSITY OF MINNESOTA DULUTH BY Soleh K Dib IN
More informationPhysics of the rhythmic applause
PHYSICAL REVIEW E VOLUME 61, NUMBER 6 JUNE 2000 Physics of the rhythmic applause Z. Néda and E. Ravasz Department of Theoretical Physics, Babeş-Bolyai University, strada Kogălniceanu nr.1, RO-3400, Cluj-Napoca,
More informationKuramoto model with uniformly spaced frequencies: Finite-N asymptotics of the locking threshold
Kuramoto model with uniformly spaced frequencies: Finite-N asymptotics of the locking threshold grows continuously from 0 as γ decreases through γ c. Most strikingly, Kuramoto showed that the onset of
More informationNonlinear Dynamics: Synchronisation
Nonlinear Dynamics: Synchronisation Bristol Centre for Complexity Sciences Ian Ross BRIDGE, School of Geographical Sciences, University of Bristol October 19, 2007 1 / 16 I: Introduction 2 / 16 I: Fireflies
More informationPredicting Phase Synchronization for Homoclinic Chaos in a CO 2 Laser
Predicting Phase Synchronization for Homoclinic Chaos in a CO 2 Laser Isao Tokuda, Jürgen Kurths, Enrico Allaria, Riccardo Meucci, Stefano Boccaletti and F. Tito Arecchi Nonlinear Dynamics, Institute of
More informationnonlinear oscillators. method of averaging
Physics 4 Spring 7 nonlinear oscillators. method of averaging lecture notes, spring semester 7 http://www.phys.uconn.edu/ rozman/courses/p4_7s/ Last modified: April, 7 Oscillator with nonlinear friction
More informationChapter 14 Semiconductor Laser Networks: Synchrony, Consistency, and Analogy of Synaptic Neurons
Chapter 4 Semiconductor Laser Networks: Synchrony, Consistency, and Analogy of Synaptic Neurons Abstract Synchronization among coupled elements is universally observed in nonlinear systems, such as in
More informationUniversity of Bristol - Explore Bristol Research. Early version, also known as pre-print
Erzgraber, H, Krauskopf, B, & Lenstra, D (2004) Compound laser modes of mutually delay-coupled lasers : bifurcation analysis of the locking region Early version, also known as pre-print Link to publication
More informationCluster synchrony in systems of coupled phase oscillators with higher-order coupling
PHYSICAL REVIEW E 84, 68 () Cluster synchrony in systems of coupled phase oscillators with higher-order coupling Per Sebastian Skardal,,* Edward Ott, and Juan G. Restrepo Department of Applied Mathematics,
More informationMultiobjective Optimization of an Extremal Evolution Model
Multiobjective Optimization of an Extremal Evolution Model Mohamed Fathey Elettreby Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Reprint requests to M. F. E.;
More informationarxiv: v1 [nlin.cd] 23 Nov 2015
Different kinds of chimera death states in nonlocally coupled oscillators K. Premalatha 1, V. K. Chandrasekar 2, M. Senthilvelan 1, M. Lakshmanan 1 1 Centre for Nonlinear Dynamics, School of Physics, Bharathidasan
More informationProjective synchronization of a complex network with different fractional order chaos nodes
Projective synchronization of a complex network with different fractional order chaos nodes Wang Ming-Jun( ) a)b), Wang Xing-Yuan( ) a), and Niu Yu-Jun( ) a) a) School of Electronic and Information Engineering,
More informationFirefly Synchronization
Firefly Synchronization Hope Runyeon May 3, 2006 Imagine an entire tree or hillside of thousands of fireflies flashing on and off all at once. Very few people have been fortunate enough to see this synchronization
More informationAutomatic control of phase synchronization in coupled complex oscillators
Physica D xxx (2004) xxx xxx Automatic control of phase synchronization in coupled complex oscillators Vladimir N. Belykh a, Grigory V. Osipov b, Nina Kuckländer c,, Bernd Blasius c,jürgen Kurths c a Mathematics
More informationarxiv: v1 [nlin.cd] 1 Sep 2012
Amplitude and phase dynamics in oscillators with distributed-delay coupling Y.N. yrychko,.b. Blyuss arxiv:129.133v1 [nlin.cd] 1 Sep 212 Department of Mathematics, University of Sussex, Falmer, Brighton,
More informationSynchronization in complex networks
Synchronization in complex networks Alex Arenas, 1, 2,3 Albert Díaz-Guilera, 4,2 Jurgen Kurths, 5 Yamir Moreno, 2,6 and Changsong Zhou 7 1 Departament d Enginyeria Informàtica i Matemàtiques, Universitat
More informationConstructive vs. destructive interference; Coherent vs. incoherent interference
Constructive vs. destructive interference; Coherent vs. incoherent interference Waves that combine in phase add up to relatively high irradiance. = Constructive interference (coherent) Waves that combine
More informationControlling the cortex state transitions by altering the oscillation energy
Controlling the cortex state transitions by altering the oscillation energy Valery Tereshko a and Alexander N. Pisarchik b a School of Computing, University of Paisley, Paisley PA 2BE, Scotland b Centro
More informationMetropolis Monte Carlo simulation of the Ising Model
Metropolis Monte Carlo simulation of the Ising Model Krishna Shrinivas (CH10B026) Swaroop Ramaswamy (CH10B068) May 10, 2013 Modelling and Simulation of Particulate Processes (CH5012) Introduction The Ising
More informationarxiv: v1 [nlin.cd] 1 Feb 2012
Amplitude death in systems of coupled oscillators with distributed-delay coupling Y.N. Kyrychko, K.B. Blyuss Department of Mathematics, University of Sussex, Brighton, BN1 9QH, United Kingdom arxiv:122.226v1
More informationPART 2 : BALANCED HOMODYNE DETECTION
PART 2 : BALANCED HOMODYNE DETECTION Michael G. Raymer Oregon Center for Optics, University of Oregon raymer@uoregon.edu 1 of 31 OUTLINE PART 1 1. Noise Properties of Photodetectors 2. Quantization of
More informationSUPPLEMENTARY INFORMATION
Superconducting qubit oscillator circuit beyond the ultrastrong-coupling regime S1. FLUX BIAS DEPENDENCE OF THE COUPLER S CRITICAL CURRENT The circuit diagram of the coupler in circuit I is shown as the
More information