PHASE-LOCKED SOLUTIONS IN A HUB CONNECTED OSCILLATOR RING NETWORK
|
|
- Harriet Strickland
- 5 years ago
- Views:
Transcription
1 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 Indies, St. Augustine, Trinidad Abstract. In this work, we analyze a type of system called the hub connected oscillator ring (HCOR) network. The model consists of 3 sub-rings each coupled to a central hub oscillator with a bi-directional coupling constant, α. When uncoupled from the hub, each substructure is characterized by 3 identical oscillators with natural frequency ω i and bi-directional coupling constant, α i. The conditions necessary for existence of phase-locked solutions were derived and their associated stability criteria were determined. The synchronization tree for phase-locked solutions was also developed. A bifurcation analysis of the system was conducted. Finally simulations were carried out to validate the analytically derived results; these results agreed well with the theory. Keywords: Phase-only oscillators, nonlinear dynamics, bifurcations 1. INTRODUCTION Coupled limit cycle oscillators have been widely used to model both physical (laser arrays, Josephson junctions) and engineering systems (communications and electricity distribution networks). In particular, there has been much interest in the development of collective behavior of these systems, particularly phase-locking ( Bridge et.al., 29; Chopra and Spong, 25; D Huys et.al.,28; Jadbabaie et.al., 24; Jeanne et.al., 25; Kuramoto, 1974; Mirollo and Strogatz, 25; Rogge and Aeyels, 24; Winfree, 198). Several factors dictate the type of responses that arise in a given system of oscillators, the most significant being the nature of the individual oscillators and the topology of their connections. Researchers have noted that many of the key phenomena observed in these systems may be recovered by investigating a phase-only equivalent systems. The Kuramoto (phase-only) model has therefore been widely researched. The most common topologies examined have been the (a) all-to-all (or global network) where each oscillator is coupled to every other oscillator in the system, (b) nearest neighbor coupled rings and (c) chains of coupled oscillators. However, a typical engineering topology would be more likely to consist of several groups of interacting motifs (D Huys et.al.,28) with limited communication via a hub. This is particularly important in communications. To begin to understand the dynamics of such systems, we analyze a type of system called the hub connected oscillator ring (HCOR) network. We look at a model which consists of 3 sub-rings each coupled to a central hub oscillator with a bi-directional coupling constant, α. When uncoupled from the hub, each substructure is characterized by 3 identical oscillators with natural frequency ω i and bi-directional coupling constant, α i. In this work, it was shown that phase-locked solutions only exist when each intra-ring coupling coefficients (α i ) exceeds a ring-specific critical value. Using numericall integration, we show that there is a heirarchy of synchronization of phase-locked solutions. A bifurcation analysis of the system was conducted. Finally simulations were carried out to validate the analytically derived results; these results agreed well with the theory. 2. EXISTENCE OF PHASE-LOCKED SOLUTIONS Consider 3 coupled oscillator rings, each coupled to a central hub oscillator with a bidirectional coupling constant α, as shown in Fig. (1). When uncoupled from the other rings, each substructure consists of three identical oscillators with natural frequency (ω i ) and distinct bidirectional coupling constant (α i ). We consider the case of phase only oscillators which evolve in time according to the differential equation: θ i,j = ω i + α sin(θ θ i,j )δ j1 + α i (sin(θ i,j+1 θ i,j ) + sin(θ i,j 1 θ i,j )) (1) where i, j = 1, 2, 3. The hub oscillator has its own natural frequency of ω and its phase is denoted by θ, so θ = ω + α 3 sin(θ i,1 θ ) 1 where δ j1 = 1 if j = 1 and otherwise. If we define the phase differences by: φ i,j = (θ i,j+1 θ i,j ) mod 2π φ i,3 = (θ i,1 θ i,3 ) mod 2π and ψ i = (θ θ i,1 ) mod 2π then the equations may be rewritten as: θ i,j = ω i + α sin(ψ i )δ j1 + α i (sin(φ i,j ) sin(φ i,j 1 )) (3) (2)
2 Copyright c 29 by ABCM Figure 1. An example of a hub connected topology. Each dot is a particle and each line is a bi-directional coupling link. and θ = ω α 3 sin(ψ i ) 1 (4) 2.1 The Phase-locked frequency A network is said to be phase locked when the phase difference between any two oscillators is constant with time, i.e. we can define θ = Ωt θ i,j (t) = Ωt + k i,j (5) (6) where k i,j is the steady-state phase difference and Ω is the phase locked frequency. If we sum the governing equations (1) and (2) for the oscillators, we determine that the phase-locked frequency is the weighted average of the oscillator frequencies, i.e. Ω = ω + 3ω i 1 (7) 2.2 Critical Intra-Ring Coupling Coefficients, α i s Differentiating equations (5) and (6) and substituting into (3) and (4), yields the phase locking equation Ω = ω i + α sin(ψ i )δ j1 + α i (sin(φ i,j ) sin(φ i,j 1 )) (8) For the three element subring, we may rearrange equation (8) to give: Ω ω i = α i (sin(φ i,2 ) sin(φ i,1 )) (9) Ω ω i = α i (sin(φ i,3 ) sin(φ i,2 )) (1) But φ i,3 + φ i,2 + φ i,1 = mod 2π (11) so substituting this relationship in equation (9) and (1) and simplifying the trigonometric relationships we have, (α i s + i ) 2 ( 4 α 2 i s α i i s i s 2 3 α 2 i s 2 6 α i i s i ) = (12) where s = sin(φ i,1 ) and i = (Ω ω i )
3 Copyright c 29 by ABCM Figure 2. (a) Typical bifurcation diagram showing the emergence of "possible" phase-locked solutions in a sub-ring. (b) Synchronization tree as α varies for fixed parameters ω 1 = 2, ω 2 = 3, ω 3 = 4, ω = 1, α 1 = α 2 = 1, α 3 = 2 When α i =, equation (12) has no real roots for s, while as α i, the equation yields 6 real roots (2 simple roots at ± 3/2 and a quadruple root at ). Therefore, as α i is increased the number of real roots of the equation increases from to 6. There is therefore a minimum value of α i for which a real solution will exist. Note the number of roots changes when there is a double root. We differentiate equation (12) to obtain 4 (α i s + i ) ( 6 αi 3 s αi 2 i s α i 2 i s 2 3 αi 3 s i s 6 αi 2 i s + 3 α i 2 ) i (13) The term (s + i α i ) = satisfies both (12) and (13). This has a real solution iff sin(φ i,j ) 1 i.e. Ω ωi α i 1, implying that a pair of critical values of α i occur at ±(Ω ω i ). To determine when the two other pairs of roots emerge, we eliminate s from (12) using the relationship in (13). The resulting equation is The equation has double roots, indicating that at two pairs of new roots emerge α i,cra = (Ω ω i ) α 6 i 2 i (4 i α i ) 2 (4 i + α i ) 2 ( 2 i + 3α 2 i ) 2 = α i,cr = ±4(Ω ω i ) For this case, s is given by sin(φ i,1 ) = (Ω ω i )/α i, substituting into equation (9), we obtain sin(φ i,2 ) = i.e. φ i,2 =, π and (14) sin(φ i,3 ) = (Ω ω i) α i = sin(φ i,1 ) (15) The solutions corresponding to φ i,2 = are equivalent to θ 1,2 = θ 1,3, i.e. the solution is spatially symmetric. For one of these symmetric solutions all the cos φ i,j terms were positive while the other has two negative cos φ i,j terms. The other solution φ i,2 = π and the consistency condition, implies that φ i,3 = π φ i,1. But for this relationship, sin(φ i,3 ) = sin(φ i,1 ) which negates equation(15); hence these solutions cannot exist. Thus on exceeding the lower critical coupling coefficient, only two symmetricphase-locked solution candidates appear α i,crb = 4(Ω ω i ) At the second critical value of α i, the resulting trajectory is not spatially symmetric. The corresponding bifurcation diagram for the onset of synchronization within a sub-ring is as shown in Figure 2(a). However, imposing the consistency equation results yields a combination of phase differences within the subring with at least one term φ i,j < π 2.
4 Copyright c 29 by ABCM 2.3 Critical Hub Coupling Coefficients, α The equation governing the asscoiated phase differrences of the j = 1 oscillators which are connected to the hub oscillator is Ω = ω i + α sin(ψ i ) + α i (sin(φ i,1 ) sin(φ i,3 )) (16) But from summing equations (1) and (11), we know that sin(φ i,1 ) sin(φ i,3 ) = 2(Ω ω i ), so the equation is transformed into: 3(Ω ω i ) = α sin(ψ i ) (17) These equations then give us the following value for the critical link coupling coefficient: α cr = max {3(Ω ω i )} for i = 1, 2, 3 (18) 3. STABILITY OF PHASE-LOCKED SOLUTIONS In order to study the stability of the phase-locked solutions, we transform the equations using θ i,j = θ i,j Ωt. The periodic orbits are thus transformed into equilibrium points. Linearizing about these points and substituting the corresponding phase locked conditions yields a 1 x 1 matrix, [B], whose eigenvalues will yield the stability of the original system. Note that α N j=1 cos ψ i α cos ψ 1 α cos ψ 2 α cos ψ 3 [B] = α cos ψ 1 α cos ψ 2 α cos ψ 3 [B 1 ] [O] [O] [O] [B 2 ] [O] [O] [O] [B 3 ] with [O] being the 3 x 3 null matrix and α i (cos φ i,1 + cos φ i,3 ) α i cos φ i,1 α i cos φ i,3 [Bi α ] = α i cos φ i,1 α i (cos φ i,2 + cos φ i,1 ) α i cos φ i,2 + α i cos φ i,3 α i cos φ i,2 α i (cos φ i,3 + cos φ i,2 ) α cos ψ i We describe an oscillator ring with phase difference vector, Φ i = {φ i,1, φ i,2, φ i,3 } as a potential sink, source or saddle, respectively if the eigenvalues of [Bi ] = [Bα= i ] evaluated at Φ are all negative semi-definite, all positive semi-definite or mixed, respectively; we describe the hub connections with phase difference vector, Ψ = {ψ 1, ψ 2, ψ 3 } similarly. Using graph theory, it can be shown that the complete system will be stable iff all the oscillator rings and the hub connections are potential sinks. Using Gershgorin s theorem, for α i >, the symmetric solution with φ i,1 < π 2 is a potential sink and the other symmetric solution is a potential saddle; conversely, when α i <, the symmetric solution with φ i,1 < π 2 is a potential source and the other symmetric solution is a potential saddle. Thus for positive α i, there is only one potential sink in each sub-ring, yielding one stable limit cycle. For the asymmetric solutions, the phase-locked solutions have at least one of φ i,j > π 2. Gershgorin s theorem implies that with α i >, these will correspond to either potential sources (all φ i,j > π 2 ) or potential saddles, while α i < yields either potential sinks (all φ i,j > π 2 ) or potential saddles. Numerical investigations show that there are two potential sources per subring for α i > and two potential sinks per subring for α i < yielding two stable limit cycles. 4. NUMERICAL RESULTS In this section we simulate the HCOR network with 3 sub-rings each characterised by its own natural frequency and containing 3 elements. We use the following values in the simulation ω 1 = 2, ω 2 = 3, ω 3 = 4, ω = 1 (units in rad/s). 4.1 Parameters associated with the onset of synchronization From the theory developed above, the critical parameters associated with the onset of synchronization are: Ω = 2.8, α 1,cr =.8, α 2,cr =.2, α 3,cr = 1.2, while α cr = 3.6
5 Copyright c 29 by ABCM Table 1. Possible Phase-Locked Combinations per component Hub OSC. Subring 1 Subring 2 Subring 3 { Symmetric solutions Symmetric solutions Symmetric solutions.6435 ψ 1 = (.9273,,.9273) (.214,,.214) (.6435,,.6435) (4.689,, 4.689) (2.942,, 2.942) (2.4981,, ) {.156 ψ 2 = Asymmetric solutions Asymmetric solutions (1.578, , ) { ( , , 1.578) ψ 3 = (.6892, 2.696, 2.935) ( 2.935, 2.696,.6892) Asymmetric solutions Note that α 2,cr < α 1,cr < α 3,cr. The individual sub-ring coefficient couplings were chosen to be α 1 = 1, α 2 = 1 and α 3 = 2, respectively. These values are greater than the individual critical synchronization values for their respective oscillator rings. The hub coupling coefficient was chosen to vary with α (, 4) and simulations were performed. The corresponding synchronization tree is shown in Fig. 2(b). Before the first partial synchronization coupling coefficient α = α 1,ps =.6, we note that there is no synchronization between the different sub-rings. Although the phases within each sub-ring remain close to each other, the phases between different sub-rings diverge. As α is increased above α 1,ps = 3α 2,cr =.6, there is partial synchronization; the oscillators in sub-ring 2 and the hub oscillator converge to the same operating frequency, while the other two sub-rings operate at their own separate frequencies. Further increases in the hub coupling coefficient (to α = α 2,ps = 3α 1,cr = 2.4) result in partial synchronization of sub-rings 1 and 2 with the hub oscillator and finally (at α = α cr = 3α 3,cr = 3.6) full synchronization. At full synchronization, the angular frequencies of all oscillators converge on the mean frequency Ω = Analysis of Phase-locked solution Let us consider the system with α 1 = 1, α 2 = 1, α 3 = 2, α = 4. We will determine the phase-locked solutions and investigate the stability of each system. Based on the analysis of the previous section, Note that we know that at least one solution exists. Within a sub-ring, the symmetric phase-locked solution corresponds to sin(φ i,2 ) = ; sin(φ i,3 ) = sin(φ i,1 ) = Ω ω i ; sin(ψ i ) = 3(Ω ω i) α i α while the asymmetric solutions solve (12). Only complex conjugate pairs of roots exist for subrings 1 and 3 (no asymmetric phase-locked solutions exist), while subring 2 has real roots; consequently, the possible combinations of phase differences which correspond to the frequency and coupling coefficient values given above are given in Table 1. Any combination of an element from each column gives a phase-locked solution. These phase differences represent 252 (=2*2*2*6*2*2) possible limit cycle solutions for the given combination of frequencies and coupling coefficients. Since α i >, i = 1, 2, 3, there is only one stable trajectory ; the stable orbit can be seen from numerical integration (see Fig. 3). Note that the values obtained for the relative phases in the numerical integration correspond to the values determined analytically (mod 2π). 5. CONCLUDING REMARKS In this work, we analyzed a hub connected oscillator ring (HCOR) network consisting of 3 sub-rings each coupled to a central hub oscillator with a bi-directional coupling constant, α. We showed that if a phase-locked solution exists, its frequency is Ω =.1ω i=1 ω i. We determined that a pair of symmetric phase-locked solutions emerge within each subring when the coupling coefficient, α i exceeds Ω ω i. When α i > 4 Ω ω i, four additional asymmetric phase locked solutions emerge. For positive α i, we noted that one of the symmetric solutions was stable while the other was a saddle; for negative α i, the solutions were both unstable, one being a saddle, the other a source. The synchronization tree for phase-locked solutions was also developed numerically. Finally simulations were carried out to validate the analytically derived results; these results agreed well with the theory.
6 Copyright c 29 by ABCM Figure 3. Phase differences relative to the hub oscillator for arbitrary input. (a) Phase differences for the communication oscillators; (b), (c), (d) Phase differences in subrings 1, 2 and 3, respectively. Note that the system is symmetric since θ i,2 = θ i,3 6. REFERENCES Bridge, J., Rand R., and Sah, S. Dynamics of a ring network of phase-only oscillators, Communications in Nonlinear Science and Numerical Simulation 14 (29), Chopra, N. and Spong, M.W., On synchronization of Kuramoto oscillators, Proc. of 44 th IEEE Conf. Decision and Control and European Control Conference (25), D Huys, O., Vicente, R., Erneux, T., Danckaert, J. and Fischer, I., Synchronization properties of network motifs: Influence of coupling delay and symmetry, Chaos (18), (28) Jadbabaie, A., Motee, N., Barahona, M. On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. of the ACC. (24), Jeanne, J., Leonard, N.E., Paley, D. Collective motion of ring-coupled planar particles, Proc. of 44 th IEEE Conf. Decision and Control and European Control Conference (25), Kuramoto, Y., in International Symposium on Mathematical Problems in Theoretical Physics, ed. by H. Araki, Lecture Notes in Physics, vol. 39, (Springer, New York, 1975), 42. Mirollo, R.E., Strogatz, S.H. The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D 25 (25) Rogge, J.A., and Aeyels, D., Stability of phase locking in a ring of unidirectionally coupled oscillators, J. Phys. A: Math. Gen. 37, (24) Winfree, A.T., The Geometry of Biological Time, Springer, New York (198) 7. RESPONSIBILITY NOTICE The author is the only responsible for the printed material included in this paper
Survey 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 informationSynchronization in delaycoupled bipartite networks
Synchronization in delaycoupled bipartite networks Ram Ramaswamy School of Physical Sciences Jawaharlal Nehru University, New Delhi February 20, 2015 Outline Ø Bipartite networks and delay-coupled phase
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 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 informationOn the Critical Coupling Strength for Kuramoto Oscillators
On the Critical Coupling Strength for Kuramoto Oscillators Florian Dörfler and Francesco Bullo Abstract The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made
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 informationCollective Motion of Ring-Coupled Planar Particles
th IEEE Conf. Decision and Control and European Control Conference, 00 Collective Motion of Ring-Coupled Planar Particles James Jeanne Electrical Engineering Princeton University Princeton, J 08, USA jjeanne@princeton.edu
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 informationarxiv: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 informationCONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS
International Journal of Bifurcation and Chaos, Vol. 12, No. 6 (22) 1417 1422 c World Scientific Publishing Company CONTROLLING IN BETWEEN THE LORENZ AND THE CHEN SYSTEMS JINHU LÜ Institute of Systems
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 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 informationDynamics of four coupled phase-only oscillators
Communications in Nonlinear Science and Numerical Simulation 13 (2008) 501 507 www.elsevier.com/locate/cnsns Short communication Dynamics of four coupled phase-only oscillators Richard Rand *, Jeffrey
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 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 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 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 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 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 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 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 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 informationCalculus and Differential Equations II
MATH 250 B Second order autonomous linear systems We are mostly interested with 2 2 first order autonomous systems of the form { x = a x + b y y = c x + d y where x and y are functions of t and a, b, c,
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 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 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 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 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 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 informationarxiv: v1 [nlin.ao] 3 May 2015
UNI-DIRECTIONAL SYNCHRONIZATION AND FREQUENCY DIFFERENCE LOCKING INDUCED BY A HETEROCLINIC RATCHET arxiv:155.418v1 [nlin.ao] 3 May 215 Abstract A system of four coupled oscillators that exhibits unusual
More informationMath 312 Lecture Notes Linear Two-dimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Two-dimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
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 informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
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 informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
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 informationLecture 37: Principal Axes, Translations, and Eulerian Angles
Lecture 37: Principal Axes, Translations, and Eulerian Angles When Can We Find Principal Axes? We can always write down the cubic equation that one must solve to determine the principal moments But if
More informationDynamics of Decision Making in Animal Group. Motion
Dynamics of Decision Making in Animal Group Motion Benjamin Nabet, Naomi E Leonard, Iain D Couzin and Simon A Levin June, 007 Abstract We present a continuous model of a multi-agent system motivated by
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 informationOn the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
1 On the Stability of the Kuramoto Model of Coupled onlinear Oscillators Ali Jadbabaie, ader Motee, and Mauricio Barahona arxiv:math/0504419v1 [math.oc] 20 Apr 2005 Abstract We provide an analysis of the
More informationON THE CRITICAL COUPLING FOR KURAMOTO OSCILLATORS
ON THE CRITICAL COUPLING FOR KURAMOTO OSCILLATORS FLORIAN DÖRFLER AND FRANCESCO BULLO Abstract. The celebrated Kuramoto model captures various synchronization phenomena in biological and man-made dynamical
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 informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationSynchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time Systems
Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 871 876 c International Academic Publishers Vol. 48, No. 5, November 15, 2007 Synchronization and Bifurcation Analysis in Coupled Networks of Discrete-Time
More informationSynchronization in Quotient Network Based on Symmetry
Send Orders for Reprints to reprints@benthamscience.ae The Open Cybernetics & Systemics Journal, 2014, 8, 455-461 455 Synchronization in Quotient Network Based on Symmetry Open Access Tao Shaohua,1, Feng
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 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 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 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 information4 Second-Order Systems
4 Second-Order Systems Second-order autonomous systems occupy an important place in the study of nonlinear systems because solution trajectories can be represented in the plane. This allows for easy visualization
More informationSynchronization and Transient Stability in Power Networks and Non-Uniform Kuramoto Oscillators
Synchronization and Transient Stability in Power Networks and Non-Uniform Kuramoto Oscillators Florian Dörfler and Francesco Bullo Abstract Motivated by recent interest for multi-agent systems and smart
More informationarxiv: v1 [nlin.ao] 19 May 2017
Feature-rich bifurcations in a simple electronic circuit Debdipta Goswami 1, and Subhankar Ray 2, 1 Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA
More informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More informationCluster Synchronization in Networks of Kuramoto Oscillators
Cluster Synchronization in Networks of Kuramoto Oscillators Chiara Favaretto Angelo Cenedese Fabio Pasqualetti Department of Information Engineering, University of Padova, Italy (e-mail: chiara.favaretto.2@phd.unipd.it,angelo.cenedese@unipd.it)
More informationModels Involving Interactions between Predator and Prey Populations
Models Involving Interactions between Predator and Prey Populations Matthew Mitchell Georgia College and State University December 30, 2015 Abstract Predator-prey models are used to show the intricate
More informationGroup Coordination and Cooperative Control of Steered Particles in the Plane
1 1 Group Coordination and Cooperative Control of Steered Particles in the Plane Rodolphe Sepulchre 1, Derek Paley 2, and Naomi Ehrich Leonard 2 1 Electrical Engineering and Computer Science Université
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 informationPeriod-One Rotating Solutions of Horizontally Excited Pendulum Based on Iterative Harmonic Balance
Advances in Pure Mathematics, 015, 5, 413-47 Published Online June 015 in Scies. http://www.scirp.org/journal/apm http://dx.doi.org/10.436/apm.015.58041 Period-One otating Solutions of Horizontally Excited
More informationLecture 3 : Bifurcation Analysis
Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October - December 2008 D. Sumpter & S.C. Nicolis General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state
More informationStabilization of Collective Motion of Self-Propelled Particles
Stabilization of Collective Motion of Self-Propelled Particles Rodolphe Sepulchre 1, Dere Paley 2, and Naomi Leonard 2 1 Department of Electrical Engineering and Computer Science, Université de Liège,
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 informationOn the Stability of the Kuramoto Model of Coupled Nonlinear Oscillators
On the Stability of the Kuramoto Model of Coupled onlinear Oscillators Ali Jadbabaie, ader Motee and Mauricio Barahona Department of Electrical and Systems Engineering and GRASP Laboratory, University
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 information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationKristina Lerman USC Information Sciences Institute
Rethinking Network Structure Kristina Lerman USC Information Sciences Institute Università della Svizzera Italiana, December 16, 2011 Measuring network structure Central nodes Community structure Strength
More informationPhysics 235 Chapter 4. Chapter 4 Non-Linear Oscillations and Chaos
Chapter 4 Non-Linear Oscillations and Chaos Non-Linear Differential Equations Up to now we have considered differential equations with terms that are proportional to the acceleration, the velocity, and
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 informationLecture 38: Equations of Rigid-Body Motion
Lecture 38: Equations of Rigid-Body Motion It s going to be easiest to find the equations of motion for the object in the body frame i.e., the frame where the axes are principal axes In general, we can
More informationNonlinear Oscillators: Free Response
20 Nonlinear Oscillators: Free Response Tools Used in Lab 20 Pendulums To the Instructor: This lab is just an introduction to the nonlinear phase portraits, but the connection between phase portraits and
More informationChapter 14 Three Ways of Treating a Linear Delay Differential Equation
Chapter 14 Three Ways of Treating a Linear Delay Differential Equation Si Mohamed Sah and Richard H. Rand Abstract This work concerns the occurrence of Hopf bifurcations in delay differential equations
More informationUSING COUPLED OSCILLATORS TO MODEL THE SINO-ATRIAL NODE IN THE HEART
USING COUPLED OSCILLATORS TO MODEL THE SINO-ATRIAL NODE IN THE HEART A Thesis Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree
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 informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More information898 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 17, NO. 6, DECEMBER X/01$ IEEE
898 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 17, NO. 6, DECEMBER 2001 Short Papers The Chaotic Mobile Robot Yoshihiko Nakamura and Akinori Sekiguchi Abstract In this paper, we develop a method
More informationDynamical modelling of systems of coupled oscillators
Dynamical modelling of systems of coupled oscillators Mathematical Neuroscience Network Training Workshop Edinburgh Peter Ashwin University of Exeter 22nd March 2009 Peter Ashwin (University of Exeter)
More informationBifurcations of phase portraits of pendulum with vibrating suspension point
Bifurcations of phase portraits of pendulum with vibrating suspension point arxiv:1605.09448v [math.ds] 9 Sep 016 A.I. Neishtadt 1,,, K. Sheng 1 1 Loughborough University, Loughborough, LE11 3TU, UK Space
More informationThe projects listed on the following pages are suitable for MSc/MSci or PhD students. An MSc/MSci project normally requires a review of the
The projects listed on the following pages are suitable for MSc/MSci or PhD students. An MSc/MSci project normally requires a review of the literature and finding recent related results in the existing
More informationarxiv: v1 [math.ds] 20 Sep 2016
THE PITCHFORK BIFURCATION arxiv:1609.05996v1 [math.ds] 20 Sep 2016 Contents INDIKA RAJAPAKSE AND STEVE SMALE Abstract. We give development of a new theory of the Pitchfork bifurcation, which removes the
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 informationMath 5BI: Problem Set 6 Gradient dynamical systems
Math 5BI: Problem Set 6 Gradient dynamical systems April 25, 2007 Recall that if f(x) = f(x 1, x 2,..., x n ) is a smooth function of n variables, the gradient of f is the vector field f(x) = ( f)(x 1,
More informationMath 345 Intro to Math Biology Lecture 7: Models of System of Nonlinear Difference Equations
Math 345 Intro to Math Biology Lecture 7: Models of System of Nonlinear Difference Equations Junping Shi College of William and Mary, USA Equilibrium Model: x n+1 = f (x n ), here f is a nonlinear function
More informationStabilization of a 3D Rigid Pendulum
25 American Control Conference June 8-, 25. Portland, OR, USA ThC5.6 Stabilization of a 3D Rigid Pendulum Nalin A. Chaturvedi, Fabio Bacconi, Amit K. Sanyal, Dennis Bernstein, N. Harris McClamroch Department
More informationOn the Trajectories and Coordination of Steered Particles with Time-Periodic Speed Profiles
Proceedings of the 9 American Control Conference, St. Louis, MO, June 9 On the Trajectories and Coordination of Steered Particles with Time-Periodic Speed Profiles Daniel T. Swain, Naomi Ehrich Leonard
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 informationConsensus Protocols for Networks of Dynamic Agents
Consensus Protocols for Networks of Dynamic Agents Reza Olfati Saber Richard M. Murray Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 e-mail: {olfati,murray}@cds.caltech.edu
More informationProblem Set Number 2, j/2.036j MIT (Fall 2014)
Problem Set Number 2, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Mon., September 29, 2014. 1 Inverse function problem #01. Statement: Inverse function
More informationBifurcation and Chaos in Coupled Periodically Forced Non-identical Duffing Oscillators
APS/13-QED Bifurcation and Chaos in Coupled Periodically Forced Non-identical Duffing Oscillators U. E. Vincent 1 and A. Kenfack, 1 Department of Physics, Olabisi Onabanjo University, Ago-Iwoye, Nigeria.
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 information8.1 Bifurcations of Equilibria
1 81 Bifurcations of Equilibria Bifurcation theory studies qualitative changes in solutions as a parameter varies In general one could study the bifurcation theory of ODEs PDEs integro-differential equations
More informationMulti-Pendulum Synchronization Using Constrained Agreement Protocols
Multi-Pendulum Synchronization Using Constrained Agreement Protocols Rahul Chipalatty, Magnus Egerstedt, and Shun-Ichi Azuma 2 gtg5s@gatechedu,magnus@ecegatechedu,sazuma@iyoto-uacjp Georgia Institute of
More informationName :. Roll No. :... Invigilator s Signature :.. CS/B.TECH (NEW)(CSE/IT)/SEM-4/M-401/ MATHEMATICS - III
Name :. Roll No. :..... Invigilator s Signature :.. 202 MATHEMATICS - III Time Allotted : 3 Hours Full Marks : 70 The figures in the margin indicate full marks. Candidates are required to give their answers
More informationEE222 - Spring 16 - Lecture 2 Notes 1
EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued
More informationEffects of Interactive Function Forms in a Self-Organized Critical Model Based on Neural Networks
Commun. Theor. Phys. (Beijing, China) 40 (2003) pp. 607 613 c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Effects of Interactive Function Forms in a Self-Organized Critical Model
More informationOn Bifurcations. in Nonlinear Consensus Networks
On Bifurcations in Nonlinear Consensus Networks Vaibhav Srivastava Jeff Moehlis Francesco Bullo Abstract The theory of consensus dynamics is widely employed to study various linear behaviors in networked
More informationChaos Theory. Namit Anand Y Integrated M.Sc.( ) Under the guidance of. Prof. S.C. Phatak. Center for Excellence in Basic Sciences
Chaos Theory Namit Anand Y1111033 Integrated M.Sc.(2011-2016) Under the guidance of Prof. S.C. Phatak Center for Excellence in Basic Sciences University of Mumbai 1 Contents 1 Abstract 3 1.1 Basic Definitions
More informationChaos in generically coupled phase oscillator networks with nonpairwise interactions
Chaos in generically coupled phase oscillator networks with nonpairwise interactions Christian Bick, Peter Ashwin, and Ana Rodrigues Centre for Systems, Dynamics and Control and Department of Mathematics,
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 informationChaos Control of the Chaotic Symmetric Gyroscope System
48 Chaos Control of the Chaotic Symmetric Gyroscope System * Barış CEVHER, Yılmaz UYAROĞLU and 3 Selçuk EMIROĞLU,,3 Faculty of Engineering, Department of Electrical and Electronics Engineering Sakarya
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 informationEffects of Scale-Free Topological Properties on Dynamical Synchronization and Control in Coupled Map Lattices
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 361 368 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Effects of Scale-Free Topological Properties on Dynamical Synchronization
More information