Routes to complex dynamics in a ring of unidirectionally coupled systems

Abstract

We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

Multistable behavior above synchronization in a locally coupled Kuramoto model

“…Zheng et al [12] already in 1998 pointed out that the behavior of the order parameter "indicates the coexistence of multiple attractors of phase locking states" above the synchronization critical coupling, K s . Even when synchronization with coexistence of attractors has been reported by different authors and in different fields below K s [12][13][14][20][21][22][23][24][25], nobody, to the best of our knowledge, has pursued the matter above full synchronization. If we have the intention to simulate real systems, mostly in technological applications, it is important to know whether we can move freely below and also above synchronization within stable solutions and to know whether they are unique.…”

Nonlinear resonance and synchronization in the ring of unidirectionally coupled Toda oscillators

“…It has been shown, that increasing the coupling strength leads to selfoscillation birth [2,3]. Hence, the system which is investigated in our paper, represents a special type of generator under external force.…”

“…Oscillator chains can be divided into two groups: self-excited, which demonstrate stable limit cycle without external driving [1][2][3][4][5] and strictly dissipative chains, which have stable steady state without forcing [6][7][8][9]. For the ring of unidirectionally coupled Duffing oscillators [2] and for the ring of unidirectionally coupled Toda oscillators with nonlinear coupling function [3], it has been shown that with the increase of coupling coefficient the stable fixed point undergoes Andronov-Hopf bifurcation where the stable limit cycle may appear [3,4].…”

“…For the ring of unidirectionally coupled Duffing oscillators [2] and for the ring of unidirectionally coupled Toda oscillators with nonlinear coupling function [3], it has been shown that with the increase of coupling coefficient the stable fixed point undergoes Andronov-Hopf bifurcation where the stable limit cycle may appear [3,4]. Hence, the system can reach a stable fixed point (when the coupling coefficient of the system is small) or stable limit cycle (when the coupling coefficient is above the threshold of Andronov-Hopf bifurcation).…”

The dynamics of the two mutually coupled autonomous Van der Pol Oscillator

“…Such as drive-response or activepassive of the system, direct bi-directional (mutual) coupling or unidirectional (master-slave) diffusive coupling between the oscillators [14][15][16]. In this connection we introduce a linear feedback coupling [17] of the individual system and coupled mutually of the two systems. For a particular choice of parameter, we confirm the existence of the anticipatory synchronization via phase synchronization the coupling co-efficient is varied.…”