Phaselocked patterns and amplitude death in a ring of delay coupled limit cycle oscillators

Abstract

We study the existence and stability of phaselocked patterns and amplitude death states in a closed chain of delay coupled identical limit cycle oscillators that are near a supercritical Hopf bifurcation. The coupling is limited to nearest neighbors and is linear. We analyze a model set of discrete dynamical equations using the method of plane waves. The resultant dispersion relation, which is valid for any arbitrary number of oscillators, displays important differences from similar relations obtained from continuum models. We discuss the general characteristics of the equilibrium states including their dependencies on various system parameters. We next carry out a detailed linear stability investigation of these states in order to delineate their actual existence regions and to determine their parametric dependence on time delay. Time delay is found to expand the range of possible phaselocked patterns and to contribute favorably toward their stability. The amplitude death state is studied in the parameter space of time delay and coupling strength. It is shown that death island regions can exist for any number of oscillators N in the presence of finite time delay. A particularly interesting result is that the size of an island is independent of N when N is even but is a decreasing function of N when N is odd.

23 pages, 12 figures (3 of the figures in PNG format, separately from TeX); minor additions; typos corrected

References in corpus (7)

• Time Delay in the Kuramoto Model of Coupled Oscillators
• Time Delay Induced Death in Coupled Limit Cycle Oscillators
• Distributed Delays Facilitate Amplitude Death of Coupled Oscillators
• Experimental Evidence of Time Delay Induced Death in Coupled Limit Cycle Oscillators
• Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation
• Total and partial amplitude death in networks of diffusively coupled oscillators
• Identification of parameters in amplitude equations describing coupled wakes

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• Embedding the dynamics of a single delay system into a feed-forward ring
• Exceptional points in coupled dissipative dynamical systems
• Pattern selection in a ring of Kuramoto oscillators
• Delay induced Turing-like waves for one species reaction-diffusion model on a network
• Synchronization in the presence of memory
• Amplitude death in networks of delay-coupled delay oscillators
• Stability of networks of delay-coupled delay oscillators
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• Coupled complex Ginzburg-Landau systems with saturable nonlinearity and asymmetric cross-phase modulation
• Amplitude death in a ring of nonidentical nonlinear oscillators with unidirectional coupling
• Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators
• Realization of critical eigenvalues for scalar and symmetric linear delay-differential equations