Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling

Song, Yongli; Xu, Jian; Zhang, Tonghua

doi:10.1063/1.3578046

Cited by 29 publications

(17 citation statements)

References 42 publications

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“…One application is to the stabilization of fixed points [60,61,62]. Velocity delayed coupling has been also explored in the context of amplitude death [63,64] (see Sec. 2.7).…”

Section: Delay Interactionmentioning

confidence: 99%

Amplitude death: The emergence of stationarity in coupled nonlinear systems

2012

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When nonlinear dynamical systems are coupled, depending on the intrinsic dynamics and the manner in which the coupling is organized, a host of novel phenomena can arise. In this context, an important emergent phenomenon is the complete suppression of oscillations, formally termed amplitude death (AD). Oscillations of the entire system cease as a consequence of the interaction, leading to stationary behavior. The fixed points that the coupling stabilizes can be the otherwise unstable fixed points of the uncoupled system or can correspond to novel stationary points. Such behaviour is of relevance in areas ranging from laser physics to the dynamics of biological systems. In this review we discuss the characteristics of the different coupling strategies and scenarios that lead to AD in a variety of different situations, and draw attention to several open issues and challenging problems for further study.

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“…One application is to the stabilization of fixed points [60,61,62]. Velocity delayed coupling has been also explored in the context of amplitude death [63,64] (see Sec. 2.7).…”

Section: Delay Interactionmentioning

confidence: 99%

Amplitude death: The emergence of stationarity in coupled nonlinear systems

2012

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“…Zhang et al [18] studied multiple Hopf bifurcations of three coupled van der Pol oscillators with delay. Song et al [19] studied bifurcation and amplitude death in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling. Barrón et al [20] studied synchronization of four coupled van der Pol oscillators.…”

Section: Introductionmentioning

confidence: 99%

Weak and non-resonant double Hopf bifurcations in m coupled van der Pol oscillators with delay coupling

Wang

Chen

2015

Applied Mathematical Modelling

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Keywords:The method of multiple scales Van der Pol oscillator Double Hopf bifurcation Time delay a b s t r a c t A system of m coupled van der Pol oscillators is considered. The dynamical behavior around 1 : cðc 2 R þ ; c -1; 2; 3Þ double Hopf bifurcation point is studied analytically. By using the method of multiple scales, the amplitude equations are obtained. Two parameters including time delay and the coupling strength are chosen as the bifurcation parameters, and then the dynamical behavior arising from the bifurcation is classified qualitatively in two-parameter plane. Some interesting dynamical phenomena including amplitude death, periodic solution and coexisting periodic solutions are obtained. By fixing m or c, the effects of the changes of c or m on the dynamics are investigated, respectively.

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“…Burić et al [33] investigated a pair of FitzHugh-Nagumo systems with time delays and obtained the bifurcational relations among coupling time lag and coupling constant for different values of the internal time lags. Song et al [34] investigated the bifurcation, amplitude death, and oscillation patterns induced by the coupling time delay in a system of three coupled van der Pol oscillators, which are coupled through the damping terms. Ge and Xu [35][36][37][38] studied the stability switches, synchronization, fold-Hopf, and double Hopf bifurcations in different BAM neural networks consisting of coupled neurons in two layers and delayed couplings.…”

Section: Introductionmentioning

confidence: 99%

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

Mao

Wang

2015

Nonlinear Dyn

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This paper reveals the dynamical behaviors of a coupled network consisting of four neural subnetworks and multiple two-way couplings of neurons between the individual sub-networks. Different types of time delays are introduced into the internal connections within the individual sub-networks and the couplings between the sub-networks. Stability switches of the network equilibrium and Hopf bifurcation are analyzed by discussing the associated characteristic equation. The conditions for the existence of different patterns of oscillations are discussed. By using the Lyapunov's second method, the global stability of the network equilibrium is studied. Numerical simulations are performed to validate the theoretical results, and rich dynamical behaviors are observed, such as synchronous oscillations, multiple stability switches between the rest state and oscillations, phase-locked oscillations, asynchronous oscillations, and the coexistence of different patterns of bifurcated oscillations.

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Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling

Cited by 29 publications

References 42 publications

Amplitude death: The emergence of stationarity in coupled nonlinear systems

Amplitude death: The emergence of stationarity in coupled nonlinear systems

Weak and non-resonant double Hopf bifurcations in m coupled van der Pol oscillators with delay coupling

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

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