Stability and Hopf bifurcation in van der Pol oscillators with state-dependent delayed feedback

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

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Hopf bifurcation of hybrid Van der Pol oscillators

Nonlinear Analysis Hybrid Systems ◽

10.1016/j.nahs.2017.05.003 ◽

2017 ◽

Vol 26 ◽

pp. 225-238 ◽

Cited By ~ 3

Author(s):

Leonardo Herrera ◽

Oscar Montano ◽

Yury Orlov

Keyword(s):

Hopf Bifurcation ◽

Van Der Pol ◽

Van Der Pol Oscillators

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Hopf bifurcation and spatio-temporal patterns in delay-coupled van der Pol oscillators

Nonlinear Dynamics ◽

10.1007/s11071-010-9799-y ◽

2010 ◽

Vol 63 (1-2) ◽

pp. 223-237 ◽

Cited By ~ 18

Author(s):

Yongli Song

Keyword(s):

Hopf Bifurcation ◽

Temporal Patterns ◽

Van Der Pol ◽

Van Der Pol Oscillators ◽

Spatio Temporal

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Death Behavior of Mean-Field Coupled van der Pol Oscillators Induced by Time-Delayed Feedback

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127420500406 ◽

2020 ◽

Vol 30 (03) ◽

pp. 2050040

Author(s):

Qin Guo ◽

Xige Yang ◽

Jiankang Liu ◽

Wei Xu

Keyword(s):

Time Delay ◽

Mean Field ◽

Delayed Feedback ◽

Van Der Pol ◽

Coupled Nonlinear Oscillators ◽

Oscillatory State ◽

Feedback Strength ◽

Death State ◽

Van Der Pol Oscillators ◽

Transition Points

Explosive death in coupled nonlinear oscillators has been an active area of extensive research in nonlinear dynamics in the recent decades. Depending on proper choice of network topology, coupling scenarios, and feedback strength, explosive death can be revealed. In this work, for the first time, we report the effect of delayed feedback on the death behavior in an ensemble of identical mean-field coupled van der Pol oscillators. In both systems with or without time delay, the normalized amplitude exhibits an abrupt transition between the oscillatory state and the death state. Intriguingly, the presence of time delay in the coupling may induce the normalized amplitude of all oscillators in the network to experience a step-like descent with small jumps in approaching the death state, pulling back the forward and backward transition points. The backward transition point has been explicitly obtained, which is confirmed by the numerical results.

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Modulation of contractions in the small intestine indicate desynchronization via supercritical Andronov–Hopf bifurcation

Scientific Reports ◽

10.1038/s41598-020-71999-4 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Sean P. Parsons ◽

Jan D. Huizinga

Keyword(s):

Small Intestine ◽

Hopf Bifurcation ◽

Coupled Oscillators ◽

Interstitial Cells Of Cajal ◽

Van Der Pol ◽

Physical Systems ◽

Fourier Frequency ◽

Van Der Pol Oscillators ◽

A Chain ◽

Cells Of Cajal

Abstract The small intestine is covered by a network of coupled oscillators, the interstitial cells of Cajal (ICC). These oscillators synchronize to generate rhythmic phase waves of contraction. At points of low coupling, oscillations desynchronise, frequency steps occur and every few waves terminates as a dislocation. The amplitude of contractions is modulated at frequency steps. The phase difference between contractions at a frequency step and a proximal reference point increased slowly at first and then, just at the dislocation, increased rapidly. Simultaneous frequency and amplitude modulation (AM/FM) results in a Fourier frequency spectrum with a lower sideband, a so called Lashinsky spectrum, and this was also seen in the small intestine. A model of the small intestine consisting of a chain of coupled Van der Pol oscillators, also demonstrated simultaneous AM/FM at frequency steps along with a Lashinsky spectrum. Simultaneous AM/FM, together with a Lashinsky spectrum, are predicted to occur when periodically-forced or mutually-coupled oscillators desynchronise via a supercritical Andronov–Hopf bifurcation and have been observed before in other physical systems of forced or coupled oscillators in plasma physics and electrical engineering. Thus motility patterns in the intestine can be understood from the viewpoint of very general dynamical principles.

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