Hopf bifurcation of Van der Pol oscillators operating under unilateral constraints

2016 ◽  
Author(s):  
Yury Orlov ◽  
Oscar Montano ◽  
Leonardo Herrera
Keyword(s):  
Hopf Bifurcation ◽  
Unilateral Constraints ◽  
Van Der Pol ◽  
Van Der Pol Oscillators

Hopf bifurcation in delayed van der Pol oscillators

2012 ◽  
Vol 71 (3) ◽  
pp. 555-568 ◽  
Author(s):  
Ling Zhang ◽  
Shangjiang Guo
Keyword(s):  
Hopf Bifurcation ◽  
Van Der Pol ◽  
Van Der Pol Oscillators

Coupled Oscillatory Systems with 𝔻4 Symmetry and Application to van der Pol Oscillators

2016 ◽  
Vol 26 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Adrian C. Murza ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Invariant Manifolds ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Oscillatory Systems ◽  
Weakly Coupled ◽  
Form Theory ◽  
Van Der Pol Oscillators

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.

Hopf bifurcation of hybrid Van der Pol oscillators

2017 ◽  
Vol 26 ◽  
pp. 225-238 ◽  
Author(s):  
Leonardo Herrera ◽  
Oscar Montano ◽  
Yury Orlov
Keyword(s):  
Hopf Bifurcation ◽  
Van Der Pol ◽  
Van Der Pol Oscillators

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

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.

Asymptotic Wave Solutions for the Model of a Medium with Van Der Pol Oscillators

2014 ◽  
Vol 59 (9) ◽  
pp. 932-938
Author(s):  
V.A. Danylenko ◽  
◽  
S.I. Skurativskyi ◽  
I.A. Skurativska ◽  
◽  
...  
Keyword(s):  
Van Der Pol ◽  
Wave Solutions ◽  
Van Der Pol Oscillators

scholarly journals Modal synchronization of coupled bistable van der Pol oscillators

2021 ◽  
Vol 143 ◽  
pp. 110555
Author(s):  
I.B. Shiroky ◽  
O.V. Gendelman
Keyword(s):  
Van Der Pol ◽  
Van Der Pol Oscillators

scholarly journals Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

2017 ◽  
Vol 4 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Mohit Sinha ◽  
Florian Dorfler ◽  
Brian B. Johnson ◽  
Sairaj V. Dhople
Keyword(s):  
Nonlinear Dynamics ◽  
Droop Control ◽  
Van Der Pol ◽  
Control Laws ◽  
Van Der Pol Oscillators
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