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

Studies of nonlinear dynamical systems with many degrees of freedom show that the behavior of these systems is significantly different as compared with the behavior of systems with less than two degrees of freedom. These findings motivated us to carry out a survey of research focusing on the behavior of high-dimensional chaos, which include onset of chaos, routes to chaos and the persistence of chaos. This paper reports on various methods of generating and investigating nonlinear, dissipative and driven dynamical systems that exhibit high-dimensional chaos, and reviews recent results in this new field of research. We study high-dimensional Lorenz, Duffing, Rössler and Van der Pol oscillators, modified canonical Chua's circuits, and other dynamical systems and maps, and we formulate general rules of high-dimensional chaos. Basic techniques of chaos control and synchronization developed for high-dimensional dynamical systems are also reviewed.

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Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators

Nelineinaya Dinamika ◽

10.20537/nd180402 ◽

2018 ◽

Vol 14 (4) ◽

pp. 435-451

Author(s):

V.M. Doroshenko ◽

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V.P. Kruglov ◽

S.P. Kuznetsov ◽

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...

Keyword(s):

Van Der Pol ◽

Van Der Pol Oscillators

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Coupled Van Der Pol oscillators utilised as Central pattern generators for quadruped locomotion

2009 Chinese Control and Decision Conference ◽

10.1109/ccdc.2009.5192385 ◽

2009 ◽

Cited By ~ 2

Author(s):

Chengju Liu ◽

Qijun Chen ◽

Jiaqi Zhang

Keyword(s):

Central Pattern Generators ◽

Van Der Pol ◽

Quadruped Locomotion ◽

Van Der Pol Oscillators ◽

Pattern Generators

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Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2003/vib-48593 ◽

2003 ◽

Author(s):

Erika Camacho ◽

Richard Rand ◽

Howard Howland

Keyword(s):

Software Package ◽

Bifurcation Theory ◽

Floquet Theory ◽

Van Der Pol Oscillator ◽

Direct Connection ◽

Periodic Motions ◽

Van Der Pol ◽

Existence And Stability ◽

Van Der Pol Oscillators ◽

The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

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