The stability of entrainment conditions forRLC coupled van der pol oscillators used as a model for intestinal electrical rhythms

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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Dynamics of Two Coupled Van der Pol Oscillators With Delay Coupling

Volume 1C: 16th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc97/vib-4019 ◽

1997 ◽

Author(s):

Stephen Wirkus ◽

Richard Rand

Keyword(s):

Three Dimensions ◽

Slow Flow ◽

Method Of Averaging ◽

Van Der Pol ◽

Laser Oscillators ◽

Van Der Pol Oscillators ◽

The Stability

Abstract We investigate the dynamics of a system of two van der Pol oscillators with delayed velocity coupling. We use the method of averaging to reduce the problem to the study of a slow-flow in three dimensions. In particular we study the stability of the in-phase and out-of-phase modes, and the bifurcations associated with changes in their stability. Our interest in this system is due to its relevance to coupled laser oscillators.

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An Investigation of Coupled van der Pol Oscillators

Journal of Vibration and Acoustics ◽

10.1115/1.1553469 ◽

2003 ◽

Vol 125 (2) ◽

pp. 162-169 ◽

Cited By ~ 12

Author(s):

Lesley Ann Low ◽

Per G. Reinhall ◽

Duane W. Storti

Keyword(s):

Coupled Oscillators ◽

Stability Boundary ◽

Numerical Results ◽

Van Der Pol ◽

Coupling Parameters ◽

Van Der Pol Oscillators ◽

The Stability ◽

And Behavior

In this paper we present findings from an investigation of synchronization of linearly diffusively coupled van der Pol oscillators. The stability boundary of the in-phase mode of two identical oscillators in terms of the two coupling parameters is determined numerically. We show that in addition to the out-of-phase and in-phase motions of the oscillators there exist two other phase-locked motions and behavior that appears chaotic. The effect of detuning the oscillators from each other is also presented. Finally, the analytical and numerical results from an investigation of the in-phase mode system of n coupled oscillators is presented.

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Phase-Locked Mode Stability for Coupled van der Pol Oscillators

Journal of Vibration and Acoustics ◽

10.1115/1.1302314 ◽

1999 ◽

Vol 122 (3) ◽

pp. 318-323 ◽

Cited By ~ 14

Author(s):

Duane W. Storti ◽

Per G. Reinhall

Keyword(s):

Variational Equation ◽

Stability Boundary ◽

Series Expansions ◽

Van Der Pol ◽

Stability Boundaries ◽

Power Series Expansions ◽

Mode Stability ◽

Van Der Pol Oscillators ◽

The Stability ◽

The Hill

The critical variational equation governing the stability of phase-locked modes for a pair of diffusively coupled van der Pol oscillators is presented in the form of a linear oscillator with a periodic damping coefficient that involves the van der Pol limit cycle. The variational equation is transformed into a Hill’s equation, and stability boundaries are obtained by analytical and numerical methods. We identify a countable set of resonances and obtain expressions for the associated stability boundaries as power series expansions of the associated Hill determinants. We establish an additional “zero mean damping” condition and express it as a Pade´ approximant describing a surface that combines with the Hill determinant surfaces to complete the stability boundary. The expansions obtained are evaluated to visualize the first three resonant surfaces which are compared with numerically determined slices through the stability boundaries computed over the range 0.4<ε<5. [S0739-3717(00)00502-X]

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The stability of entrainment conditions for coupled van der pol oscillators used as a model for intestinal electrical rhythms

Bulletin of Mathematical Biology ◽

10.1016/s0092-8240(77)80073-6 ◽

1977 ◽

Vol 39 (3) ◽

pp. 359-372 ◽

Cited By ~ 2

Author(s):

D LINKENS

Keyword(s):

Van Der Pol ◽

Van Der Pol Oscillators ◽

The Stability

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Vibration of Harmonically Excited Oscillators With Asymmetric Constraints

Journal of Applied Mechanics ◽

10.1115/1.2899502 ◽

1992 ◽

Vol 59 (2S) ◽

pp. S284-S290 ◽

Cited By ~ 24

Author(s):

S. Natsiavas ◽

H. Gonzalez

Keyword(s):

Piecewise Linear ◽

Periodic Motions ◽

Van Der Pol ◽

System Parameters ◽

State Response ◽

Damping Coefficients ◽

Restoring Forces ◽

Van Der Pol Oscillators ◽

The Stability ◽

Stiffness And Damping

investigation is carried out for a class of piecewise linear oscillators with asymmetric characteristics. The damping and restoring forces are general trilinear functions of the system velocity and displacement, respectively, while the excitation is harmonic in time. First, an analysis is presented which determines harmonic and subharmonic steady-state response. Then, a special formulation is employed in examining the stability of located periodic motions. Finally, numerical results are presented for several representative sets of the system parameters. Effects of asymmetries in the response due to unequal gaps as well as unequal stiffness and damping coefficients are analyzed in detail. Asymmetric response of a system with symmetric technical characteristics is also investigated. The behavior of the systems examined resembles response of similar nonlinear systems with continuous characteristics, like the response of the Duffing and van der Pol oscillators. Complicated nonperiodic response is also encountered and analyzed.

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Analog Simulation of Phase Locked Modes of Diffusively Coupled van der Pol Oscillators

Volume 7B: 17th Biennial Conference on Mechanical Vibration and Noise ◽

10.1115/detc99/vib-8013 ◽

1999 ◽

Author(s):

Lesley Ann Low ◽

Per G. Reinhall ◽

Duane W. Storti

Keyword(s):

Limit Cycle ◽

Limit Cycles ◽

Physical System ◽

Coupled Oscillators ◽

Electric Circuit ◽

Van Der Pol ◽

Analog Simulation ◽

Coupling Parameters ◽

Van Der Pol Oscillators ◽

The Stability

Abstract Limit cycle oscillators arise in a wide variety of mechanical, electrical and biological systems. Recently, emphasis has been placed on the study of systems of coupled limit cycles, such as cardiac oscillations. Synchronization criteria have remained a focus of most investigations. One area of investigation in the field of coupled limit cycles is studying the behavior of a pair of linearly coupled van der Pol oscillators (Low, 1998; Rand, 1980; Sliger, 1997). Previous investigations (Storti, 1993; Storti, 1996) found the stability regions of the coupled oscillators for their in-phase and out-of-phase modes numerically. The coupled oscillators can be viewed as a mechanical system, where the coupling parameters are equivalent to a spring and damper attached between two masses. With positive coupling (positive damping) a region was found (Storti, 1993; Storti, 1996) where the in-phase mode is unstable. This counter intuitive result is yet to be discovered in a physical system of two coupled limit cycle oscillators. This research focuses on finding the region with positive coupling parameters where the in-phase mode is unstable using a physical model of two linearly coupled van der Pol oscillators. The coupled van der Pol oscillators were modeled using an analog electric circuit.

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