An Investigation of Coupled van der Pol Oscillators

2003 ◽  
Vol 125 (2) ◽  
pp. 162-169 ◽  
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.

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

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.

Phase-Locked Mode Stability for Coupled van der Pol Oscillators

1999 ◽  
Vol 122 (3) ◽  
pp. 318-323 ◽  
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]

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

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 &gt; 0 and k &gt; 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.

DYNAMICS AND MODULATED CHAOS FOR TWO COUPLED OSCILLATORS

2004 ◽  
Vol 14 (01) ◽  
pp. 337-346 ◽  
Author(s):  
QINSHENG BI
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
The Other ◽  
Van Der Pol ◽  
Parametrically Excited ◽  
Van Der Pol Oscillators ◽  
Period Doubling Bifurcations

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

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.

Dynamics of Two Coupled Van der Pol Oscillators With Delay Coupling

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.

scholarly journals Oscillations and Synchronization in a System of Three Reactively Coupled Oscillators

2016 ◽  
Vol 26 (01) ◽  
pp. 1650010 ◽  
Author(s):  
Alexander P. Kuznetsov ◽  
Ludmila V. Turukina ◽  
Nikolai Yu. Chernyshov ◽  
Yuliya V. Sedova
Keyword(s):  
Lyapunov Exponent ◽  
Bifurcation Analysis ◽  
Coupled Oscillators ◽  
Coupling Parameter ◽  
Van Der Pol ◽  
Coupling Phase ◽  
Proper Order ◽  
Van Der Pol Oscillators ◽  
Reactive Coupling

We consider a system of three interacting van der Pol oscillators with reactive coupling. Phase equations are derived, using proper order of expansion over the coupling parameter. The dynamics of the system is studied by means of the bifurcation analysis and with the method of Lyapunov exponent charts. Essential and physically meaningful features of the reactive coupling are discussed.

Vibration of Harmonically Excited Oscillators With Asymmetric Constraints

1992 ◽  
Vol 59 (2S) ◽  
pp. S284-S290 ◽  
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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