Bifurcation of periodic motions in two weakly coupled van der Pol oscillators

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501418 ◽

2016 ◽

Vol 26 (08) ◽

pp. 1650141 ◽

Cited By ~ 1

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.

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The transition from phase locking to drift in a system of two weakly coupled van der pol oscillators

International Journal of Non-Linear Mechanics ◽

10.1016/0020-7462(88)90034-0 ◽

1988 ◽

Vol 23 (5-6) ◽

pp. 369-376 ◽

Cited By ~ 63

Author(s):

Tapesh Chakraborty ◽

Richard H. Rand

Keyword(s):

Phase Locking ◽

Van Der Pol ◽

Weakly Coupled ◽

Van Der Pol Oscillators

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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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The Principal Parametric Resonance of Coupled van der Pol Oscillators under Feedback Control

Shock and Vibration ◽

10.1155/2012/507378 ◽

2012 ◽

Vol 19 (3) ◽

pp. 365-377 ◽

Cited By ~ 1

Author(s):

Xinye Li ◽

Huabiao Zhang ◽

Lijuan He

Keyword(s):

Time Delay ◽

Feedback Control ◽

Parametric Resonance ◽

Relaxation Oscillation ◽

Periodic Motions ◽

Van Der Pol ◽

Flow Equations ◽

Principal Parametric Resonance ◽

Van Der Pol Oscillators ◽

And Control

The principal parametric resonance of two van der Pol oscillators under coupled position and velocity feedback control with time delay is investigated analytically and numerically on the assumption that only one of the two oscillators is parametrically excited and the feedback control is linear. The slow-flow equations are obtained by the averaging method and simplified by truncating the first term of Taylor expansions for those terms with time delay. It is found that nontrivial solutions corresponding to periodic motions exist only for one oscillator if no feedback control is applied although the two oscillators are nonlinearly coupled. Based on Levenberg-Marquardt method, the effects of excitation and control parameters on the amplitude of periodic solutions of the system are graphically given. It can be seen that both of the two oscillators can be excited in periodic vibration with proper feedback. However, the amplitudes of the periodic vibrations are independent of the sign of feedback gains. In addition, the influence of time delay on the response of the system is periodic. In terms of numerical simulations, it is shown that both of the two oscillators can also have quasi-periodic motions, periodic motions about a new equilibrium position and other complex motions such as relaxation oscillation when feedback control is considered.

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