Systematic designing of bi-rhythmic and tri-rhythmic models in families of Van der Pol and Rayleigh oscillators

Abstract

Van der Pol and Rayleigh oscillators are two traditional paradigms of nonlinear dynamics. They can be subsumed into a general form of Liénard-Levinson-Smith(LLS) system. Based on a recipe for finding out maximum number of limit cycles possible for a class of LLS oscillator, we propose here a scheme for systematic designing of generalised Rayleigh and Van der Pol families of oscillators with a desired number of multiple limit cycles. Numerical simulations are explicitly carried out for systematic search of the parameter space for birhythmic and tri-rhythmic systems and their higher order variants.

Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

“…Tang et al [8] studied the vibration response and its generation mechanism in the van der Pol-Rayleigh system under slowvarying periodic excitation, and analyzed the excitation hysteresis behavior and its generation mechanism of the system. Saha et al [9] classified the van der Pol oscillator and Rayleigh oscillator into the general form of the Liénard-Levinson-Smith (LLS) system, and thus design a generalized van der Pol and Rayleigh oscillator family system with multiple limit cycles. Hasegawa [10] studied the Jarzynski equality in van der Pol and Rayleigh oscillators to which a ramp force with a duration τ is applied.…”

Where the Liénard--Levinson--Smith (LLS) theorem cannot be applied for a generalised Liénard system

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We have examined a class of Liénard-Levinson-Smith (LLS) system having a stable limit cycle which demonstrates the case where the LLS theorem cannot be applied. The problem has been partly raised in a recent communication by Saha et al [1] (last para of sec 4.2.2). Here we have provided a physical approach to address this problem using the concept of energy consumption per cycle.

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“…As a general ground of vibration beyond the sounds of stretched strings, bars, membranes and plates such subjects as ocean tides, not to speak of optics, and literally extended to any cyclic events where such novelty of treatment and results are followed with detailed consideration [3][4][5]. In open systems a limit cycle plays an important role as a feedback loop in dynamics in various kind of physical, chemical and biological processes, such as van der Pol oscillator [1,[3][4][5][6][7], Glycolytic oscillator (Selkov model) [6,[8][9][10][11][12][13], Belousov-Zhabotinsky reaction [13], Brusselator model for oscillatory chemical reactions [6,13,14] and Circadian oscillator [8,13,[15][16][17] are some of the major examples. The variants of van der Pol oscillator for physical circuits whereas circadian oscillator for biological rhythms [8,[15][16][17], are prototypical testing grounds for isolated closed trajectories where the origin of such competition between instability and damping can be investigated.…”

“…Liénard-Levinson-Smith (LLS) [5,6,14,18,21,22], proposed a more general form with the damping coefficient function depending on position as well as momentum of particle of the form [1],…”

“…This steady state determines the amplitude of the limit cycle (cf. [1,29,30]) and the stability of the limit cycle will defined the sign of the d ṙ dr | r=rss (cf. [1,29,30]).…”