Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van Der Pol Oscillator

Yamapi, R.; Nbendjo, B. R. Nana; Kadji, H. G. Enjieu

doi:10.1142/s0218127407017847

Cited by 31 publications

(19 citation statements)

References 22 publications

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“…[13,14], while the possibility that introducing an active control of chaos can be tamed for an appropriate choice of the coupling parameters has been considered in Ref. [32]. Eq.…”

Section: A the Multi-limit Cycle Van Der Pol Oscillatormentioning

confidence: 99%

Global stability analysis of birhythmicity in a self-sustained oscillator

Yamapi

Филатрелла

Aziz-Alaoui

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We analyze global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit cycles variation of the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients α and β. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time τ from one limit-cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation of the escape time τ versus the inverse noise-intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters. Some models employed to describe natural systems, such as for instance glycolysis reactions and circadian proteins rhythmics, exhibit spontaneous oscillations at two distinct frequencies. The phenomenon is known as birhythmicity, and the underlying dynamical structure is characterized by the coexistence of two stable attractors, each displaying a different frequency. Being the attractors locally stable, the system would however stay at a single frequency, the one selected by the choice of the initial conditions, unless an external source disturbs the evolution and causes a switch to the other attractor. To investigate such process, we have focused on a particular system of biological interest, a modified van der Pol oscillator (that displays birhythmicity), to determine the global stability properties of the attractors under the influence of noise. More specifically, we have characterized the stability of the attractors with the escape times, or the average time that the system requires to switch from an attractor to the other under the influence of random fluctuations. Such analysis reveals that the two attractors can possess very different properties, with very different relative residence times. Even excluding the most asymmetric cases, the system can spend something like 10 years on one attractor for each second spent on the other. We conclude that although a system can be structurally biorhythmic for the contemporary presence of two locally stable attractors at two different frequencies, actual switch from one frequency to the other could be very difficult to observe. A global stability analysis can therefore help to determine the region of the parameter space in which birhythmic behavior will be genuinely observed.

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“…[13,14], while the possibility that introducing an active control of chaos can be tamed for an appropriate choice of the coupling parameters has been considered in Ref. [32]. Eq.…”

Section: A the Multi-limit Cycle Van Der Pol Oscillatormentioning

confidence: 99%

Global stability analysis of birhythmicity in a self-sustained oscillator

Yamapi

Филатрелла

Aziz-Alaoui

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Extending van der Pol oscillator model with a nonlinear function of higher order polynomial, Kaiser(15;16;36;37;38;39;17;40) has described bi-rythmicity with the nonlinear equation,…”

Section: Three-cycles Case: Kaiser Bi-rythmicity Modelmentioning

confidence: 99%

Reduction of Kinetic Equations to Liénard–Levinson–Smith Form: Counting Limit Cycles

Saha

Gangopadhyay

Ray

2019

Int. J. Appl. Comput. Math

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We have presented an unified scheme to express a class of system of equations in two variables into a Liénard -Levinson -Smith(LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Liénard systems for arbitrary polynomial functions of damping and restoring force. Krylov-Boguliubov(K-B) method is implemented to determine the maximum number of limit cycles admissible for a LLS oscillator atleast in the weak damping limit. Scheme is illustrated by a number of model systems with single cycle as well as the multiple cycle cases.

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“…From Eq. (3) one can infer that, for µ = 0 and d = 0, the harmonic solution may be given by [23] x(t) = A cos(t + φ),…”

Section: Control Of Birhythmicity Through Conjugate Self-feedbacmentioning

confidence: 99%

Control of birhythmicity through conjugate self-feedback: Theory and experiment

2016

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Birhythmicity arises in several physical, biological, and chemical systems. Although many control schemes have been proposed for various forms of multistability, only a few exist for controlling birhythmicity. In this paper we investigate the control of birhythmic oscillation by introducing a self-feedback mechanism that incorporates the variable to be controlled and its canonical conjugate. Using a detailed analytical treatment, bifurcation analysis, and experimental demonstrations, we establish that the proposed technique is capable of eliminating birhythmicity and generates monorhythmic oscillation. Further, the detailed parameter space study reveals that, apart from monorhythmicity, the system shows a transition between birhythmicity and other dynamical forms of bistability. This study may have practical applications in controlling birhythmic behavior of several systems, in particular in biochemical and mechanical processes.

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Dynamics and Active Control of Motion of a Driven Multi-Limit-Cycle Van Der Pol Oscillator

Cited by 31 publications

References 22 publications

Global stability analysis of birhythmicity in a self-sustained oscillator

Global stability analysis of birhythmicity in a self-sustained oscillator

Reduction of Kinetic Equations to Liénard–Levinson–Smith Form: Counting Limit Cycles

Control of birhythmicity through conjugate self-feedback: Theory and experiment

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