Research paper
Control of chaotic behavior in the dynamics of generalized Bonhoeffer-van der Pol system: Effect of asymmetric parameter

https://doi.org/10.1016/j.cnsns.2021.106017Get rights and content

Highlights

  • The Dynamics of both forced GBVdP and asymmetric DVdP systems is investigated.

  • Pulses describing separatrix are derived and depends on the asymmetric parameter.

  • The asymmetric parameter increases the domain range of possible dynamics chaos.

  • The basic properties of unforced system influence the irregular dynamics.

  • The asymmetric parameter can be used to control this irregular dynamics.

Abstract

The effect of the asymmetric parameter in the dynamics, both of the forced Duffing van der Pol (DVdP) system and of the forced generalized Bonhoeffer-van der Pol (BVdP) system is investigated, namely the possibility for these systems to execute dynamics chaos even for large values of this asymmetric parameter. We have shown that the dynamics of the associated non dissipative and unforced system can be interpreted by means of an effective energy potential which may exhibit a two-hump or a two-well configuration according to the magnitude of the parameters of the system. In the two-well configuration, a pair of pulses of different amplitudes describing homoclinic orbits useful in the prediction of irregular dynamics for the associated forced system are analytically derived as well as the expression of the single pulse describing the single homoclinic orbit in the case of two-hump configuration and which leads to a kink-antikink pair (heteroclinic orbits) in the absence of the asymmetric parameter. By means of the Melnikov theory, the conditions for the existence of the transverse intersection of stable and unstable orbits or dynamics chaos are also derived. In the particular case for which the system can exhibit a pair of homoclinic orbits, the presence of the asymmetric parameter induces two types of domains among which a more favorable domain for the existence of chaotic behavior. In both cases, the asymmetric parameter reduces the domain range in which the system can execute regular dynamics and consequently increases its domain of a possible dynamics chaos. The accuracy of these analytical results is checked through the bifurcation diagrams and the corresponding Lyapunov exponents resulting from numerical simulations. It appears that the background properties of the unforced system, namely its capacity to exhibit autonomous oscillations and the stability of its equilibrium points which are closely connected to the magnitude of the asymmetric parameter, influence the accuracy of the analytical predictions.

Introduction

Despite the density of works devoted to the nonlinear systems operating as a nonlinear relaxation oscillator, these systems continue to benefit the attraction of many researchers due both to their great potential application and their capacity to exhibit new phenomena. Among these nonlinear oscillators, the Duffing-van der Pol (DVdP) oscillator is one of the most popular and exhibits the ability to describe several phenomena in various fields of sciences namely in physics, biology, chemistry or in engineering sciences. In physics, this system can be used to model the dynamics of self-excited bridges [1], the dynamics of sliding behavior of a body [2], or to describe the dynamics of a charged particle in the plasma medium [3], [4], to name just a few. In chemistry, the DVdP system can be used to describe the kinetic dynamics of a chemical reaction  [5], [6]. In biology, this system in certain condition is used to model and simulate the normal cardiac pulse  [7]. The generalized Bonhoeffer-van der Pol (BVdP) system which is an equivalent form of the DVdP system have not escape to these applications. Like the ordinary BVdP system, the generalized BVdP were satisfactorily used in medicine to reproduce the physiological states of nerve fiber namely, the resting state, active, refractory, enhanced and depressed states [8]. Several dynamics survey have been done on the VdPD system as well as on the BVdP system, and the results of these studies revel that these systems are able to exhibit the resonance phenomena, the frequency doubling, the mixed mode oscillations and the chaotic oscillation.

The basic DVdP system and as well as its equivalent form the BVdP system have been also enriched by the introduction of a constant term denoted as the asymmetric term. This concept of asymmetry of the system refers to the fact that the potential energy of the system is defined differently in two symmetric domains. This is, for example, the case of the asymmetry Duffing-van der Pol (ADVdP) oscillator and the equivalent form of the BVdP system with a constant impulse in which the constant impulse induces an asymmetry in the phase plane plot of the system. In many cases, the asymmetric parameter is applied to the system as an external action and can then be used to control its dynamics. This is the case of the electrical circuit executing the dynamics of the DVdP in which the asymmetric parameter corresponds to the applied dc voltage [9]. It is also the case of model describing the kinetic dynamics of a chemical reaction in which the asymmetric parameter corresponds to the feedback constraint of the autocatalytic step  [5], [6], [10]. Similarly, in the BVdP oscillator which describes the propagation of an electrical impulse or voltage pulse along the membrane of nerve cell, the asymmetric term corresponds to the bias of the membrane current [11], [12], [13]. Some works oriented in the control of the chaotic dynamics of nonlinear systems have introduced the asymmetric term as a control parameter in the classical Duffing model [14]. An asymmetric coupling has also been introduced between two identical Duffing oscillators in order to suppress their chaotic dynamics [15]. The notion of asymmetry relating to a piecewise nonlinearity has also been mentioned in the case of Duffing-van der Pol oscillator with three different asymmetric potentials driven by narrow-band frequency modulated force, and the effect of the asymmetry parameter in the appearence of the horseshoe chaos bring out [16]. Let us note that the generation of chaotic motion of periodically driven particle in an asymmetric potential well [17], asymmetric oscillation in a planar pendulum [18], characteristics of stochastic resonance in asymmetric Duffing oscillator [19] and the occurrence of symmetric breaking of fluctuation dynamics by noise in asymmetrical bistable oscillators [20] were also reported. Up to now, excepted few studies, the works conducted in these systems have considered the asymmetric term as a small quantity, that is, as a perturbation in the dynamics of the basic system leading to this fact to the undervaluation of its influence on the properties of the system. For example, in the regime of irregular dynamics, the analytical results allowing to predict the appearance of dynamics chaos is always independent on the asymmetric parameter [11]. On the other hand, despite the density of works done in the BVdP system, it appears nevertheless that the equation modelling its dynamics is commonly used in its original form which is described with three constant coefficients, limiting the extent of the physical phenomena that it could present. Recently we have shown that the extension of the basic model, well-known as the model equations with three coefficients, to that of six independent constant coefficients will change qualitatively and quantitatively the behavior of this system [9]. In fact it has been also shown that although the classical VdP equation presents similar qualitative features to those of the heart actuation potential, it does not allow the possibility of change the values of spontaneous depolarization time and refraction time independently. In this light, Grudzinski and Zebrowski have introduced a modified VdP equation which take into account these two aspects in order to model and simulate significant physiological features of the action potentials [21]. The equivalent BVdP form of this equation can be written by means of six independent coefficients. Similarly, it has been also demonstrated that the nonlinear kinetics dynamics of the chemical reaction can be modeled rather with four independent coefficients, but not with the classical three parameters of the BVdP’s system. One of the main question is that what can be the influence of this asymmetric parameter on the appearance of irregular dynamics of this new system and can it be used to control its appearance? Let us note that the notion of “control” hereafter will consist to drive the system, by judicious choice of the magnitude of the asymmetric parameter, toward a particular working regime: either the regular regime or the regime of dynamics chaos which is under consideration. Similarly, the asymmetric parameter will also be used to choose the more favorable domain for the manifestation of this chaotic phenomena according to the magnitude of the value of the system’s parameters.

In fact, Recently, we have considered a generalized form of the autonomous Bonhoeffer-van der Pol (BVdP) system consisting into a second-order dynamical system with six independent parameters instead of three usually used [9]. The analysis shown that the system may exhibit one or three steady states when it is driven by an external constant impulse taken as a main control parameter. Domain ranges in which the system can function as well as monostable system as a bistable system were derived. In addition, by means of the theory of Hopf Bifurcation, there are large possibilities for the system to work as self-sustained oscillator, forced oscillator or other possibilities for which the system does not operate, indicating the richness of this generalized form of the BVdP system. Applying these analytical investigations to the electronic circuit modelling the dynamics of the basic BVdP system, two distinct working regimes were highlighted, depending on the magnitude of the capacitor with respect to a threshold value function of the characteristic parameters both of the self and of the nonlinear resistor. At this stage of research, another question which can be asked is: does the basic properties of the unforced system may influence the irregular dynamics of the forced system? The answer of these two question are the main objectives of this work.

The paper is organized as follows: In Section 2, we describe the model of the generalized BVdP (GBVdP) system under consideration which is built from the electrical circuit executing the dynamics of the VdPD system and that of the BVdP system with three parameters. Similarly, the relations allowing to pass from the generalized asymmetric DVdP (GAVdPD) system to the generalized BVdP system are established and the effect of the asymmetric parameter on its equilibrum points revisited. In Section 3, the Energetic approach is developed under the influence of the asymmetric parameter as well as the resulting localized nonlinear excitation, describing either the homoclinic orbit or the heteroclinic orbit, in the phase plane plot. In Section 4, by means of the Melnikov theory, the condition for the existence of the transverse intersection of stable and unstable orbits are investigated and the effect of asymmetric parameter examined. In Section 5, based on the results from the forced ADVdP system, the condition for the appearance of chaotic behavior in the forced GBVdP system are also derived and the influence of the constant impulse studied. Finally, Section 6 is devoted to concluding remarks.

Section snippets

GFBVdP oscillator and the corresponding ADVP equation

To start let us consider the electrical circuit of Fig. 1(a) usually used as the circuit diagram of the basic forced BVdP [22], [23], [24], [25]. This circuit is constituted of three branches connected in parallel; the first branch contains a self of inductance L in serie with a resistor R and a sinusoidal current voltage source e(t) associated with a direct current (dc) voltage source E; the second branch is a single capacitor C while the third branch is the nonlinear element (NLR). The

Preliminaries: Energetic approach

The forced GBVdP system and its corresponding forced ADVdP equation exhibit different type of solutions namely the small amplitude periodic solution, the limit cycle solution and the large amplitude solution. By making use of the appropriate change of variables, y˜=β˜y, t˜=ω0t with β˜2=γ/ω02, Eq. (8) is rewritten in the following more suggestive formd2y˜dt˜2+ϵ1y˜+ϵ2y˜3δ˜=μ˜(1σ˜y˜2)dy˜dt˜+e˜msinΩ˜t˜in whichμ˜=μ/ω0,σ˜=(ω02/γ)σ,δ˜=(γ/ω03)δ,e˜m=(γ/ω03)em,andΩ˜=Ω/ω0.With this change of variables,

Basic formulations

Let us consider the mathematical model of the ADVdP system defined by Eq(15) in the formdXdt˜=f(X)+ϵg(X,t˜),where X=[x1,x2]T. f=[f1,f2]T and g=[g1,g2]T are vectors, respectively, defined byf1=x2,f2=ϵ1x1ϵ2x13+δ˜,g1=0,andg2=μ˜(1σ˜x12)x2+em˜sinΩ˜t˜and where ϵ1 is a measure of the smallness of g(X,t˜) compared to f(X,t˜). Accordingly, g(X,t˜) can be treated as perturbation of the dynamics of the system described by Eq. (39). It is important to note that, in general, the asymmetric parameter is

Large orbits of the system

Recent studies on the dynamics of the GBVdP system has pointed out that this system may exhibit one or three equilibria according to the relative magnitude of its parameters with regard to the value of constant impulse c1 which acts here as the asymmetric one. Thus for af(aebd)>0 with any value of parameter c1 or af(aebd)<0 with |c1|>c1cr1, the system exhibits only a single equilibrium point which may be structurally stable if af>0 and unstable if af<0. However for af(aebd)<0 with |c1|c1cr1

Conclusion

In this paper, we have studied the influence of the asymmetric parameter or the constant impulse respectively in the behavior of the generalized asymmetric Duffing-van der Pol (ADVdP) system and that of the generalized forced Bonhoeffer-van der Pol (BVdP) system, namely the possibility for these systems to execute chaotic dynamics even for large value of this asymmetric parameter. We have considered the generalized form of these unforced systems and have shown that the dynamics of the system

Credit Author Statement

For the manuscript entitled ǣControl of chaotic behavior in the dynamics of generalized Bonhoeffer-van der pol system: Effect of asymmetric parameterǥ by Armel Viquit SONNA and David. This original work is ours and has not yet been published somewhere

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could appeared to influence the work reported in this paper.

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