Dynamics of the fractional-order Van der Pol oscillator

Abstract

-In this paper we propose a modified version of the classical unforced Van der Pol oscillator that occurs when introducing a fractional-order time derivative in the state space equations that describes its dynamics. The resulting fractional-order Van der Pol oscillator is analyzed in the time and frequency domains, for several values of order's fractional derivative and, consequently, of the total system order. It is shown that the system can exhibit different output behavior depending on the total system order. Several numerical simulations and performance indices illustrate the fractional dynamics.

Analysis of a fractional order Van der Pol-like oscillator via describing function method

“…Analysis of an extended version of the Van der Pol oscillator which contains a fractional power of y or its derivative has been done in [22,23]. Also, numerical and theoretical analyses of a modified version of Van der Pol oscillator containing derivatives of fractional order have been respectively given in [13][14][15]. The approach employed in [15] to analyze this system is based on stability analysis of incommensurate fractional order systems.…”

“…In recent years, the study of oscillatory behaviors in fractional order dynamical systems has been a subject of increasing attention [11][12][13][14][15][16][17][18]. These studies provide powerful tools to deepen our knowledge about complex behaviors of fractional order dynamical systems [15].…”

“…Nowadays, it has been known that some fractional order systems can generate regular or irregular (chaotic) oscillations. For example, the fractional order Brusselator system [12], fractional order Van der Pol system [13][14][15], fractional order BonhoefferVan der Pol system [16], fractional order Chen system [17,18], and fractional order Lu system [17,18] are some of these systems which have been studied in the literature in the sense of their abilities in generating regular or irregular (chaotic) oscillations. Thus far, different variants of the Van der Pol oscillator have been introduced, and their properties have been studied in the literature [13-15, 22, 23].…”

“…The growing interest in the application of fractional order differential equations necessitates more attention to and careful study on phenomena that might be observed in the systems represented by these equations. To this end, in recent years fractional order systems have been studied from different aspects of their applications and characteristics, including stability analysis [2,3], system identification [4,5], system approximation [6], control [7,8], synchronization [9,10], dynamical behavior analysis [11][12][13][14][15][16][17][18], and so on. The subject of this paper is related to dynamical behavior analysis of a fractional order system.…”

Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation

“…Ahmad et al [17] described the stabilization of three different types of chaotic fractional order systems, Cao et al [18] presented an optimization method based on genetic algorithms for the design of a P I λ D δ controller, and fractional adaptive control is described by Ladaci and Charef [19]. Fractional order derivatives have also been applied to the analysis of the van der Pol equation [20,21]. Finally, several schemes for the solution of multi-order fractional differential equations have been proposed [22,23].…”