Dynamical Response of a Van der Pol System with an External Harmonic Excitation and Fractional Derivative
We examined the Van der Pol system with external forcing and a memory possessing fractional damping term. Calculating the basins of attraction we showed broad spectrum of nonlinear behaviour connected with sensitivity to the initial conditions. To quantif
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Dynamical Response of a Van der Pol System with an External Harmonic Excitation and Fractional Derivative Arkadiusz Syta and Grzegorz Litak
Abstract We examined the Van der Pol system with external forcing and a memory possessing fractional damping term. Calculating the basins of attraction we showed broad spectrum of nonlinear behaviour connected with sensitivity to the initial conditions. To quantify dynamical response of the system we propose the statistical 0–1 test. The results have been confirmed by bifurcation diagrams, phase portraits and Poincare sections. Keywords Van der Pol system • Fractional derivative • 0–1 test • Chaos detection
6.1 Introduction The system with fractional damping dependent on the velocity history has focused a lot of interest and was extensively studied in the last decade [1–6]. To model complex energy dissipation with minimum number of parameters in presence of hysteresis and memory effect, the fractional order derivative in the damping term is proposed. In such systems the damping force is proportional to a fractional derivative of the displacement instead of the classical case (first order derivative of the displacement). The memory of the system was noted to be important factor in different areas [5, 6]. Van der Pol systems, describing relaxation-oscillations are characterized by a non-viscous composite damping term [7, 8] which is small value, negative for small amplitude oscillations and changes the sign to positive
A. Syta () • G. Litak Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, PL-20-618 Lublin, Poland e-mail: [email protected]; [email protected] J.A.T. Machado et al. (eds.), Discontinuity and Complexity in Nonlinear Physical Systems, Nonlinear Systems and Complexity 6, DOI 10.1007/978-3-319-01411-1__6, © Springer International Publishing Switzerland 2014
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for increasing amplitude. This system property is reflected by dynamical response of limit cycle [9]. Comparing to viscous nonlinear systems this implies type of bifurcations and transition to chaos including hopf bifurcations [10, 11]. Recently, Van der Pol systems have been studied in a series of papers [12–15]. Pinto and Machado proposed the complex order van der Pol oscillator [12] reporting the changes in the system response spectrum with varying the fractional order of derivative in the damping term. Attari et al. [13] focused on periodic solutions and studied system parameters for their stability. Suchorsky and Rand [14] investigated the synchronization by a fractional coupling of two Van der Pol systems. Finally, Chen and Chen [15] studied a fractionally damped van der Pol equation with harmonic external forcing. They focus on the effect of fractional damping influence on the dynamic quasi-periodic and chaotic responses. In particular, the transition from quasi-periodic to chaotic motion was demonstrated. In the present paper we continue the analysis of chaotic motion proposing an efficient method for chaotic solution identification by m
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