Analysing chaos in fractional-order systems with the harmonic balance method

A generalized Duffing–van der Pol oscillator with nonlinear fractional order damping is introduced and investigated by the residue harmonic homotopy. The cubic displacement involved in fractional operator is used to describe the higher-order viscoelastic behavior of materials and of aerodynamic damping. The residue harmonic balance method is employed to analytically generate higher-order approximations for the steady state responses of an autonomous system. Nonlinear dynamic behaviors of the harmonically forced oscillator are further explored by the harmonic balance method along with the polynomial homotopy continuation technique. A parametric investigation is carried out to analyze the effects of fractional order of damping and the effect of the magnitude of imposed excitation on the system using amplitude-frequency curves. Jump avoidance conditions are addressed. Neimark bifurcations are captured to delineate regions of instability. The existence of even harmonics in the Fourier expansions implies symmetry-breaking bifurcation in certain combinations of system parameters. Numerical simulations are given by comparing with analytical solutions for validation purpose. We find that all Neimark bifurcation points in the response diagram always exist along a straight line.

Download Full-text

Harmonic Balance Method for Chaotic Dynamics in Fractional-Order Rössler Toroidal System

Discrete Dynamics in Nature and Society ◽

10.1155/2013/593856 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6

Author(s):

Huijian Zhu

Keyword(s):

Theoretical Analysis ◽

Numerical Simulations ◽

Fractional Order ◽

Harmonic Balance ◽

Chaotic Dynamics ◽

Chaotic Behavior ◽

Harmonic Balance Method ◽

Balance Method ◽

Chaotic Motions ◽

Behavior Based

This paper deals with the problem of determining the conditions under which fractional order Rössler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations.

Download Full-text

Oscillations in an x(2n+2)/(2n+1) Potential via He’s Frequency-Amplitude Formulation

Zeitschrift für Naturforschung A ◽

10.1515/zna-2009-1217 ◽

2009 ◽

Vol 64 (12) ◽

pp. 877-878 ◽

Cited By ~ 2

Author(s):

Abd Elhalim Ebaid

Keyword(s):

Periodic Solution ◽

Fractional Order ◽

Harmonic Balance ◽

Nonlinear Oscillator ◽

Harmonic Balance Method ◽

Solution Procedure ◽

Present Approach ◽

Balance Method

A recent technique, known as He’s frequency-amplitude formulation approach, is proposed in this letter to obtain an analytical approximate periodic solution to a nonlinear oscillator equation with potential of arbitrary fractional order. The solution procedure of the present approach is very simple and more convenient in comparison with the harmonic balance method

Download Full-text