Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method

2013 ◽  
Vol 89 ◽  
pp. 1-12 ◽  
Author(s):  
Min Xiao ◽  
Wei Xing Zheng ◽  
Jinde Cao
Keyword(s):  
Fractional Order ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Approximate Expressions

NEIMARK BIFURCATIONS OF A GENERALIZED DUFFING–VAN DER POL OSCILLATOR WITH NONLINEAR FRACTIONAL ORDER DAMPING

2013 ◽  
Vol 23 (11) ◽  
pp. 1350177 ◽  
Author(s):  
A. Y. T. LEUNG ◽  
H. X. YANG ◽  
P. ZHU
Keyword(s):  
Fractional Order ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Viscoelastic Behavior ◽  
Higher Order ◽  
Balance Method ◽  
Van Der Pol Oscillator ◽  
Homotopy Continuation ◽  
Van Der Pol ◽  
Steady State Responses

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.

The Residue Harmonic Balance Method for Duffing-Van Der Pol Oscillator with Fractional Derivative

2013 ◽  
Vol 774-776 ◽  
pp. 103-106
Author(s):  
Xin Xue ◽  
Lian Zhong Li ◽  
Dan Sun
Keyword(s):  
Fractional Derivative ◽  
Harmonic Balance ◽  
Numerical Solutions ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Solution Procedure ◽  
Balance Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Orders

Duffing-van der Pol oscillator with fractional derivative was constructed in this paper. The solution procedure was proposed with the residue harmonic balance method. The effect of different fractional orders on resonance responses of the system in steady state were analyzed for an example without parameters. The approximate solutions were contrasted with numerical solutions. The results show that the residue harmonic balance method to Duffing-van der Pol differential equation with fractional derivative is very valid.

scholarly journals Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

2004 ◽  
Vol 26 (3) ◽  
pp. 157-166
Author(s):  
Nguyen Van Khang ◽  
Thai Manh Cau
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Incremental Harmonic Balance ◽  
The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

Dynamic response of Mathieu–Duffing oscillator with Caputo derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jianhua Tang ◽  
Chuntao Yin
Keyword(s):  
Dynamic Response ◽  
Caputo Derivative ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Expression ◽  
Balance Method ◽  
Composite Function ◽  
Derivatives Of ◽  
Approximate Expressions

Abstract In this paper, the harmonic balance method and its variants are used to analyze the response of Mathieu–Duffing oscillator with Caputo derivative. First, the exact and approximate expressions of the Caputo derivatives of trigonometric function and composite function are derived. Next, using the approximate expression of the Caputo derivative of the composite function, the resonance of Duffing oscillator with Caputo derivative is analyzed by the harmonic balance method. Finally, Mathieu–Duffing oscillator with Caputo derivative is approximated by three kinds of methods, i.e., the harmonic balance method, the residue harmonic balance method and the improved harmonic balance method. The corresponding numerical simulations are given to illustrate the performance of these methods as well. The results show that the residue harmonic balance method is more precise than the harmonic balance method and the improved harmonic balance method in analyzing the dynamic response of Mathieu–Duffing oscillator with Caputo derivative.

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

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.

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

2009 ◽  
Vol 64 (12) ◽  
pp. 877-878 ◽  
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

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