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.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Periodic Motions ◽  
Conditions Of Stability ◽  
Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

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.

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

scholarly journals Comparison Analysis Based on the Cubic Spline Wavelet and Daubechies Wavelet of Harmonic Balance Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jing Gao
Keyword(s):  
Theoretical Analysis ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Harmonic Balance Method ◽  
Cubic Spline ◽  
Nonlinear Circuits ◽  
Balance Method ◽  
Daubechies Wavelet ◽  
Comparison Analysis ◽  
Spline Wavelet

This paper develops a theoretical analysis of harmonic balance method, based on the cubic spline wavelet and Daubechies wavelet, for steady state analysis of nonlinear circuits under periodic excitation. The properties of the resulting Jacobian matrix for harmonic balance are analyzed. Numerical experiments illustrate the theoretical analysis.

Laser induced microwave oscillator under the influence of interference

2014 ◽  
Vol 6 (6) ◽  
pp. 581-590 ◽  
Author(s):  
Arindum Mukherjee ◽  
Somnath Chatterjee ◽  
Nikhil Ranjan Das ◽  
Baidyanath Biswas
Keyword(s):  
Theoretical Analysis ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Microwave Oscillator ◽  
Experimental Results ◽  
Balance Method ◽  
Optoelectronic Oscillator ◽  
System Equation ◽  
Novel Technique ◽  
Weak Interference

An optoelectronic oscillator (OEO) under the influence of a weak interference has been investigated. The system equation of the OEO under the influence of interference has been derived. A novel technique for calculating the lock range of the oscillator using the harmonic balance method in presence of interference has been demonstrated. Theoretical analysis coupled with experimental results has been presented.

Period-3 Motions and Bifurcation in a Softenning Duffing Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Harmonic Amplitude ◽  
Stable Period

In this paper, analytical solutions for period-3 motions in the periodically forced softening Duffing oscillator are developed by the harmonic balance method. From the Hopf bifurcation of period-3 motions, period-6 motions can be analytically determined. Numerical simulations for stable period-3 motions are completed and the corresponding harmonic amplitude spectrums are presented.

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