ON SEMI-ANALYTICAL PROCEDURE FOR DETECTING LIMIT CYCLE BIFURCATIONS

2004 ◽  
Vol 14 (03) ◽  
pp. 951-970 ◽  
Author(s):  
FEDERICO I. ROBBIO ◽  
DIEGO M. ALONSO ◽  
JORGE L. MOIOLA
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Analytical Approximations ◽  
Approximation Formulas ◽  
Accurate Procedure ◽  
Nonlinear Flows

This paper reports some computation of periodic solutions arising from Hopf bifurcations in order to build up a more accurate procedure for semi-analytical approximations to detect limit cycle bifurcations. The approximation formulas are derived using nonlinear feedback systems theory and the harmonic balance method. The monodromy matrix is computed for several simple nonlinear flows to detect the first bifurcation of the cycles in the neighborhood of the original Hopf bifurcation.

DETECTION OF LIMIT CYCLE BIFURCATIONS USING HARMONIC BALANCE METHODS

2004 ◽  
Vol 14 (10) ◽  
pp. 3647-3654 ◽  
Author(s):  
FEDERICO I. ROBBIO ◽  
DIEGO M. ALONSO ◽  
JORGE L. MOIOLA
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Vector Fields ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Zero Eigenvalue ◽  
Complex Singularities

In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can be promising in analyzing limit cycle bifurcations arising in the unfoldings of other complex singularities.

scholarly journals Harmonic Balance Solution Of Coupled Nonlinear Non-Conservative Differential Equation

2013 ◽  
Vol 32 ◽  
pp. 1-14
Author(s):  
M Saifur Rahman ◽  
M Majedur Rahman ◽  
M Sajedur Rahaman ◽  
M Shamsul Alam
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Limit Cycle ◽  
Nonlinear Differential Equation ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Balance Solution ◽  
Good Agreement

A modified harmonic balance method is employed to determine the second approximate solutions to a coupled nonlinear differential equation near the limit cycle. The solution shows a good agreement with the numerical solution. DOI: http://dx.doi.org/10.3329/ganit.v32i0.13640 GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 32 (2012) 1 – 14

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.

scholarly journals APPROXIMATE ANALYTICAL SOLUTIONS FOR THE RELATIVISTIC OSCILLATOR USING A LINEARIZED HARMONIC BALANCE METHOD

2009 ◽  
Vol 23 (04) ◽  
pp. 521-536 ◽  
Author(s):  
A. BELÉNDEZ ◽  
D. I. MÉNDEZ ◽  
M. L. ALVAREZ ◽  
C. PASCUAL ◽  
T. BELÉNDEZ
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Oscillation Period ◽  
Analytical Approximations ◽  
Method Of Harmonic Balance ◽  
Approximate Analytical Solutions ◽  
Approximate Frequency ◽  
Relativistic Oscillator ◽  
Odd Nonlinearity

The analytical approximate technique developed by Wu et al. for conservative oscillators with odd nonlinearity is used to construct approximate frequency-amplitude relations and periodic solutions to the relativistic oscillator. By combining Newton's method with the method of harmonic balance, analytical approximations to the oscillation period and periodic solutions are constructed for this oscillator. The approximate periods obtained are valid for the complete range of oscillation amplitudes, A, and the discrepancy between the second approximate period and the exact one never exceeds 1.24%, and it tends to 1.09% when A tends to infinity. Excellent agreement of the approximate periods and periodic solutions with the exact ones are demonstrated and discussed.

Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

On Bifurcation Control in Time Delay Feedback Systems

1998 ◽  
Vol 08 (04) ◽  
pp. 713-721 ◽  
Author(s):  
M. Basso ◽  
A. Evangelisti ◽  
R. Genesio ◽  
A. Tesi
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Bifurcation Control ◽  
Flip Bifurcation ◽  
Delay Model ◽  
Feedback Systems ◽  
First Order ◽  
Balance Analysis ◽  
Feedback Control Systems ◽  
Limit Cycle Bifurcation

The paper addresses bifurcations of limit cycles for a class of feedback control systems depending on parameters. A set of simple approximate analytical conditions characterizing all the generic limit cycle bifurcations is determined via a first-order harmonic balance analysis in a suitable frequency band. Based on the results of this analysis, an approach to limit cycle bifurcation control is proposed. In particular, an example concerning a biological delay model is developed, where a flip bifurcation control is designed via a modified Pyragas technique.

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