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.

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

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.

On Multiple Hopf Bifurcations of Airflow Excited Vibration of a Translating String

2007 ◽  
Author(s):  
Yuefang Wang ◽  
Lefeng Lu¨ ◽  
Yingxi Liu
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Equilibrium Configuration ◽  
Transverse Motion ◽  
Hopf Bifurcations ◽  
Flip Bifurcation ◽  
Excited Vibration ◽  
Incremental Harmonic Balance ◽  
The Stability ◽  
Bifurcation Behavior

This paper presents the stability and bifurcation of transverse motion of translating strings excited by a steady wind flowfield. The stability of the equilibrium configuration is presented for loss of stability and generation of limit cycles via the Hopf bifurcation. It is demonstrated that there are single, double and quadruple Hopf bifurcations in the parametric space that lead to the limit cycle motion. The method of Incremental Harmonic Balance is used to solve the limit cycle response of which the stability is determined by computation of the Floquet multipliers. For the forced vibration, it is pointed out that the periodic and quasi-periodic motions exist as parameters are changed. The quench frequency and the Neimark-Sacker (NS) bifurcation and flip bifurcation are obtained. The continuity software MATCONT is adopted and the Resonance 1:1, 1:3 and 1:4 as well as NS to NS bifurcations are presented. The bifurcation behavior reveals the complexity of the string’s motion response induce by aerodynamic excitations.

On the limit cycle bifurcation of a polynomial system from a global center

2014 ◽  
Vol 12 (03) ◽  
pp. 251-268 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Polynomial System ◽  
Melnikov Function ◽  
Quadratic System ◽  
Hopf Bifurcations ◽  
First Order ◽  
Global Center ◽  
Limit Cycle Bifurcation

In this paper, we consider a quadratic system with a global center under polynomial perturbations of degree n(n ≥ 1). By using the first-order Melnikov function, we study Poincaré and Poincaré–Andronov–Hopf bifurcations. We prove that both Poincaré and Hopf cyclicity are n for n ≥ 2 up to the first order in ε.

scholarly journals Limit Cycle Bifurcations Near a Cuspidal Loop

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1425
Author(s):  
Pan Liu ◽  
Maoan Han
Keyword(s):  
Hamiltonian System ◽  
Limit Cycle ◽  
Limit Cycles ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation

In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center.

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