scholarly journals Normal form and limit cycle bifurcation of piecewise smooth differential systems with a center

2016 ◽  
Vol 261 (2) ◽  
pp. 1399-1428 ◽  
Author(s):  
Lijun Wei ◽  
Xiang Zhang
Keyword(s):  
Normal Form ◽  
Limit Cycle ◽  
Piecewise Smooth ◽  
Differential Systems ◽  
Limit Cycle Bifurcation

ON THE SLIDING BIFURCATION OF A CLASS OF PLANAR FILIPPOV SYSTEMS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350040 ◽  
Author(s):  
DINGHENG PI ◽  
JIANG YU ◽  
XIANG ZHANG
Keyword(s):  
Limit Cycles ◽  
Homoclinic Bifurcation ◽  
Qualitative Theory ◽  
Piecewise Smooth ◽  
The Other ◽  
Heteroclinic Cycle ◽  
Differential Systems ◽  
Sliding Bifurcation ◽  
Bifurcation Phenomena ◽  
Limit Cycle Bifurcation

In this paper, we study the sliding bifurcation phenomena of a class of planar piecewise smooth differential systems consisting of linear and quadratic subsystems. Using the differential inclusion and the qualitative theory of ordinary differential equations, we find some new interesting phenomena appearing in the piecewise smooth differential systems. In brief, we prove that the system may have sliding homoclinic bifurcation, sliding cycle bifurcation, semistable limit cycle bifurcation and heteroclinic cycle bifurcation. In addition, the mentioned systems can have at most two limit cycles, and the maximal number of limit cycles can be realized and central nested with one bifurcated from the sliding–crossing bifurcation of a sliding cycle and the other from the saddle homoclinic bifurcation. These two limit cycles collide and then both disappear. This novel scenario is verified by our systems.

Stability and Limit Cycle Bifurcation for Two Kinds of Generalized Double Homoclinic Loops in Planar Piecewise Smooth Systems

2014 ◽  
Vol 24 (12) ◽  
pp. 1450153
Author(s):  
Feng Liang ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Upper Bound ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Homoclinic Loop ◽  
Limit Cycle Bifurcation

In this paper, we present two kinds of generalized double homoclinic loops in planar piecewise smooth systems. For their stability a criterion is provided. Under nondegenerate conditions, we prove that for each case there are at most five limit cycles which can be bifurcated from the generalized double homoclinic loop. Especially, we construct two concrete systems to show that the upper bound can be achieved in both cases.

scholarly journals Asymptotic Expansion of the Heteroclinic Bifurcation for the Planar Normal Form of the 1:2 Resonance

2016 ◽  
Vol 26 (01) ◽  
pp. 1650017 ◽  
Author(s):  
Luci A. F. Roberto ◽  
Paulo R. da Silva ◽  
Joan Torregrosa
Keyword(s):  
Asymptotic Expansion ◽  
Normal Form ◽  
Phase Portrait ◽  
Limit Cycle ◽  
Heteroclinic Bifurcation ◽  
Differential Systems ◽  
The Family ◽  
Heteroclinic Connection ◽  
Parameter Values

We consider the family of planar differential systems depending on two real parameters [Formula: see text] This system corresponds to the normal form for the 1:2 resonance which exhibits a heteroclinic connection. The phase portrait of the system has a limit cycle which disappears in the heteroclinic connection for the parameter values on the curve [Formula: see text] [Formula: see text] We significantly improve the knowledge of this curve in a neighborhood of the origin.

scholarly journals Limit Cycles of a Class of Piecewise Smooth Liénard Systems

2016 ◽  
Vol 26 (01) ◽  
pp. 1650009 ◽  
Author(s):  
Lijuan Sheng
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Function Theory ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Piecewise Polynomial ◽  
Multiple Parameters ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation

In this paper, we study the problem of limit cycle bifurcation in two piecewise polynomial systems of Liénard type with multiple parameters. Based on the developed Melnikov function theory, we obtain the maximum number of limit cycles of these two systems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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.

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