Periodic Orbits for Planar Piecewise Smooth Systems with a Line of Discontinuity

2014 ◽  
Vol 26 (4) ◽  
pp. 1049-1078 ◽  
Author(s):  
L. Dieci ◽  
C. Elia
Keyword(s):  
Periodic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Line Of Discontinuity

Bifurcation of Periodic Orbits Crossing Switching Manifolds Multiple Times in Planar Piecewise Smooth Systems

2019 ◽  
Vol 29 (12) ◽  
pp. 1950160
Author(s):  
Zhihui Fan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Sufficient Conditions ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Smooth Curves ◽  
Switching Curve ◽  
Smooth System

In this paper, we discuss the bifurcation of periodic orbits in planar piecewise smooth systems with discontinuities on finitely many smooth curves intersecting at the origin. We assume that the unperturbed system has either a limit cycle or a periodic annulus such that the limit cycle or each periodic orbit in the periodic annulus crosses every switching curve transversally multiple times. When the unperturbed system has a limit cycle, we give the conditions for its stability and persistence. When the unperturbed system has a periodic annulus, we obtain the expression of the first order Melnikov function and establish sufficient conditions under which limit cycles can bifurcate from the annulus. As an example, we construct a concrete nonlinear planar piecewise smooth system with three zones with 11 limit cycles bifurcated from the periodic annulus.

Bifurcation Analysis of Planar Piecewise Smooth Systems with a Line of Discontinuity

2016 ◽  
Vol 26 (06) ◽  
pp. 1650104
Author(s):  
Dingheng Pi ◽  
Shihong Xu
Keyword(s):  
Hamiltonian System ◽  
Linear System ◽  
Qualitative Analysis ◽  
Bifurcation Analysis ◽  
Dynamical Behavior ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Differential Systems ◽  
Line Of Discontinuity ◽  
Smooth System

In this paper, we consider bifurcations of a class of planar piecewise smooth differential systems constituted by a general linear system and a quadratic Hamiltonian system. The linear system has four parameters. When the parameters vary in different regions, the left linear system can have a saddle, a node or a focus. For each case, we provide a completely qualitative analysis of the dynamical behavior for this piecewise smooth system. Our results generalize and improve the results in this direction.

Diverting homoclinic chaos in a class of piecewise smooth oscillators to stable periodic orbits using small parametrical perturbations

2015 ◽  
Vol 149 ◽  
pp. 1587-1595 ◽  
Author(s):  
Huaqing Li ◽  
Xiaofeng Liao ◽  
Junjian Huang ◽  
Guo Chen ◽  
Zhaoyang Dong ◽  
...  
Keyword(s):  
Periodic Orbits ◽  
Piecewise Smooth ◽  
Homoclinic Chaos ◽  
Stable Periodic

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

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