On the Boundaries of Basins of Attraction in Piecewise Smooth Systems

2010 ◽  
Author(s):  
Alessandro Colombo ◽  
Ugo Galvanetto
Keyword(s):  
Basins Of Attraction ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems

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.

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.

Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650030 ◽  
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Homoclinic Solution ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Stable And Unstable Manifolds ◽  
Straight Line ◽  
Unstable Manifolds ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

scholarly journals Grazing and Border-Collision in Piecewise-Smooth Systems: A Unified Analytical Framework

2001 ◽  
Vol 86 (12) ◽  
pp. 2553-2556 ◽  
Author(s):  
M. di Bernardo ◽  
C. J. Budd ◽  
A. R. Champneys
Keyword(s):  
Analytical Framework ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems
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