scholarly journals The Melnikov Method and Subharmonic Orbits in a Piecewise-Smooth System

2012 ◽  
Vol 11 (3) ◽  
pp. 801-830 ◽  
Author(s):  
A. Granados ◽  
S. J. Hogan ◽  
T. M. Seara
Keyword(s):  
Melnikov Method ◽  
Piecewise Smooth ◽  
Piecewise Smooth System ◽  
Smooth System ◽  
Subharmonic Orbits

Melnikov Method for a Three-Zonal Planar Hybrid Piecewise-Smooth System and Application

2016 ◽  
Vol 26 (01) ◽  
pp. 1650014
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Periodic Orbits ◽  
Melnikov Method ◽  
Perturbation Parameter ◽  
Melnikov Function ◽  
Displacement Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
First Order ◽  
Piecewise Smooth System ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three domains separated by two switching manifolds [Formula: see text] and [Formula: see text]. The dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we wish to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter [Formula: see text] near [Formula: see text]. We denote the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.

scholarly journals Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism

2018 ◽  
Vol 18 (1) ◽  
pp. 315-332
Author(s):  
J.-P. Françoise ◽  
Hongjun Ji ◽  
Dongmei Xiao ◽  
Jiang Yu
Keyword(s):  
Piecewise Smooth ◽  
Global Dynamics ◽  
Lactate Metabolism ◽  
Piecewise Smooth System ◽  
Smooth System ◽  
Brain Lactate

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.

A Lorenz-type attractor in a piecewise-smooth system: Rigorous results

2019 ◽  
Vol 29 (10) ◽  
pp. 103108 ◽  
Author(s):  
Vladimir N. Belykh ◽  
Nikita V. Barabash ◽  
Igor V. Belykh
Keyword(s):  
Piecewise Smooth ◽  
Rigorous Results ◽  
Piecewise Smooth System ◽  
Smooth System

scholarly journals Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fang Wu ◽  
Lihong Huang ◽  
Jiafu Wang
Keyword(s):  
Periodic Solutions ◽  
Displacement Function ◽  
Piecewise Smooth ◽  
Parameter Family ◽  
Piecewise Smooth System ◽  
Closed Orbits ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Smooth System ◽  
Two Parameter

<p style='text-indent:20px;'>In this paper, we consider the nonsmooth bifurcation around a class of critical crossing cycles, which are codimension-2 closed orbits composed of tangency singularities and regular orbits, for a two-parameter family of planar piecewise smooth system with two zones. By the construction of suitable displacement function (equivalently, Poincar<inline-formula><tex-math id="M1">\begin{document}$ {\rm\acute{e}} $\end{document}</tex-math></inline-formula> map), the stability and the existence of periodic solutions under the variation of the parameters inside this system are characterized. More precisely, we obtain some parameter regions on the existence of crossing cycles and sliding cycles near those loops. As applications, several examples are given to illustrate our main conclusions.</p>

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