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 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 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 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>

STRANGE NONCHAOTIC ATTRACTORS IN A QUASIPERIODICALLY-FORCED PIECEWISE SMOOTH SYSTEM WITH FAREY TREE

Fractals ◽  
2019 ◽  
Vol 27 (07) ◽  
pp. 1950118 ◽  
Author(s):  
YUNZHU SHEN ◽  
YONGXIANG ZHANG
Keyword(s):  
Lyapunov Exponents ◽  
Scaling Laws ◽  
Power Spectra ◽  
Piecewise Smooth ◽  
Largest Lyapunov Exponent ◽  
Piecewise Smooth System ◽  
Finite Time Lyapunov Exponents ◽  
Farey Tree ◽  
Strange Nonchaotic Attractors ◽  
Smooth System

The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

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