scholarly journals Maximum number of limit cycles for certain piecewise linear dynamical systems

2015 ◽  
Vol 82 (3) ◽  
pp. 1159-1175 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Linear Dynamical Systems

Poincaré Maps of “ <”-Shape Planar Piecewise Linear Dynamical Systems with a Saddle

2019 ◽  
Vol 29 (12) ◽  
pp. 1950165
Author(s):  
Qianqian Zhao ◽  
Jiang Yu
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Normal Form ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Inflection Point ◽  
Piecewise Linear ◽  
Poincaré Maps ◽  
Linear Dynamical Systems ◽  
Poincare Maps

It is important in the study of limit cycles to investigate the properties of Poincaré maps of discontinuous dynamical systems. In this paper, we focus on a class of planar piecewise linear dynamical systems with “[Formula: see text]”-shape regions and prove that the Poincaré map of a subsystem with a saddle has at most one inflection point which can be reached. Furthermore, we show that one class of such systems with a saddle-center has at least three limit cycles; a class of such systems with saddle and center in the normal form has at most one limit cycle which can be reached; and a class of such systems with saddle and center at the origin has at most three limit cycles with a lower bound of two. We try to reveal the reasons for the increase of the number of limit cycles when the discontinuity happens to a system.

BIMODAL PIECEWISE LINEAR DYNAMICAL SYSTEMS: REDUCED FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2795-2808 ◽  
Author(s):  
JOSEP FERRER ◽  
M. DOLORS MAGRET ◽  
MARTA PEÑA
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Equivalence Classes ◽  
State Variables ◽  
Linear Dynamical Systems ◽  
Piecewise Linear Systems ◽  
Wide Range ◽  
General Systems ◽  
Reduced Forms

Piecewise linear systems constitute a class of nonlinear systems which have recently attracted the interest of researchers because of their interesting properties and the wide range of applications from which they arise. Different authors have used reduced forms when studying these systems, mostly in the case where they are observable. In this work, we focus on bimodal continuous dynamical systems (those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane) depending on two or three state variables, which are the most common piecewise linear systems found in practice. Reduced forms are obtained for general systems, not necessarily observable. As an application, we calculate the dimension of the equivalence classes.

Limit Cycles in Discontinuous Planar Piecewise Linear Systems Separated by a Nonregular Line of Center–Center Type

2021 ◽  
Vol 31 (09) ◽  
pp. 2150136
Author(s):  
Qianqian Zhao ◽  
Cheng Wang ◽  
Jiang Yu
Keyword(s):  
Dynamical Systems ◽  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
The Other ◽  
Natural Phenomena ◽  
Separation Line ◽  
Piecewise Linear Systems

Many natural phenomena can be modeled as discontinuous dynamical systems separated by a nonregular line. The number and distribution of limit cycles in discontinuous linear systems are important topics for research. In this paper, we focus on the limit cycles created by discontinuous planar piecewise linear systems separated by a nonregular line of center–center type, and prove that such systems have at most two limit cycles, which can be reached. Furthermore, the two limit cycles are nested and intersect the separation line at two points or four points, that is, either both intersect the separation line at two points or one intersects the separation line at two points and the other one at four points.

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