Limit cycles and global dynamics of planar piecewise linear refracting systems of focus–focus type

2021 ◽  
Vol 58 ◽  
pp. 103228
Author(s):  
Haihua Liang ◽  
Shimin Li ◽  
Xiang Zhang
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Global Dynamics ◽  
Focus Type

Limit Cycles Induced by Threshold Nonlinearity in Planar Piecewise Linear Systems of Node-Focus or Node-Center Type

2020 ◽  
Vol 30 (11) ◽  
pp. 2050160
Author(s):  
Jiafu Wang ◽  
Su He ◽  
Lihong Huang
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Displacement Function ◽  
Differential Systems ◽  
Exact Number ◽  
Minimum Number ◽  
Focus Type ◽  
The Stability

In this paper, we investigate limit cycles induced by threshold nonlinearity of piecewise linear (PWL) differential systems, which are node-focus type or node-center type with the focus or the center being virtual or boundary. To get the number and stability of limit cycles, we adopt a new displacement function with a better configuration than usual. For a given parameter subregion, we exhibit the exact number or the minimum number of limit cycles. In particular, sufficient conditions are established ensuring that there are exactly two limit cycles. When the focus is boundary, we not only show that the maximum number is two, but also verify that the exact number is zero, one or two by varying parameter subregions. Finally, the exact number as well as the stability are obtained in different parameter regions for the PWL differential systems of node-center type.

scholarly journals Limit cycles from a monodromic infinity in planar piecewise linear systems

2021 ◽  
Vol 496 (2) ◽  
pp. 124818
Author(s):  
Emilio Freire ◽  
Enrique Ponce ◽  
Joan Torregrosa ◽  
Francisco Torres
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Piecewise Linear Systems

scholarly journals Canards in piecewise-linear systems: explosions and super-explosions

2013 ◽  
Vol 469 (2154) ◽  
pp. 20120603 ◽  
Author(s):  
Mathieu Desroches ◽  
Emilio Freire ◽  
S. John Hogan ◽  
Enrique Ponce ◽  
Phanikrishna Thota
Keyword(s):  
Phase Space ◽  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Single Parameter ◽  
Van Der Pol ◽  
Piecewise Linear Systems ◽  
Homoclinic Connections ◽  
Limiting Case

We show that a planar slow–fast piecewise-linear (PWL) system with three zones admits limit cycles that share a lot of similarity with van der Pol canards, in particular an explosive growth. Using phase-space compactification, we show that these quasi-canard cycles are strongly related to a bifurcation at infinity. Furthermore, we investigate a limiting case in which we show the existence of a continuum of canard homoclinic connections that coexist for a single-parameter value and with amplitude ranging from an order of ε to an order of 1, a phenomenon truly associated with the non-smooth character of this system and which we call super-explosion .

Global Dynamics in the Leslie–Gower Model with the Allee Effect

2018 ◽  
Vol 28 (12) ◽  
pp. 1850151 ◽  
Author(s):  
Valery A. Gaiko ◽  
Cornelis Vuik
Keyword(s):  
Singular Point ◽  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Biomedical System

We complete the global bifurcation analysis of the Leslie–Gower system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations of limit cycles, we prove that such a system can have at most two limit cycles surrounding one singular point.

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
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