scholarly journals On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

2013 ◽  
Vol 14 (5) ◽  
pp. 2002-2012 ◽  
Author(s):  
Jaume Llibre ◽  
Manuel Ordóñez ◽  
Enrique Ponce
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Piecewise Linear ◽  
Piecewise Linear Systems ◽  
Uniqueness Of Limit Cycles

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 .

scholarly journals Saddle-node of limit cycles in planar piecewise linear systems and applications

2019 ◽  
Vol 39 (9) ◽  
pp. 5275-5299
Author(s):  
Victoriano Carmona ◽  
◽  
Soledad Fernández-García ◽  
Antonio E. Teruel ◽  
◽  
...  
Keyword(s):  
Linear Systems ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Piecewise Linear Systems ◽  
Saddle Node

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.

A BIPARAMETRIC BIFURCATION IN 3D CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH TWO ZONES: APPLICATION TO CHUA'S CIRCUIT

2007 ◽  
Vol 17 (02) ◽  
pp. 445-457 ◽  
Author(s):  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS
Keyword(s):  
Hopf Bifurcation ◽  
Linear Systems ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Continuous Systems ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Piecewise Linear Systems ◽  
Limit Cycle Bifurcation

In this paper, a possible degeneration of the focus-center-limit cycle bifurcation for piecewise smooth continuous systems is analyzed. The case of continuous piecewise linear systems with two zones is considered, and the coexistence of two limit cycles for certain values of parameters is justified. Finally, the Chua's circuit is shown to exhibit the analyzed bifurcation. The obtained bifurcation set in the parameter plane is similar to the degenerate Hopf bifurcation for differentiable systems.

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