Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

2020 ◽  
Vol 54 ◽  
pp. 103112 ◽  
Author(s):  
E. Freire ◽  
E. Ponce ◽  
J. Ros ◽  
E. Vela ◽  
A. Amador
Keyword(s):  
Hopf Bifurcation ◽  
Linear Systems ◽  
Piecewise Linear ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
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 .

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.

BIFURCATION SETS OF SYMMETRICAL CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH THREE ZONES

2002 ◽  
Vol 12 (08) ◽  
pp. 1675-1702 ◽  
Author(s):  
EMILIO FREIRE ◽  
ENRIQUE PONCE ◽  
FRANCISCO RODRIGO ◽  
FRANCISCO TORRES
Keyword(s):  
Linear Systems ◽  
Canonical Form ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Oscillatory Behavior ◽  
Van Der Pol ◽  
Linear Vector ◽  
Piecewise Linear Systems

Planar symmetrical continuous piecewise linear vector fields with three zones are thoroughly analyzed. Special emphasis is placed on their oscillatory behavior. The study is made by using a Van der Pol canonical form which captures the most interesting dynamics and minimizes the number of parameters to be dealt with. This work is a continuation of a previous paper [Freire et al., 1998] and uses the same approach and techniques.

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

Nondegenerate Piecewise Linear Systems: A Finite Newton Algorithm and Applications in Machine Learning

2012 ◽  
Vol 24 (4) ◽  
pp. 1047-1084 ◽  
Author(s):  
Xiao-Tong Yuan ◽  
Shuicheng Yan
Keyword(s):  
Linear Systems ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Optimization Methods ◽  
Coefficient Matrix ◽  
Learning Problems ◽  
Support Vector ◽  
Data Sets ◽  
Piecewise Linear Systems ◽  
Vector Machines

We investigate Newton-type optimization methods for solving piecewise linear systems (PLSs) with nondegenerate coefficient matrix. Such systems arise, for example, from the numerical solution of linear complementarity problem, which is useful to model several learning and optimization problems. In this letter, we propose an effective damped Newton method, PLS-DN, to find the exact (up to machine precision) solution of nondegenerate PLSs. PLS-DN exhibits provable semiiterative property, that is, the algorithm converges globally to the exact solution in a finite number of iterations. The rate of convergence is shown to be at least linear before termination. We emphasize the applications of our method in modeling, from a novel perspective of PLSs, some statistical learning problems such as box-constrained least squares, elitist Lasso (Kowalski & Torreesani, 2008 ), and support vector machines (Cortes & Vapnik, 1995 ). Numerical results on synthetic and benchmark data sets are presented to demonstrate the effectiveness and efficiency of PLS-DN on these problems.

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