scholarly journals Existence of homoclinic orbits and heteroclinic cycle in a class of three-dimensional piecewise linear systems with three switching manifolds

2020 ◽  
Vol 30 (12) ◽  
pp. 123143
Author(s):  
Bin Zhu ◽  
Zhouchao Wei ◽  
R. J. Escalante-González ◽  
Nikolay V. Kuznetsov
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Cycle ◽  
Piecewise Linear Systems

scholarly journals Periodic orbits and invariant cones in three-dimensional piecewise linear systems

2015 ◽  
Vol 35 (1) ◽  
pp. 59-72 ◽  
Author(s):  
Victoriano Carmona ◽  
◽  
Emilio Freire ◽  
Soledad Fernández-García ◽  
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Piecewise Linear Systems ◽  
Invariant Cones

scholarly journals Homoclinic Bifurcations in Planar Piecewise-Linear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Bin Xu ◽  
Fenghong Yang ◽  
Yun Tang ◽  
Mu Lin
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Piecewise Linear Systems ◽  
Homoclinic Bifurcations ◽  
Two Parameters ◽  
Necessary And Sufficient

The problem of homoclinic bifurcations in planar continuous piecewise-linear systems with two zones is studied. This is accomplished by investigating the existence of homoclinic orbits in the systems. The systems with homoclinic orbits can be divided into two cases: the visible saddle-focus (or saddle-center) case and the case of twofold nodes with opposite stability. Necessary and sufficient conditions for the existence of homoclinic orbits are provided for further study of homoclinic bifurcations. Two kinds of homoclinic bifurcations are discussed: one is generically related to nondegenerate homoclinic orbits; the other is the discontinuity induced homoclinic bifurcations related to the boundary. The results show that at least two parameters are needed to unfold all possible homoclinc bifurcations in the systems.

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

2005 ◽  
Vol 15 (10) ◽  
pp. 3153-3164 ◽  
Author(s):  
V. CARMONA ◽  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS ◽  
F. TORRES
Keyword(s):  
Linear Systems ◽  
Limit Cycle ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Periodic Oscillations ◽  
Bounded Linear ◽  
Piecewise Linear Systems ◽  
Chua’S Oscillator ◽  
Center Configuration ◽  
Limit Cycle Bifurcation

The generic case of three-dimensional continuous piecewise linear systems with two zones is analyzed. From a bounded linear center configuration we prove that the periodic orbit which is tangent to the separation plane becomes a limit cycle under generic conditions. Expressions for the amplitude, period and characteristic multipliers of the bifurcating limit cycle are given. The obtained results are applied to the study of the onset of asymmetric periodic oscillations in Chua's oscillator.

ON PERIODIC ORBITS OF 3D SYMMETRIC PIECEWISE LINEAR SYSTEMS WITH REAL TRIPLE EIGENVALUES

2009 ◽  
Vol 19 (07) ◽  
pp. 2391-2399 ◽  
Author(s):  
E. PONCE ◽  
J. ROS
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Nonlinear Behavior ◽  
High Gain ◽  
Approximate Methods ◽  
Saturated Control ◽  
Piecewise Linear Systems

The counter-intuitive appearance of stable periodic orbits in three-dimensional piecewise linear systems has been recently reported for stable saturated control systems with high gain and real triple eigenvalues [Moreno & Suárez 2004]. In this letter using several mathematical tools, the reported sudden phenomenon is explained, and the analysis is completed by providing a global bifurcation diagram of the symmetric periodic orbits. Approximate methods are used only to illustrate the nonlinear behavior with respect to the bifurcation parameter. Analytical methods are employed to rigorously prove the main result. The employed techniques are useful not only for the family studied but also for generic three-dimensional symmetric piecewise linear systems.

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.

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 .

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