Existence and Stability of Invariant Cones in 3-Dim Homogeneous Piecewise Linear Systems with Two Zones

2017 ◽  
Vol 27 (01) ◽  
pp. 1750007 ◽  
Author(s):  
Song-Mei Huan
Keyword(s):  
Linear Systems ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Piecewise Linear Systems ◽  
Proper Form ◽  
Bifurcation Phenomena ◽  
Existence And Stability ◽  
Linear Subsystem ◽  
Invariant Cones ◽  
Nonsmooth Dynamical Systems

We mainly investigate the existence, stability and number of invariant cones in 3-dim homogeneous piecewise linear systems with two zones separated by a plane containing the 1-dim invariant manifold of each linear subsystem. By transforming the system into a proper form with the 1-dim invariant manifolds on the separation plane either coincident or perpendicular, we obtain complete results on the existence, stability and number of invariant cones and show that the maximum number of invariant cones is two. The explicit parameter relations obtained here contribute to understanding and investigating bifurcation phenomena occurring in nonsmooth dynamical 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

Saddle–node bifurcation of invariant cones in 3D piecewise linear systems

2012 ◽  
Vol 241 (5) ◽  
pp. 623-635 ◽  
Author(s):  
Victoriano Carmona ◽  
Soledad Fernández-García ◽  
Emilio Freire
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Saddle Node Bifurcation ◽  
Piecewise Linear Systems ◽  
Saddle Node ◽  
Invariant Cones

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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