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 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

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

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 Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

2017 ◽  
Vol 2 (2) ◽  
pp. 449-464 ◽  
Author(s):  
Marina Esteban ◽  
Enrique Ponce ◽  
Francisco Torres
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Hysteretic Systems ◽  
Piecewise Linear Systems ◽  
Saddle Node

AbstractThis paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from a reduction of symmetric 3D systems with slow-fast dynamics. We concentrate our attention on the saddle dynamics cases, determining the existence of periodic orbits as well as their stability, and possible bifurcations. Dealing with reachable saddles not in the central hysteresis band, we show the existence of subcritical/supercritical heteroclinic bifurcations as well as saddle-node bifurcations of periodic orbits.

Periodic Orbit Bifurcations in Planar Hysteretic Systems without Equilibria

2020 ◽  
Vol 30 (07) ◽  
pp. 2030016
Author(s):  
Marina Esteban ◽  
Enrique Ponce ◽  
Francisco Torres
Keyword(s):  
Periodic Orbit ◽  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Symmetric Case ◽  
Hysteretic Systems ◽  
Piecewise Linear Systems ◽  
Saddle Node

This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from 3D systems with slow–fast dynamics. We focus our attention on the symmetric case without equilibria, determining the existence of periodic orbits as well as their stability, and possible bifurcations. New analytical characterizations of bifurcations in these hysteretic systems are obtained. In particular, bifurcations of periodic orbits from infinity, grazing and saddle-node bifurcations of periodic orbits are studied in detail and the corresponding bifurcation sets are provided. Finally, the study of the hysteretic systems is shown to be useful in detecting periodic orbits for some [Formula: see text]D 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.

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