Periodic Orbit Bifurcations in Planar Hysteretic Systems without Equilibria

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.

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Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

Applied Mathematics and Nonlinear Sciences ◽

10.21042/amns.2017.2.00036 ◽

2017 ◽

Vol 2 (2) ◽

pp. 449-464 ◽

Cited By ~ 10

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.

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Stability Analysis of Periodic Orbits of Nonautonomous Piecewise-Linear Systems by Mapping Approach

IEEE Transactions on Circuits & Systems II Express Briefs ◽

10.1109/tcsii.2009.2032475 ◽

2009 ◽

Vol 56 (11) ◽

pp. 845-849 ◽

Cited By ~ 6

Author(s):

Yibo Zhao ◽

Jiuchao Feng ◽

C.K. Tse

Keyword(s):

Stability Analysis ◽

Linear Systems ◽

Periodic Orbits ◽

Piecewise Linear ◽

Piecewise Linear Systems ◽

Mapping Approach

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

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2019215 ◽

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

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

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2015.35.59 ◽

2015 ◽

Vol 35 (1) ◽

pp. 59-72 ◽

Cited By ~ 6

Author(s):

Victoriano Carmona ◽

◽

Emilio Freire ◽

Soledad Fernández-García ◽

Keyword(s):

Linear Systems ◽

Periodic Orbits ◽

Piecewise Linear ◽

Three Dimensional ◽

Piecewise Linear Systems ◽

Invariant Cones

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Periodic orbits for perturbations of piecewise linear systems

Journal of Differential Equations ◽

10.1016/j.jde.2010.10.025 ◽

2011 ◽

Vol 250 (4) ◽

pp. 2244-2266 ◽

Cited By ~ 6

Author(s):

Victoriano Carmona ◽

Soledad Fernández-García ◽

Emilio Freire

Keyword(s):

Linear Systems ◽

Periodic Orbits ◽

Piecewise Linear ◽

Piecewise Linear Systems

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A Time-Domain Asymptotic Approach to Predict Saddle-Node and Period Doubling Bifurcations in Pulse Width Modulated Piecewise Linear Systems

MATEC Web of Conferences ◽

10.1051/matecconf/20141608004 ◽

2014 ◽

Vol 16 ◽

pp. 08004 ◽

Cited By ~ 7

Author(s):

A. El Aroudi

Keyword(s):

Linear Systems ◽

Time Domain ◽

Pulse Width ◽

Piecewise Linear ◽

Period Doubling ◽

Asymptotic Approach ◽

Pulse Width Modulated ◽

Piecewise Linear Systems ◽

Saddle Node ◽

Period Doubling Bifurcations

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Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127498001728 ◽

1998 ◽

Vol 08 (11) ◽

pp. 2073-2097 ◽

Cited By ~ 128

Author(s):

Emilio Freire ◽

Enrique Ponce ◽

Francisco Rodrigo ◽

Francisco Torres

Keyword(s):

Linear Systems ◽

Periodic Orbits ◽

Canonical Form ◽

Vector Fields ◽

Piecewise Linear ◽

Oscillatory Behavior ◽

Linear Vector ◽

Piecewise Linear Systems ◽

The Creation

Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409024165 ◽

2009 ◽

Vol 19 (07) ◽

pp. 2391-2399 ◽

Cited By ~ 3

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.

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