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

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 Periodic orbits for perturbations of piecewise linear systems

2011 ◽  
Vol 250 (4) ◽  
pp. 2244-2266 ◽  
Author(s):  
Victoriano Carmona ◽  
Soledad Fernández-García ◽  
Emilio Freire
Keyword(s):  
Linear Systems ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Piecewise Linear 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.

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.

Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

1998 ◽  
Vol 08 (11) ◽  
pp. 2073-2097 ◽  
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.

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 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.

Sign in / Sign up

Export Citation Format

Share Document