Victoriano Carmona

On simplifying and classifying piecewise-linear systems

A basic methodology to understand the dynamical behavior of a system relies on its decomposition into simple enough functional blocks. In this work, following that idea, we consider a family of piecewise-linear systems that can be written as a feedback structure. By using some results related to control systems theory, a simplifying procedure is given. In particular, we pay attention to obtain equivalent state equations containing both a minimum number of nonzero coefficients and a minimum number of nonlinear dynamical equations (canonical forms). Two new canonical forms are obtained, allowing to classify the members of the family in different classes. Some consequences derived from the above simplified equations are given. The state equations of different electronic oscillators with two or three state variables and two or three linear regions are studied, illustrating the proposed methodology.

BIFURCATION OF INVARIANT CONES IN PIECEWISE LINEAR HOMOGENEOUS SYSTEMS

Invariant surfaces in three-dimensional continuous piecewise linear homogeneous systems with two pieces separated by a plane are detected. The Poincare map associated to this plane transforms half-straight lines passing through the origin into half-straight lines of the same type. The invariant half-straight lines under this map determine invariant cones for which the existence, stability and bifurcation are studied. This analysis lead us to consider some questions about the topological type and stability of the origin.

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

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 Chuas oscillator.

Existence of a Reversible T-Point Heteroclinic Cycle in a Piecewise Linear Version of the Michelson System

The proof of the existence of a global connection in differential systems is generally a difficult task. Some authors use numerical techniques to show this existence, even in the case of continuous piecewise linear systems. In this paper we give an analytical proof of the existence of a reversible T-point heteroclinic cycle in a continuous piecewise linear version of the widely studied Michelson system. The principal ideas of this proof can be extended to other piecewise linear systems.

Existence of homoclinic connections in continuous piecewise linear systems

Numerical methods are often used to put in evidence the existence of global connections in differential systems. The principal reason is that the corresponding analytical proofs are usually very complicated. In this work we give an analytical proof of the existence of a pair of homoclinic connections in a continuous piecewise linear system, which can be considered to be a version of the widely studied Michelson system. Although the computations developed in this proof are specific to the system, the techniques can be extended to other piecewise linear systems.

Noose bifurcation and crossing tangency in reversible piecewise linear systems

The structure of the periodic orbits analysed in this paper was first reported in the Michelson system by Kent and Elgin (1991 Nonlinearity 4 1045–61), who gave it the name noose bifurcation. Recently, it has been found in a piecewise linear system with two linearity zones separated by a plane, which is called the separation plane. In this system, the orbits that take part in the noose bifurcation have two and four points of intersection with the separation plane, and they are arranged in two curves that are connected by a point where the periodic orbit has a crossing tangency with the separation plane. In this work, we analytically prove the local existence of the curve of periodic orbits with four intersections that emerges from the point corresponding to the crossing tangency. Moreover, we add a numerical study of the stability and bifurcations of the periodic orbits involved in the noose curve for the piecewise linear system and check that they exhibit the same configuration as that of the Michelson system.

INSTABILITY IN THE SIMPLEST CLASS OF CONTINUOUS SWITCHED LINEAR SYSTEMS WITH STABLE COMPONENTS

Abstract For discontinuous switched linear systems, even when they are built by composing stable systems, examples of unstable systems are known. Here, three-dimensional homogeneous continuous piecewise linear systems composed by two linear systems sharing a boundary plane are considered. If the two linearization matrices are non-singular, then the only equilibrium point is at the origin, which is in the separation plane of the linear regions. For the important case where both matrices have complex eigenvalues and their whole spectra are in the open-left half of the complex plane, the possible counter-intuitive instability of the origin is analytically proved. Thus, in this paper, examples of unstable continuous switched linear system with just two stable components are shown.

Some Recent Results for Continuous Switched Linear Systems

The elemental structure arising from the continuous autonomous switching of two linear systems is considered. After introducing certain canonical forms, some analytical results about limit cycle bifurcation are reported, showing that such systems generically exhibit a jump transition to oscillating behavior. Explicit expressions for quantitative characteristics of the periodic oscillation are obtained for the cases of dimension two and three. As another relevant result, it is shown that continuous n-dimensional switched linear systems whose both components are Hurwitz need not be globally asymptotically stable when n is greater or equal to 3.

Noose Structure and Bifurcations of Periodic Orbits in Reversible Three-Dimensional Piecewise Linear Differential Systems

The main goal of this work is to describe the periodic behavior of a class of three-dimensional reversible piecewise linear continuous systems. More concretely, we study an interesting structure called the noose bifurcation that was previously detected by Kent and Elgin in the Michelson system. We numerically obtain the curves of periodic orbits that appear from the bifurcations at the noose curve, where other phenomena related to different types of tangencies with the separation plane arise. Besides that, we show that some of these curves of periodic orbits wiggle around global connections when the period increases. The complete structure of periodic orbits, including the stability and bifurcations, coincides with the one observed in the Michelson system. However, we also point out the relevance of the crossing tangency and the small loop that emerges from it in the existence of the noose bifurcation.

Existence of Homoclinic and Heteroclinic Connections in Continuous Piecewise Linear Systems

In the present work, the existence of global connections in a continuous piecewise linear system is analytically proven. Concretely, by using a common technique we prove the existence of a pair of homoclinic connections and a reversible T-point heteroclinic cycle. The main ideas of this proof can be extended to other piecewise linear systems.