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

Altres ajuts: Conserjería de Educación y Ciencia de la Junta de Andalucía (TIC-0130, EXC/2005/FQM-872, P08-FQM-03770)Altres ajuts: Universitat de les Illes Balears grant UIB2005/6 and by CAIB grand number CEH-064864Numerical 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

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

Agraïments: This work was partially supported by the Consejería de Educación y Ciencia de la Junta de Andalucía (TIC-0130, P08-FQM-03770). The second author is supported by Ministerio de Educaci'on, grant AP2008-02486

On the periodic solutions of the Milchelson continuous and discontinuous piecewise linear differential system

Applying new results from the averaging theory for continuous and discontinuous differential systems, we study the periodic solutions of two distinct versions of the Michel- son differential system: a Michelson continuous piecewise linear differential system and a Michelson discontinuous piecewise linear differential system. The tools here used can be applied to general nonsmooth differential systems

Global bifurcations close to symmetry

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection - the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies - trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions

Canard Trajectories in 3D piecewise linear systems

We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold Sε. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle

On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in R3\mathbb{R}^3

In this paper we use the blow up method of Dumortier and Roussarie \cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of singularities of piecewise smooth dynamical systems \cite{filippov1988differential} in $\mathbb R^3$. Using the regularization method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the power of our approach by considering the case of a fold line. We quickly recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a simple manner. Then, for the two-fold singularity, we show that the regularized system only fully retains the features of the singular canards in the piecewise smooth system in the cases when the sliding region does not include a full sector of singular canards. In particular, we show that every locally unique primary singular canard persists the regularizing perturbation. For the case of a sector of primary singular canards, we show that the regularized system contains a canard, provided a certain non-resonance condition holds. Finally, we provide numerical evidence for the existence of secondary canards near resonance.Comment: To appear in SIAM Journal of Applied Dynamical System

Construcción de familias de sistemas caóticos lineales por partes

"En el estudio de sistemas dinámicos hay gran interés en la generación de sistemas caóticos con propiedades particulares, ya sea simplicidad de estructural, atractores con múltiples enroscados, entre otras. En los métodos propuestos para generar estos sistemas hay una cosa en común, esto es, la falta de pruebas rigurosas de caos. Por tal razón en esta tesis proponemos métodos para construir sistemas tridimensionales simples para los cuales sean demostrable que su comportamiento dinámico es caótico. En particular, proponemos que la demostración de caos sea mediante el método de Shilnikov. El principal dificultad para esta demostración es el garantizar la existencia de órbitas homoclínicas o ciclos heteroclínicos. Un camino para construir sistemas caóticos es asegurando que los sistemas construidos posean órbitas homoclínicas/heteroclínicas. En esta tesis proponemos tres familias de sistemas lineales por partes para las cuales se puede demostrar que tienen dinámica caótica usando el método de Shilnikov. Aprovechamos la simpleza geométrica de las descripciones lineales alrededor de los puntos de equilibrio del sistema para proponer un algoritmo de construcción que presentan una serie de pautas para intersectar los eigenespacios y el plano de swithcheo de modo que garantizamos que las conexiones entre los puntos de equilibrio son órbitas homoclínicas o ciclos heteroclínicos. Ilustramos nuestros resultados con simulaciones numéricas de diferentes realizaciones de sistemas caóticos lineales por partes construidos utilizando los algoritmos propuestos de dos y tres dominios lineales. ""Currently, in the study of dynamical systems there is a great deal of interest on the generation of chaotic systems with particular properties, such as structural simplicity, multiscroll attractors, among others. There is a common feature in most of the methods proposed for their construction, that is, the lack of rigorous proofs of chaos for the resulting systems. For that reason in this thesis we propose to construct simple three dimensional systems for which the chaotic behavior can be demonstrable. In particular, we propose that the demonstration of chaos be done using the Shilnikov method. The main di culty for this type of demonstration is to guaranty the existence of homoclinic orbits and heteroclinic cycles. A way to construct chaotic systems is to ensure that the resulting systems have homoclinic/heteroclinic orbits. In this thesis we propose three families of piecewise linear systems for which is possible to demonstrate that their dynamical behavior is chaotic using the Shilnikov method. We take advantage of the geometric simplicity of the linear descriptions of the dynamics around the equilibrium points of the system to propose a construction algorithm that gives a series of steps such that de eigenspacies and swithing planes intersect in a way that guarantees that the connections between the equilibrium points are through homoclinic or heteroclinic orbits. We illustrate our results with numerical simulations of the di erent realizations of piecewise linear chaotic systems constructed using the proposed algorithms for two and three linear domains.