An improved version of the Marotto Theorem
ABSTRACT In 1975, Li and Yorke introduced the first precise definition of discrete chaos and established a very simple criterion for chaos in one-dimensional difference equations, “period three implies chaos” for brevity. After three years. Marotto generalized this result to n-dimensional difference equations, showing that the existence of a snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is up to now the best one in predicting and analyzing discrete chaos in multidimensional difference equations. Yet, it is well known that there exists an error in the condition of the original Marotto Theorem, and several authors had tried to correct it in different ways. In this paper, we further clarify the issue, with an improved version of the Marotto Theorem derived.
- SourceAvailable from: Adelheid Ingeborg Mahla[Show abstract] [Hide abstract]
ABSTRACT: Modelling and analysis techniques for robots, with emphasis in strongly nonlinear or noncontinuous dynamics, are presented. Nonlinear characteristics -such as that are present in Coulomb friction, cubic stiffness, switching power electronics, interactions with another bodies or sudden load changes-produce a diversity of phenomena, as for example periodic cycles, bifurcations, quasiperiodicity and chaos. A modelling approach is by converting the continuous time system with discontinuous characteristics or hybrid system, in a discrete time system. This allows simplifying the stability analysis and doing the qualitative analysis of complex solutions. These phenomena and the analysis techniques are illustrated with some applications. (Invited tutorial). Resumen: Se presenta herramientas para el modelado y el análisis de robots, con énfasis en aquellos aspectos de la dinámica que son fuertemente no lineales o presentan discontinuidades, tales como soluciones complejas. Estos fenómenos y las técnicas que permiten analizarlos son ilustrados a través de algunas aplicaciones. (Tutorial invitado).LCRA 2003, First IEEE Latin American Conference on Robotics and Automation, 24-26 de Noviembre de 2003, Universidad de Las Américas, Santiago de Chile, Chile; 11/2003
Article: HETEROCLINICAL REPELLERS IMPLY CHAOS[Show abstract] [Hide abstract]
ABSTRACT: In this paper, we prove that chaos in the sense of Li–Yorke and of Devaney is prevalent in discrete systems admitting the so-called heteroclinical repellers, which are similar to the transversely heteroclinical orbits in both continuous and discrete systems and are corresponding to the snap-back repeller proposed by Marotto for proving the existence of chaos in higher-dimensional systems. In addition, the concept of heteroclinical repellers is generalized to be applicable to the case with degenerate transformations. In the end, some illustrative examples are provided to illustrate the theoretical results.International Journal of Bifurcation and Chaos 11/2011; 16(05). · 1.02 Impact Factor
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ABSTRACT: Chaotification of dynamical systems is a hot topic in the research field of chaos in recent years. Previous studies showed that (even linear) discrete or continuous dynamical systems can be chaotified by designing appropriate controllers. Here, we study chaotification of linear impulsive differential systems. First, we propose a framework for chaotification of general linear impulsive differential systems that can be transformed into discrete maps. Then, we give technical details for how to chaotify several typical linear impulsive differential systems that are actually canonical forms, including how to design appropriate quadratic impulsive controllers, how to find snapback repellers in the Marotto theorem, etc. As one of the main theoretical results, we rigorously prove the existence of chaos in all the considered impulsive systems. In addition, numerical examples are used to verify the theoretical prediction in each case. We are expecting that our proposed approach can have practical applications in the engineering field.International Journal of Bifurcation and Chaos 01/2013; 22(12). · 1.02 Impact Factor