An improved version of the Marotto Theorem

Department of Mathematics, Shanghai University, Shanghai, Shanghai Shi, China
Chaos Solitons & Fractals (Impact Factor: 1.45). 09/2003; 18(1):69-77. DOI: 10.1016/S0960-0779(02)00605-7


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.

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    • "In addition, in (Li & Chen, 2003), they showed that this norm can be chosen to be the Euclidean "

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    • "The idea of a snap-back repeller, a natural extension of homoclinic orbits to noninvertible maps, was first considered by Marotto [12] and there has been renewed interest recently in the light of a small technical error in Marotto's original paper (an error which does not change the basic topological argument of the paper [9] [13]) and a desire to apply the idea to more general systems and to more complicated orbits [3] [10] [11] [15]. The definitions used below are motivated by a wish to develop a bifurcation theory of snap-back repellers, so the most general definitions are not necessarily the most appropriate for this setting – in particular we want the definition of a regular snap-back repeller to be such that such objects are robust to small perturbations of the system. "
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    ABSTRACT: The bifurcation theory of snap-back repellers in hybrid dynamical sys- tems is developed. Innite sequences of bifurcations are shown to arise due to the creation of snap-back repellers in non-invertible maps. These are analogous to the cascades of bifurcations known to occur close to homoclinic tangencies for dieo- morphisms. The theoretical results are illustrated with reference to bifurcations in the normal form for border-collision bifurcations.
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    • "m (see Lemma 5 of Li and Chen [3] "
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    ABSTRACT: We consider a one-parameter family of maps F λ on ℝ m ×ℝ n with the singular map F 0 having one of the two forms (i) F 0 (x,y)=(f(x),g(x)), where f:ℝ m →ℝ m and g:ℝ m →ℝ n are continuous, and (ii) F 0 (x,y)=(f(x),g(x,y)), where f:ℝ m →ℝ m and g:ℝ m ×ℝ n →ℝ n are continuous and g is locally trapping along the second variable y. We show that if f is one-dimensional and has a positive topological entropy, or if f is high-dimensional and has a snap-back repeller, then F λ has a positive topological entropy for all λ close enough to 0.
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