Multivalued maps, selections and dynamical systems

Sánchez-Gabites, J. J.; Sanjurjo, José Manuel Rodríguez

doi:10.1016/j.topol.2006.10.016

Cited by 4 publications

(1 citation statement)

References 14 publications

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“…A multivalued function Γ from a set X to a set X, denoted by Γ : X ⇒ X, is a function from X to the set 2 X of all subsets of X. The theory of such maps is well developed [2,11] and have important applications in rigorous numerics [12], economics [5], dynamical systems [17], and differential relations [1].…”

Section: Introductionmentioning

confidence: 99%

Optimal chaotic selectors

Boyarsky

Góra

2015

Dynamical Systems

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Multivalued maps have many applications. We consider one dimensional multivalued maps whose graphs are defined by lower and upper boundary maps. Let I = [0, 1] and let P be a partition of I into a finite number of intervals. Let τ , τu : I → I be two piecewise expanding maps on P such that τ ≤ τu. Let G ⊂ I × I be the region bounded by the graphs of τ and τu. Any map η : I → I that takes values in G is called a selector of the multivalued map defined by G. We assume that τ and τu as well as all the selectors we consider have invariant distribution functions. Let F * be a target distribution. We prove the existence of a selector η * which minimizes the functional J(η) = R I (Fη (t)−F * (t)) 2 dt, where η has invariant distribution Fη . Other results pertain to the functional J 1 (η) = R I (Pη F * (t)−F * (t)) 2 dt, where Pη is the Frobenius-Perron operator of η acting on distribution functions. We present an algorithm for finding selectors which minimize J 1 (η).

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Section: Introductionmentioning

confidence: 99%

Optimal chaotic selectors

Boyarsky

Góra

2015

Dynamical Systems

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Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Sun

Zeng

et al. 2018

Acta. Math. Sin.-English Ser.

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An approach to the shape Conley index without index pairs

Sánchez-Gabites

2010

Rev Mat Complut

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Robbin and Salamon proved that the shape of the homotopy Conley index of an isolated invariant set K coincides with the shape of the one-point compactification of the unstable region W u (K) of K endowed with a certain topology which they called the intrinsic topology. In this paper an equivalent, simplified definition of the latter is given in elementary terms, without resorting to index pairs whatsoever. We then show how our approach allows for a development of the shape index and its basic properties in an index pair-free fashion and simplifies some classical constructions, such as that of the Morse equations of a Morse decomposition. As a final application, some embrionary work about the geometry of W u (K)/K is presented together with an argument about how this may contribute to our understanding of the relations between how an attractor sits in its basin of attraction and the complexity of the dynamics in the latter.

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Multivalued maps, selections and dynamical systems

Cited by 4 publications

References 14 publications

Optimal chaotic selectors

Optimal chaotic selectors

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

An approach to the shape Conley index without index pairs

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