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An algorithm to automatically detect the Smale horseshoes. (English) Zbl 1253.37005
Summary: Smale horseshoes, curvilinear rectangles and their U-shaped images patterned on Smale’s famous example, provide a rigorous way to study chaos in dynamical systems. The paper is devoted to constructing them in two-dimensional diffeomorphisms with the existence of transversal homoclinic saddles. We first propose an algorithm to automatically construct “horizontal” and “vertical” sides of the curvilinear rectangle near to segments of the stable and of the unstable manifolds, respectively, and then apply it to four classical chaotic maps (the Duffing map, the Hénon map, the Ikeda map, and the Lozi map) to verify its effectiveness.
MSC:
37-04Machine computation, programs (Dynamical systems and ergodic theory)
37E30Homeomorphisms and diffeomorphisms of planes and surfaces
37D45Strange attractors, chaotic dynamics
37C29Homoclinic and heteroclinic orbits