Smale's horseshoe
WebbIn the early 1960s, the young American mathematician Steve Smale(1930-...) was working on the beach of Copacabana when he made a discovery ...He discovered a horseshoe! Not a real one, but a mathematical object : a transformation combining a scaling and a contraction, but also a folding. It transforms a square into the shape of a horseshoe.
Smale's horseshoe
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Webb斯梅尔马蹄(Smale's horseshoe)由斯梅尔（Smale , S.)构造的形状类似于马蹄的结构稳定的离散动力系统.这个系统对高维结构稳定系统的特征提供了一个具体模型，并说明高维结 … Webb29 mars 2012 · 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.
Webb7 maj 2014 · We consider the family of quadratic Hénon diffeomorphisms of the plane ${\\bf R}^2$. A map will be said to be a "horseshoe" if its restriction to the nonwandering set is hyperbolic and conjugate to the full 2-shift. We give a criterion for being a horseshoe based on an auxiliary coding which describes positions of points relative to the stable … Webb23 feb. 2024 · Stephen Smale, (born July 15, 1930, Flint, Michigan, U.S.), American mathematician who was awarded the Fields Medal in 1966 for his work on topology in higher dimensions. Smale grew up in a rural area near Flint. From 1948 to 1956 he attended the University of Michigan, obtaining B.S., M.S., and Ph.D. degrees in mathematics. As an …
In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is … Visa mer The horseshoe map f is a diffeomorphism defined from a region S of the plane into itself. The region S is a square capped by two semi-disks. The codomain of $${\displaystyle f}$$ (the "horseshoe") is a proper subset of its … Visa mer • Baker's map • Hénon map Visa mer • "Smale Horseshoe". Scholarpedia. • Evgeny Demidov (2007). "Homoclinic structures in the standard map". ibiblio.org. Retrieved 2016-07-11. • ChaosBook.org Chapter "Stretch, fold, prune" Visa mer The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit. The neighborhood is chosen to be a small disk perpendicular to the orbit. As the system evolves, points in this disk remain close to the … Visa mer Webb1 apr. 2006 · As a consequence, the anti-integrability of the Hénon map H a,b can be correlated with that of the Smale horseshoe. We show the dynamics of a horseshoe which has uniformly contraction rate ϵ and expansion rate 1/ ϵ behaves the same as that of Fig. 1: the limit lim a →∞ H a , b corresponds to the limit ϵ → 0.
Webb10 nov. 2011 · 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. This program is written according to the algorithm proposed in the . paper, which has submitted to DDSN “An algorithm to automatically detect Smale . …
Webb20 okt. 2024 · In 1960 Stephen Smale (building on work by David Birkhoff) proved that the horseshoe is in some sense universal. A large class of dynamical systems, defined by mathematical equations, contain the dynamics of the horseshoe map and therefore also contain its chaos. Everything hinges on the existence of a particular point. inmate roster crawford countyWebb19 mars 2015 · It has been proved that, in the classical planar circular restricted three-body problem, the degenerate saddle point processes transverse homoclinic orbits. Since the standard Smale-Birkhoff theorem cannot be directly applied to indicate the chaotic dynamics of the Smale horseshoe type, we in this note alternatively apply the Conley … modbus opc ua gatewayWebb26 jan. 2005 · arXiv: Dynamical Systems. We consider the complexification of the Henon family of quadratic diffeomorphisms of the plane, and the region of parameter space … inmate roommateWebb11 maj 2024 · The horseshoe map of the system is topologically conjugate with the shift map of the symbolic dynamics. The system can be identified with the Smale horseshoe by means of its symbolic descriptions. Furthermore, since the Smale horseshoe map implies sensitive dependence on the initial condition, homoclinic intersection of stable and … modbus on tcpWebbAppDynSys : Smale Horseshoe : Iteration Prof Ghrist Math 20.3K subscribers Subscribe 4.8K views 3 years ago Applied Dynamical Systems Simulations If you iterate the … modbus openwrthttp://scholarpedia.org/article/Smale_horseshoe modbus opc-uaWebb13 dec. 2024 · The horseshoe map is continuous and injective on R 2. In W, it contracts distances horizontally by δ < 1 / 2 and expands vertically by 1 / δ. Outside W, the only … inmate roaster payette county