Relevant bibliographies by topics / Smale-Birkhoff theorem

Academic literature on the topic 'Smale-Birkhoff theorem'

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Published: 14 March 2023

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Journal articles on the topic "Smale-Birkhoff theorem"

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Cheng, Xuhua, and Zhikun She. "A Note on the Existence of a Smale Horseshoe in the Planar Circular Restricted Three-Body Problem." Abstract and Applied Analysis 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/965829.

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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-Moser conditions to analytically prove the existence of a Smale horseshoe in this classical restricted three-body problem.
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Mrowka, T. "A short proof of the Birkhoff-Smale theorem." Proceedings of the American Mathematical Society 93, no. 2 (February 1, 1985): 377. http://dx.doi.org/10.1090/s0002-9939-1985-0770559-5.

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Cheng, Xuhua, and Zhikun She. "Study on Chaotic Behavior of the Restricted Four-Body Problem with an Equilateral Triangle Configuration." International Journal of Bifurcation and Chaos 27, no. 02 (February 2017): 1750026. http://dx.doi.org/10.1142/s0218127417500262.

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In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map [Formula: see text] and check that [Formula: see text] is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.
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Mrowka, T. "Shorter Notes: A Short Proof of the Birkhoff-Smale Theorem." Proceedings of the American Mathematical Society 93, no. 2 (February 1985): 377. http://dx.doi.org/10.2307/2044784.

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Rothe, Franz. "An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings." Aequationes Mathematicae 47, no. 2-3 (April 1994): 328. http://dx.doi.org/10.1007/bf01832965.

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Rothe, Franz. "An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings." Aequationes Mathematicae 48, no. 1 (August 1994): 1–23. http://dx.doi.org/10.1007/bf01837976.

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MESSIAS, MARCELO. "Periodic perturbations of quadratic planar polynomial vector fields." Anais da Academia Brasileira de Ciências 74, no. 2 (June 2002): 193–98. http://dx.doi.org/10.1590/s0001-37652002000200001.

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In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.
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Sulalitha Priyankara, K. G. D., Sanjeeva Balasuriya, and Erik Bollt. "Quantifying the Role of Folding in Nonautonomous Flows: The Unsteady Double-Gyre." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1750156. http://dx.doi.org/10.1142/s0218127417501565.

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We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the “stretching+folding=chaos” mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale–Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times.
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BROWN, RAY, and LEON CHUA. "GENERALIZING THE TWIST-AND-FLIP PARADIGM." International Journal of Bifurcation and Chaos 01, no. 02 (June 1991): 385–416. http://dx.doi.org/10.1142/s0218127491000312.

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In this paper we generalize the horseshoe twist theorem of Brown and Chua [1991] and derive a wide class of ODEs, with and without dissipation terms, for which the Poincare map can be expressed in closed form as FTFT where T is a generalized twist. We show how to approximate the Poincaré maps of nonlinear ODEs with continuous periodic forcing by Poincare maps which have a closed-form expression of the form FT 1 T 2 … T n where the T i are twists. We extend the twist-and-flip map to three dimensions with and without damping. Further, we demonstrate how to use the square-wave analysis to argue for the existence of a twist-and-flip paradigm for the Poincare map of the van der Pol equation with square-wave forcing. We apply this analysis to the cavitation bubble oscillator that appears in Parlitz et al. [1991] and prove a variation of the horseshoe twist theorem for the twist-and-shift map, which models the cavitation bubble oscillator. We present illustrations of the diversity of the dynamics that can be found in the generalized twist-and-flip map, and we use a pair of twist maps to provide a specific and very simple illustration of the Smale horseshoe. Finally, we use the twist-and-shift map of the cavitation bubble oscillators to demonstrate that the addition of sufficient linear damping to a dynamical system having PBS (Poincaré–Birkhoff–Smale) chaos may cause the chaos to become detectable in computer simulations.
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Dissertations / Theses on the topic "Smale-Birkhoff theorem"

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Montecchiari, Piero. "Homoclinic Solutions for Asymptotically Periodic Second Order Hamiltonian Systems." Doctoral thesis, SISSA, 1994. http://hdl.handle.net/20.500.11767/4531.

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Book chapters on the topic "Smale-Birkhoff theorem"

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Sideris, Thomas C. "The Birkhoff Smale Homoclinic Theorem." In Atlantis Studies in Differential Equations, 199–217. Paris: Atlantis Press, 2013. http://dx.doi.org/10.2991/978-94-6239-021-8_10.

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