Academic literature on the topic 'Topological horseshoes'

This paper presents an efficient method for finding horseshoes in dynamical systems by using several simple results on topological horseshoes. In this method, a series of points from an attractor of a map (or a Poincaré map) are firstly computed. By dealing with the series, we can not only find the approximate location of each short unstable periodic orbit (UPO), but also learn the dynamics of almost every small neighborhood of the attractor under the map or the reverse map, which is very helpful for finding a horseshoe. The method is illustrated with the Hénon map and two other examples. Since it can be implemented with a computer software, it becomes easy to study the existence of chaos and topological entropy by virtue of topological horseshoe.

To confirm the numerically demonstrated chaotic behavior in a chaotic system with only one stable equilibrium reported by Wang and Chen, we resort to Poincaré map technique and present a rigorous computer-assisted verification of horseshoe chaos by virtue of topological horseshoes theory.

In this paper, chaos in the smooth Chua's equation is revisited. To confirm the chaotic behavior in the smooth Chua's equation demonstrated in numerical studies, we resort to Poincaré section and Poincaré map technique and present a computer assisted verification of existence of horseshoe chaos by virtue of topological horseshoes theory.

This paper studies the hyperchaotic dynamics in a four-dimensional Hopfield neural network by virtue of topological horseshoe. Two different horseshoes (chaotic invariant sets) are found in this network with the same parameters. Numerical studies show that the first one expands only one-dimensionally and the second one expands two-dimensionally. Computer simulation also shows that there exists a heteroclinic connection from the second horseshoe to the first one, which indicates that the chaotic set of this system can have a very complicated structure composed of different kinds of expansions.

In this tutorial paper, we present a history of Smale horseshoe and an overview of the progress of topological horseshoe theory. Then we offer a pedagogical exposition of elements of topological horseshoe theory with a lot of examples. Finally we demonstrate some typical applications of topological horseshoe theory to practical dynamical systems.

We study hyperbolic dynamics and bifurcations for generalized Hénon maps in the form [Formula: see text] (with b, α small and γ > 4). Hyperbolic horseshoes with alternating orientation, called half-orientable horseshoes, are proved to represent the nonwandering set of the maps in certain parameter regions. We show that there are infinitely many classes of such horseshoes with respect to the local topological conjugacy. We also study transitions from the usual orientable and nonorientable horseshoes to half-orientable ones (and vice versa) as parameters vary.

AbstractIn this paper, the classic Lorenz system is revisited. Some dynamical behaviors are shown with the Rayleigh number $\rho $ somewhat smaller than the critical value 24.06 by studying the basins characterization of attraction of attractors and tracing the one-dimensional unstable manifold of the origin, indicating some interesting clues for detecting the existence of hidden transient chaos. In addition, horseshoes chaos is verified in the famous system for some parameters corresponding to the hidden transient chaos by the topological horseshoe theory.

In this work we introduce a topological method for the search of fixed points and periodic points for continuous maps defined on generalized rectangles in finite dimensional Euclidean spaces. We name our technique "Stretching Along the Paths" method, since we deal with maps that expand the arcs along one direction. Such theory was developed in the planar case by Papini and Zanolin in [11,12] and it has been extended to the N-dimensional framework by the author and Zanolin in [16]. In the bidimensional setting, elementary theorems from plane topology suffice, while in the higher dimension some results from degree theory are needed, leading to the study of the so-called "Cutting Surfaces" [16]. Our method is also significant from a dynamical point of view, as it allows to detect complex dynamics. As it is well-known, a prototypical example of chaotic system is represented by the Smale horseshoe. However, in order to prove conjugacy with the shift map, it requires the verification of hyperbolicity conditions, which are difficult or impossible to prove in practical cases. For such reason more general and less stringent definitions of horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe while discarding the hyperbolicity hypotheses. This led to the study of the so-called "topological (or geometrical) horseshoes" [2,5]. In particular, different characterizations have been proposed by various authors in order to establish the presence of complex dynamics for continuous maps defined on subsets of the N-dimensional Euclidean space (see, for instance, [10,21,23] and the references therein). The tools employed in these and related works range from the Conley index [10] to the Lefschetz fixed point theory [20]. On the other hand, our approach, although mathematically rigorous, avoids the use of more advanced topological theories and it is relatively easy to apply to specific models arising in applications. For example we have employed such method to study discrete and continuous-time models arising from economics and biology [9,18]. In more details, the topics considered along the thesis can be summarized as follows. The description of the Stretching Along the Paths method and suitable variants of it can be found in Chapter 1. In Chapter 2 we discuss which are the chaotic features that can be obtained for a given map when our technique applies. In particular, we are able to prove semi-conjugacy to the Bernoulli shift and thus positivity of the topological entropy, the presence of topological transitivity and sensitivity with respect to initial conditions, density of periodic points. Moreover we show the mutual relationships among various classical notions of chaos (such as those by Devaney, Li-Yorke, etc.). We also introduce an alternative geometrical framework related to the so-called "Linked Twist Maps" [3,4,22], where it is possible to employ our method in order to detect complex dynamics. The theoretical results obtained so far find an application to discrete and continuous-time systems in Chapters 3 and 4. As regards the former, in Chapter 3 we deal with some one-dimensional and planar discrete economic models, both of the Overlapping Generation and of the Duopoly Game classes. The bidimensional models are taken from [8,19] and [1], respectively. On the other hand, in Chapter 4, with respect to continuous-time models, we study some nonlinear ODEs with periodic coefficients through a combination of a careful but elementary phase-plane analysis with the results on chaotic dynamics for Linked Twist Maps from Chapter 2. In more details, we consider a modified version of the Volterra predator-prey model, in which a periodic harvesting is included, as well as a simplification of the Lazer-McKenna suspension bridges model [6,7] from [13,14]. When dealing with ODEs with periodic coefficients, our method is applied to the associated Poincaré map. The contents of the present thesis are based on the papers [9,13,16,17,18] and partially on [14], where maps expansive along several directions were considered. [1] H.N. Agiza and A.A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Appl. Math. Comput. 149 (2004), 843-860. [2] K. Burns and H. Weiss, A geometric criterion for positive topological entropy, Comm. Math. Phys. 172 (1995), 95-118. [3] R. Burton and R.W. Easton, Ergodicity of linked twist maps, In: Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979), pp. 35-49, Lecture Notes in Math., 819, Springer, Berlin, 1980. [4] R.L. Devaney, Subshifts of finite type in linked twist mappings, Proc. Amer. Math. Soc. 71 (1978), 334-338. [5] J. Kennedy and J.A. Yorke, Topological horseshoes, Trans. Amer. Math. Soc. 353 (2001), 2513-2530. [6] A.C. Lazer and P.J. McKenna, Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. Henry Poincar e, Analyse non lineaire 4 (1987), 244-274. [7] A.C. Lazer and P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990), 537-578. [8] A. Medio, Chaotic dynamics. Theory and applications to economics, Cambridge University Press, Cambridge, 1992. [9] A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions. A geometrical method and applications to economics, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 3283-3309. [10] K. Mischaikow and M. Mrozek, Isolating neighborhoods and chaos, Japan J. Indust. Appl. Math. 12 (1995), 205-236. [11] D. Papini and F. Zanolin, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud. 4 (2004), 71-91. [12] D. Papini and F. Zanolin, Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells, Fixed Point Theory Appl. 2004 (2004), 113-134. [13] A. Pascoletti, M. Pireddu and F. Zanolin, Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps, Electron. J. Qual. Theory Differ. Equ., Proc. 8'th Coll. Qualitative Theory of Diff. Equ. 14 (2008), 1-32. [14] A. Pascoletti and F. Zanolin, Example of a suspension bridge ODE model exhibiting chaotic dynamics: a topological approach, J. Math. Anal. Appl. 339 (2008), 1179-1198. [15] M. Pireddu and F. Zanolin, Fixed points for dissipative-repulsive systems and topological dynamics of mappings defined on N-dimensional cells, Adv. Nonlinear Stud. 5 (2005), 411-440. [16] M. Pireddu and F. Zanolin, Cutting surfaces and applications to periodic points and chaotic-like dynamics, Topol. Methods Nonlinear Anal. 30 (2007), 279-319. [17] M. Pireddu and F. Zanolin, Some remarks on fixed points for maps which are expansive along one direction, Rend. Istit. Mat. Univ. Trieste 39 (2007), 245-274. [18] M. Pireddu and F. Zanolin, Chaotic dynamics in the Volterra predator-prey model via linked twist maps, Opuscula Math. 28/4 (2008), 567-592. [19] P. Reichlin, Equilibrium cycles in an overlapping generations economy with production, J. Econom. Theory 40 (1986), 89-102. [20] R. Srzednicki, A generalization of the Lefschetz fixed point theorem and detection of chaos, Proc. Amer. Math. Soc. 128 (2000), 1231-1239. [21] R. Srzednicki and K. Wojcik, A geometric method for detecting chaotic dynamics, J. Differential Equations 135 (1997), 66-82. [22] S. Wiggins, Chaos in the dynamics generated by sequence of maps, with application to chaotic advection in flows with aperiodic time dependence, Z. angew. Math. Phys. 50 (1999), 585-616. [23] P. Zgliczy nski and M. Gidea, Covering relations for multidimensional dynamical systems, J. Differential Equations 202 (2004), 32-58.

Abstract The present paper describes the genesis of a horseshoe vortex in the symmetry plane in front of a juncture. In contrast to a previous topological investigation, the presence of the obstacle is no longer physically modelled. Instead, the pressure gradient, induced by the obstacle, has been used to represent its influence. Consequently, the results of this investigation can be applied to any symmetrical flow above a flat plate. The genesis of the vortical structure is analysed by using the theory of nonlinear differential equations and the bifurcation theory. In particular, the genesis of a horseshoe vortex can be described by the unfolding of the degenerate singularity resulting from a Jordan Normal Form with three vanishing eigenvalues and one linear term which is related to the adverse pressure gradient. The examination of this nonlinear dynamical system reveals that a horseshoe vortex emanates from a non-separating flow through two subsequent saddle-node bifurcations in different directions and the transition of a node into a focus located in the flow field.

Abstract Numerical investigation of flow and heat transfer in a rectangular duct with a built-in circular tube has been carried out for a Reynolds number of 1000 and blockage ratio of 0.44. Since the heat transfer in the duct is dictated by the flow structure, the present study is directed toward characterization of the flow structure. To this end, the topological theory shows the promise of becoming a powerful tool for the study of the flow structure. Computations show helical vortex tubes in the wake and existence of horseshoe vortices. The w component of velocity is surprisingly large in front and in the near wake of the tube. The limiting streamlines on the tube and the bottom-plate reveal a complex flow field. The separation lines as well as singularity (saddle and nodal) points have been investigated. The iso-Nusselt number contours and the span-averaged Nusselt number in the flow passage shed light on the heat transfer performance in the duct.