Ams short course lecture notes introductory survey lectures a subseries of proceedings of symposia in applied mathematics volume 39 chaos and fractals. It is further shown that these solutions draw a horseshoe map of the smale type. Local bifurcations, center manifold theory, normal forms, the averaging theorem. This chapter provides an overview of the theoretical development of homoclinic and horseshoe dynamics. For a broad class of dynamical systems in continuous time, chaotic behaviour of trajectories is caused 14 by the fact that their poincare transformation t contains a smale horseshoe configuration 57, see also 811, ya. Entropy in dynamical systems laisang young1 in this article, the word entropy is used exclusively to refer to the entropy of a dynamical system, i. The analytic solutions around a separatrix are obtained using the first order perturbation. In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Details of the construction of the smale horseshoe map. The solutions are found by displaying rectangles in a crosssection whose images under the return map produce a smale horseshoe.
Transverse homoclinic orbits and smales horseshoes. Stakes should extend 14 to 15 inches above the pit surface. In particular, it squeezes d horizontally by a factor of 1. The horseshoe is a natural consequence of a geometrical way of looking at the equations of cartwrightlittlewood and levinson. The smale horseshoe has the advantage of being continuous. Statistical properties of smooth smale horseshoes springerlink. The paper is devoted to constructing them in twodimensional diffeomorphisms with the existence of transversal homoclinic saddles. With striking geometric and analytic clarity it robustly describes the homoclinic dynamics encountered. Smale s horseshoe map via ternary numbers created date.
Therefore a good approximation num t is one for which the l2 distance k t num numk l2. The map was introduced by stephen smale while studying the behavior of. Linear and nonlinear systems, flows, poincare maps, structural stability. Topological horseshoe in a singlescroll chen system with.
Research article a 3d smale horseshoe in a hyperchaotic. This is not to be confused with other notions of entropy connected with spatial complexity. In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. Pdf entropy, chaos, and weak horseshoe for infinite.
Horseshoe map a reprise as mentioned in the introduction, a central contribution of this thesis is the identi cation and visualization of homoclinic tangles and horseshoes in turbulent geophysical ows. In this section we will introduce the smale horseshoe, which was first intro. November 5, 2018 abstract we consider an abundant class of nonuniformly hyperbolic c2h enon like di eomorphisms called strongly regular and which corresponds to benedickscarleson parameters. Since each horse has an equal chance of winning, each has probability of 110. Horseshoe theory remains controversial, but its also still an important part of the international conversation.
To construct the smale horseshoe map figure 2, start with the unit square indicated in yellow. Let us define the probabilistic structural stability, which is implied by the. Smales horseshoe map has become a standard example in the study of discrete dynamical systems. It is a core example in the study of dynamical systems. Entropy, chaos, and weak horseshoe for infinitedimensional random dynamical systems article pdf available in communications on pure and applied mathematics april 2015 with 122 reads.
Guo, introduction to the peptide binding problem of computational immunology. Here, almost all pairs means with probability one in b. Then f 1 maps d onto the rectangle r dened by 1 8 x 3 8 1 1 2 y 9 2. Smales horseshoe map via ternary numbers created date. Smale horseshoe of cellular neural networks chenghsiung hsu department of mathematics, national central university, chungli 32054, taiwan received october 22, 1999. Then compress it in the \x\ direction by more than a factor of two. This kind of geometrical horseshoe in the phase space is called smale horseshoe. The horseshoe was, smale has written, the first structurally stable dynamical system with an infinite number of periodic solutions. This creation of a horseshoe is much simpler than involved. In general, rm belongs to the intersection of a vertical and a horizontal strip. Stephen smale, american mathematician who was awarded the fields medal in 1966 for his work on topology in higher dimensions.
Fractal geometry of nonuniformly hyperbolic horseshoes. Predicting chaos 5 uncountable number of nonperiodic orbits irrational initial conditions, and there is at least one orbit that is dense in 3i. Research article a 3d smale horseshoe in a hyperchaotic discretetime system qingdu li and xiaosong yang. While there are many methods for this task, here, we only want to discuss the one of lopreore and wyatt.
In this situation, the dynamics of the kinetic density fis governed by the mean eld kinetic equation with random. It has been shown that the h enon map has a full smale horseshoe for certain parameter values, and a partiallyformed horseshoe for others 12. Smale also proved the generalized poincare conjecture for all dimensions greater than or equal to five. The horseshoe map is an axiom a diffeomorphism that serves as a model for the general behavior at a transverse homoclinic point, where the stable and unstable manifolds of a periodic point intersect. These analytic solutions are in good agreement with the numerical integrations. The evolution of the probability density function pdf of nonlinear stochastic. Chapter 23 predicting chaos california institute of technology. Note that the communication between particles, denoted by x. As such, this invariant set lies within a neighborhood n, a solid handlebody of genus two. Hyperbolic invariant sets, strange attractors, the smale horseshoe, symbolic dynamics.
Devaney and linda keen providence, rhode island, august 1988 volume 38 computational complexity theory. Its chaotic behavior is usually shown by first conjugating it to the twosided shift map. An algorithm to automatically detect the smale horseshoes. Nowadays, in view of the work of many people in particular that of franke and selgrade these can be stated very succinctly as the chain recurrent set is hyperbolic. Properties of the maximal entropy measure and geometry of. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake. A transverse homoclinic point leads to a horseshoe. The definition of smale horseshoe is given by smale in 1967.
The mathematics behind the computer graphics edited by robert l. Chapter 2 homoclinic points and the horseshoe map a reprise. He notably constructed a function, the horseshoe, that serves as a paradigm for chaos. Revised january 15, 2000 the paper shows the spatial disorder of onedimensional cellular neural networks cnn using the iteration map method. Stakes may be inclined toward each other, but not more than a 3 inch lean. It is demonstrated in 82 that smale horseshoe chaotic dynamics can arise. Smales horseshoe map in a hamiltonian system around a separatrix. It measures the rate of increase in dynamical complexity as the system evolves with time. Finding a horseshoe on the beaches of rio steve smale 1. He described that map f shrinks d verticality, stretches d horizontally, then the modified d, i. Stakes may be inclined toward each other, but not more than a. No longer is the trajectory confined to small regions of the phase space, see. Progress of theoretical physics generalization of bakers transformation chaos and stochastic process on a smales horse shoe yoji aizawa 0 chikara murakami 0 0 department of physics, university of kyoto, kyoto 606 the areapreserving bakers transformation is extended in order to include the area nonpreserving one, which is the model for a simple horse shoe structure.
The horseshoe map was designed to reproduce the chaotic dynamics of a flow in the neighborhood of a given periodic orbit. Properties of the maximal entropy measure and geometry of h. Sinai 12 posed the problem of defining statistic characteristics of trajectories inherent in the horseshoe configuration and of describing the. For example, if m 1, the set r1 is one of the four intersections vj. New results 1 journal foundations of computational mathematics 20, 34 pp. Saddlenodes and perioddoublings of smale horseshoes.
We first propose an algorithm to automatically construct horizontal and vertical sides of. The history of the discovery of the horseshoe and the state of mathematics in 1960 is described in detail by smale 1998. Smale horseshoes, curvilinear rectangles and their ushaped images patterned on smales famous example, provide a rigorous way to study chaos in dynamical systems. Mathematics math horseshoe map is an axiom a diffeomorphism that serves as a model for the general behavior at a transverse homoclinic point, where the stable and unstable manifolds of a periodic point intersect. The equations of motion of quantum mechanics produce solutions which are probabilities evolving in time. In this paper, we will show a picture of a 3d smale horseshoe for a hyperchaotic. It follows from the birkhoffsmale theorem that the 1d hyperbolic invariant set of lemma 5. First, stretch it in the \y\ direction by more than a factor of two. The horseshoe is the fully nonlinear version of what. All periodic orbits disappear through saddlenode bifurcations. Oct 21, 2011 the history of the discovery of the horseshoe and the state of mathematics in 1960 is described in detail by smale 1998. It follows from the birkhoff smale theorem that the 1d hyperbolic invariant set of lemma 5.
However because the map is smooth, we have the chance of finding it in. Siam journal on applied dynamical systems siam society for. Examples of chaotic systems in the physical sciences. The horseshoe here is an extension of smales horseshoe. Download fulltext pdf smale horseshoe in a piecewise smooth map article pdf available in international journal of bifurcation and chaos 2904.
1440 1298 1076 451 969 1058 528 1497 652 314 1168 681 937 21 456 825 1196 597 771 738 658 635 476 390 1031 1457 1163 1026 1079 742 123 1032 74 1005 793 765 1358 244 635 337