Int. J. Pure Appl. Sci. Technol., 11(1) (2012), pp. 57-66 International Journal of Pure and Applied Sciences and Technology ISSN 2229 - 6107 Available online atwww.ijopaasat.in Research Paper Hyperbolic Dynamics in Two Dimensional Maps Md. Shariful Islam Khan 1 and Md. Shahidul Islam 2,* 1 Department of Mathematics, National University, Gazipur-1704, Bangladesh 2 Department of Mathematics, University of Dhaka, Dhaka-1000, Bangladesh * Corresponding author, e-mail: (mshahidul11@yahoo.com) (Received: 21-5-12; Accepted: 28-6-12) Abstract: In this paper, we study the theory of hyperbolic dynamical systems, especially the dynamics of two-dimensional Hénon map. A remarkable fact is that the system with complicated orbit behavior (like the “horseshoe”) can be structurally stable that hyperbolicity is a necessary condition for structural stability. We discuss the horizontal and vertical slabs in a diffeomorphism on which the induced dynamics is topologically conjugate to a full shift. Using these slabs, we provide the sufficient condition for chaos on the Hénon map Keywords: Slabs, Horseshoe, Structural stability, Hyperbolic. 1. Introduction: Hyperbolic sets [1] are used as differential models for chaotic dynamical systems. The concept of hyperbolicity was introduced first by G.A. Hedlund and E. Hopf in studying geodesic flows on surfaces with negative curvature. The systematic study of hyperbolic systems was initiated by S. Smale, A. Andronov, L. Pontryagin and D.V. Anosov. The Smale horseshoe represents a first attempt to globalize the idea of a hyperbolic fixed point: the invariant set of the horseshoe mapping has two distinguished directions, one in which the differential of the mapping is consistently contracting vectors and the other in which the differential of the mapping is consistently expanding vectors. The contracting and expanding directions are given by subspaces that are preserved by the differential, and which vary continuously with the base point. In this paper, we present a method for detecting the hyperbolic invariant sets of Hénon map. This method relies on the use of geometrical objects, referred as “slab”, whose behavior under iteration can be controlled effectively and the dynamics of Hénon map is hyperbolic.