Chaotic Dynamical Systems Associated with Tilings of RN

1 Introduction

Chaos synchronization has exhibited an increasing interest in the last decade since the pioneering works reported in (Pecora & Carroll, 1990; Pecora & Carroll, 1991), and it has been advocated as a powerful tool in secure communication (Blekhman et al, 2002; Kolumban et al,1998; Nijmeijer, 1997; Kennedy & Ogorzalek, 1997).

Chaotic systems are indeed characterized by a great sensitivity to the initial conditions and a spreading out of the trajectories, two properties which are very close to the Shannon requirements of confusion and diffusion (Massey, 1992).

There are basically two approaches when using chaotic dynamical systems for secure communications purposes. The first one amounts to numerically computing a great number of iterations of a discrete chaotic system, in using e.g. the message as initial data (see (Schmitz, 2001) and the references therein). The second one amounts to hiding a message in a chaotic dynamics. Only a part of the state vector (the “output”) is conveyed through the public channel. Next, a synchronization mechanism is designed to retrieve the message at the receiver part (see (Rosier et al, 2006) and the references therein).

In both approaches, the first difficulty is to “build” a chaotic system appropriate for encryption purposes. In this context, the corresponding chaotic signals must have no patterning, a broadband power spectrum and an auto-correlation function that quickly drops to zero. In (Pecora et al, 1997), a mean for synthesizing volume-preserving or volume-expanding maps is provided. For such systems, there are several directions of expansion (stretching), while the discrete trajectories are folded back into a confined region of the phase space. Expansion can be carried out by unstable linear mappings with at least one positive Lyapunov exponent. Folding can be carried out with modulo functions through shift operations, or with triangular, trigonometric functions through reflection operations. Fully stretching piecewise affine Markov maps have also attracted interest because such maps are expanding in all directions and they have uniform invariant probability densities (see (Hasler et al, 1996; Rovatti & Setti,1998))

Besides, we observe that the word “chaotic” has not the same meaning everywhere, and that the chaotic behavior of a system is often demonstrated only by numerical evidences. The first aim of this chapter is to provide a rigorous analysis, based on the definition given by Devaney (Devaney, 2003), of the chaotic behavior of a large class of affine dynamical systems defined on the homogeneous space associated with a regular tiling of R^N. Classical piecewise affine chaotic transformations, as the tent map, belong to that class. The dimension N may be arbitrarily large in the theory developed below, but, for obvious reasons, most of the examples given here will be related to regular tilings of the plane (N = 2). The study of the subclass of (time-invariant or switched) affine systems on T^N, the N−dimensional torus, is done in (Rosier et al, 2004 ; Rosier et al, 2006). The folding for this subclass is carried out with modulo maps, which, from a geometric point of view, amounts to “fold back” R^N to [0,1)^N by means of translations by vectors in Z^N. Those translations are replaced here by all the isometries of some crystallographic group for an arbitrary regular tiling of R^N. Notice also that the fundamental domain used in the numerical implementation may be chosen with some degree of freedom. It may be a hypercube (as [0,1)^N for T^N), or a polyhedron, or a more complicated bounded, connected set in R^N.