Periodicity and Predictability in Chaotic Systems

Abstract

Most of the good mathematical models available for natural phenomena are non linear, and these are often chaotic. Sensitive dependence on initial conditions can foil any attempt at prediction. In fact, measurement errors in collecting the initial data, intrinsic to any observational method, will propagate under the computational use of the model. With this, the final results obtained from the model may be useless when compared with the real future of the phenomena. Even in the strictly computational context, the sensitive dependence on initial conditions often makes a chaotic system S unpredictable because of representation errors. In fact, most of the orbits obtained in computers are not even good approximations to the actual ones. Real number representations by computers fall under two headings: floating-point (or numerical) and algebraic (or symbolic). In the former case, given the finite amount of memory, only numbers with finite decimal representations are treated without (roundoff) errors. In the symbolic setting, exact representation extends to some alge braic irrationals, to some transcendentals, and to various algebraic combinations of these. Therefore, the development of a dynamical system inside a computer is prone to successive errors, which is usually fatal to predictions concerning chaotic systems. In this article, we look at the question of computational predictability more closely and show that unpredictability in chaotic systems is also related to the scheme in which the system is represented. For simplicity, we restrict our discussions to discrete-time dynamical systems (i.e., to systems S defined by maps / : V -> V for metric spaces V). The orbit of S corresponding to the initial condition x0 is defined as the sequence y : N -+ V given by y(n) ? xn ? fn(x0), where N = {0, 1, 2, ...}. The time variable of S is n, and V is its state space. The space V is usually a subset of M.d for some natural number d, the dimension of the system. The orbits of S may be open, eventually periodic, or periodic. We write (eventually) periodic when referring to either of the last two categories. We call S periodic if all its orbits are (eventually) periodic.