Dynamical theories of physical systems are generally based on differential equation models. Typical systems of ordinary differential equations cannot be solved analytically, and hence one must generate solutions by numerical means. In any computer simulation, the numerical solution is fraught with truncation errors introduced by discretization and roundoff errors introduced by finite-precision calculation. A natural question arises as to whether the behavior of a numerical solution is similar to any ``nearby'' true solution of the system, that is, is the solution of the perturbed system uniformly close to the solution of the unperturbed system? This problem is even more pronounced in the case of chaotic systems. The trajectories of chaotic systems exhibit sensitive dependence on initial conditions: two trajectories with initial conditions that are extremely close diverge exponentially on the average from one another. As a result, a small truncation or roundoff error made at any step during the computation will tend to be greatly magnified by future evolution of the system. In view of this, it is natural to ask under what conditions the computed trajectory will be close to a true trajectory of the model. Consideration of simple examples of nonlinear maps show that there are critical points of trajectories where roundoff errors or other noise can introduce new behavior. At such ``glitches'' the true trajectories diverge from the computed trajectory. The occurrence of a glitch is not an uncommon phenomenon, and for those simple systems, it can be attributed to folds caused by homoclinic tangencies and near tangencies of stable and unstable manifolds.
In this review, we present practical tools that will be useful in diagnosing whether certain dynamical systems are shadowable. It is our belief that in systems with high-dimensional chaos, one can only shadow the numerical trajectories for short period of times. We examine possible causes of failure in shadowability that is likely to occur in higher-dimensional chaotic systems. We find that unshadowable pseudo-trajectories are usually accompanied by a Lyapunov exponent that ``fluctuates about zero''. By this we mean that given any long trajectory, computation of finite-time Lyapunov exponent using pieces of the trajectory would yield both positive and negative values for pieces of the trajectory of arbitrary length. Since the finite-time Lyapunov exponents quantify the expansion and contraction of phase space along the the trajectory over a finite stretch of time, this implies that the trajectory is visiting areas of phase space where the number of stable and unstable directions changes. For these systems, it would be advantageous to know how long one can expect to be able to shadow a pseudo-trajectory. Using ideas from Brownian motion theory with a reflecting barrier and applying it to the finite time Lyapunov exponent distribution, we were able to make order of magnitude estimates of the length of a pseudo-trajectory which can be shadowed.
Leon Poon, Celso Grebogi, Tim Sauer, and James A. Yorke
Department of Mathematics
University of Maryland
College Park
MD 20742
Feb 26 1998