Much of what is known about specific dynamical systems is obtained from numerical experiments. Although the discretization process usually has no significant effect on the results for simple, well-behaved dynamics, acute sensitivity to changes in initial conditions is a hallmark of chaotic behavior. How confident can one be that the numerical dynamics reflects that of the original system? Do numerically calculated trajectories always shadow a true one? What role does numerical analysis play in the study of dynamical systems? And conversely, can advances in dynamical systems provide new insights into numerical algorithms? These and related issues were the focus of the workshop on Chaotic Numerics, held at Deakin University in Geelong, Australia, in July 1993. The contributions to this book are based on lectures presented during the workshop and provide a broad overview of this area of research.

Readership

Research mathematicians.

Table of Contents

• J. K. Hale -- Numerical dynamics
• R. M. Corless -- Error backward
• M. P. Calvo, A. Murua, and J. M. Sanz-Serna -- Modified equations for ODEs
• H. C. Yee and P. K. Sweby -- The dynamics of some iterative implicit schemes
• S.-N. Chow and E. S. Van Vleck -- Shadowing of lattice maps
• B. A. Coomes, H. Koçcccccak, and K. J. Palmer -- Periodic shadowing
• W.-J. Beyn -- On well-posed problems for connecting orbits in dynamical systems
• L. Debraux -- Numerical computation of a branch of invariant circles starting at a Hopf bifurcation point
• J. Lorenz -- Numerics of invariant manifolds and attractors
• P. Diamond, P. Kloeden, and A. Pokrovskii -- Interval stochastic matrices and simulation of chaotic dynamics
• C. M. Elliott, A. R. Gardiner, I. Kostin, and B. Lu -- Mathematical and numerical analysis of a mean-field equation for the Ising model with Glauber dynamics
• V. M. Gundlach -- Attractors for weakly coupled map lattices
• M. J. Ablowitz and C. M. Schober -- Effective chaos in the nonlinear Schrödinger equation
• X. Yu -- Discretisation effect on a dynamical system with discontinuity