CHAOS AND TIME-REVERSAL SYMMETRY - ORDER AND CHAOS IN REVERSIBLE DYNAMIC-SYSTEMS

Dynamical systems with independent (continuous or discrete) time variable t and phase space variable x are called reversible if they are invariant under the combination (t --> -t, x --> Gx) where G is some transformation of phase space which is an involution (G-degrees G = Identity). Reversible systems generalise classical mechanical systems possessing time-reversal symmetry and are found in ordinary differential equations, partial differential equations and diffeomorphisms (mappings) modelling many physical.problems. This report is an introduction to some of the properties of reversible systems, with particular emphasis on reversible mappings of the plane which illustrate many of their basic features. Reversible dynamical systems are shown to be similar to Hamiltonian systems because they can possess KAM tori, yet they are different because they can also have attractors and repellers. We create and study examples of these hybrid dynamical systems and discuss the question of how to recognise whether a given dynamical system is reversible.