Isochronous Systems, the Arrow of Time, and the Definition of Deterministic Chaos

It is shown how, given an arbitrary dynamical system, other systems can be manufactured which are isochronous (periodic in all their degrees of freedom with an arbitrarily assigned fixed period T) and whose dynamics coincides exactly with that of the original system over a fraction 1 - sigma of that period where sigma is a number that can be arbitrarily assigned in the range 0 < sigma < 1. The treatment is in the context of autonomous dynamical systems whose equations of motion feature infinitely differentiable functions of the dependent variables. It is argued that this possibility suggests that it would be useful to devise measures of the degree of complexity of a dynamical system which are associated with finite portions of its time evolution-hence going beyond the standard characterizations of the chaotic behavior of a dynamical system (via its Lyapunov coefficients or its ergodic properties), requiring its observation over infinite time.