Lifetime and Illness in Almost Deterministic Maps

This paper presents a completely new approach to analysing the effect of stochasticity upon the stability of sets in deterministic maps. Such stability is quantified via the definition of an expected lifetime. More relevant to dynamical systems modelling is the case where the stochasticity is small; the almost deterministic case. Formal arguments are employed to quantify the illness, through which the effect on the lifetime expectancy due to small stochasticity, is obtained. In the spirit of perturbative analysis, such equations are derived in terms of the deterministic dynamics, and the leading order stochasticity alone; the stochastic trajectories need not be computed. Of help is the definition of the ageing for mortal trajectories.and the ageing exponent for the immortal case.. Special cases attracting sets, Hamiltonian maps, etc..are analysed in some detail, leading to a better understanding of the equations derived.