Exact solutions to chaotic and stochastic systems (Q2728794)

This paper is devoted to the study explicit functions that are exact solutions to nonlinear chaotic maps. A generalization of these functions can produce truly random sequences. Even if the initial conditions are known exactly, the next values are in principle unpredictable from the previous values. Those functions cannot be expressed as a map of type NEWLINE\[NEWLINEX_{n+1}= g(X_n, X_{n-1},\dots, X_{n-r+1}).NEWLINE\]NEWLINE Using some of these functions the authors can exactly solve random maps as the following: \(X_{n+1}= f(X_n, J_n)\), where \(J_n\) is a random variable. They confirm the result that a negative Lyapunov exponent does not imply predictability in random systems and show that forecasting methods are very effective in distinguishing chaos from random time series. The authors study the influence of the level of chaos on the stochastic resonance (SR) and present explicit analytical formulas for the output signal of systems with stochastic resonance. Moreover they show the existence of a new type of solitonic stochastic resonance (SSR), where the shape of the kink is crucial.