Loss of memory in a chaotic dynamical system

Abstract

A chaotic signal loses the memory of the initial conditions with time, and the future behavior becomes unpredictable. Here we propose a method to understand the loss of memory with time from a time series. This is done by introducing time-dependent generalized exponents. The asymptotic behavior of these exponents is interesting and can distinguish between chaotic systems that lose memory of the initial conditions completely, those that partially retain the memory, and those (borderline of chaos) that fully retain the memory. We discuss these features with some illustrative examples.