Nonlinear dynamical systems: Time reversibility {\it versus} sensitivity to the initial conditions

Abstract

Time reversal of vast classes of phenomena has direct implications with predictability, causality and the second principle of thermodynamics. We analyze in detail time reversibility of a paradigmatic dissipative nonlinear dynamical system, namely the logistic map $x_{t+1}=1-ax_t^2$. A close relation is revealed between time reversibility and the sensitivity to the initial conditions. Indeed, depending on the initial condition and the size of the time series, time reversal can enable the recovery, within a small error bar, of past information when the Lyapunov exponent is non-positive, notably at the Feigenbaum point (edge of chaos), where weak chaos is known to exist. Past information is gradually lost for increasingly large Lyapunov exponent (strong chaos), notably at $a=2$ where it attains a large value. These facts open the door to diverse novel applications in physicochemical, astronomical, medical, financial, and other time series.