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

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