Fingerprints of Chaos

The asymptotic distance between trajectories $d_{\mathrm{\infty }}$, is studied in detail to characterize the occurrence of chaos. We show that this quantity is quite distinct and complementary to the Lyapunov exponents, and it allows for a quantitave estimate for the folding mechanism which keeps the motion bounded in phase space. We study the behaviour of $d_{\mathrm{\infty }}$ in simple unidimensional maps. Near a critical point $d_{\mathrm{\infty }}$ has a power law dependence on the control parameter. Furthermore, at variance with the Lyapunov exponents, it shows jumps when there are sudden changes on the available phase-space.