Analyses of transient chaotic time series - art. no. 056207

We address the calculation of correlation dimension, the estimation of Lyap unov exponents, and the detection of unstable periodic orbits. from transie nt chaotic time series. Theoretical arguments and numerical experiments sho w that the Grassberger-Procaccia algorithm can be used to estimate the dime nsion of an underlying chaotic saddle from an ensemble of chaotic transient s. We also demonstrate that Lyapunov exponents can be estimated by computin g the rates of separation of neighboring phase-space states constructed fro m each transient time series in an ensemble. Numerical experiments utilizin g the statistics of recurrence times demonstrate that unstable periodic orb its of low periods can be extracted even when noise is present. In addition , we test the scaling law for the probability of finding periodic orbits. T he scaling law implies that unstable periodic orbits of high period are unl ikely to be detected from transient chaotic time series.