TOPOLOGICAL ANALYSIS OF CHAOTIC ORBITS - REVISITING HYPERION

There is emerging interest in the possibility of chaotic evolution in astrophysical systems. To mention just one example, recent well-sample d ground-based observations of the Saturnian satellite Hyperion strong ly suggest that it is exhibiting chaotic behavior. We present a genera l technique, the method of close returns, for the analysis of data fro m astronomical objects believed to be exhibiting chaotic motion. The m ethod is based on the extraction of pieces of the evolution that exhib it nearly periodic behavior-episodes during which the object stays nea r in phase space to some unstable periodic orbit. Such orbits generall y act as skeletal features, tracing the topological organization of th e manifold on which the chaotic dynamics takes place. This method does not require data sets as lengthy as other nonlinear analysis techniqu es do and is therefore well suited to many astronomical observing prog rams. Well sampled data covering between twenty and forty characterist ic periods of the system have been found to be sufficient for the appl ication of this technique. Additional strengths of this method are its robustness in the presence of noise and the ability for a user to cle arly distinguish between periodic, random, and chaotic behavior by ins pection of the resulting two-dimensional image. As an example of its p ower, we analyze close returns in a numerically generated data set, ba sed on a model for Hyperion extensively studied in the literature, cor responding to nightly observations of the satellite. We show that with a small data set, embedded unstable periodic orbits can be extracted and that these orbits can be responsible for nearly periodic behavior lasting a substantial fraction of the observing run.