Staking Out State Space

Unstable periodic orbits provide a framework upon which the state space dynamics of chaotic systems can be understood; their identification in nature would be a major step towards quantifying the dynamics of a system. Improved methods of detection and identification of these orbits are explored, along with estimates of the duration of observations required for detection in a geophysical system. Traditional methods for locating periodic orbits tend to rely on examining a finite data set for near returns in state space after a given period; particular windows are then chosen as candidates for the period of an orbit. Our method is devised for unlimited data streams (data sets examined as a series in time, which are much too large to record) and adopts state space recurrence, rather than a temporal window, approach. The method is demonstrated using the simple ENSO model of E. Galanti and E. Tziperman [1] ; orbits are sought and observation constraints are discussed. [1] E.Galanti & E. Tziperman. ENSO's Phase Locking to the Seasonal Cycle in the Fast-SST, Fast-Wave, and Mixed-Mode Regimes. Journal of the Atmospheric Sciences, \textbf{57}:2936-2950,2000.