Abstract/Summary

Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters’ value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics’ period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighborhood of the various UPOs. Each UPO and its neighborhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix, we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets. The attractor of a chaotic system is densely populated by an infinite number of unstable periodic orbits (UPOs), which are exact periodic solutions of the evolution equations. UPOs can be used to decompose the complex phenomenology of a chaotic flow into elementary components and have shown great potential for the understanding of macroscopic features in turbulent fluid flows. Here, we investigate how a long forward trajectory of the celebrated Lorenz 1963 model featuring the classical parameters’ value can be seen as a scattering process where the scatterers are the UPOs. This process helps elucidate how a generic ensemble of initial conditions converges to the invariant measure through diffusion and provides a new interpretation of quasi-invariant sets of the system in terms of UPOs.