Empirical generating partitions of driven oscillators using optimized symbolic shadowing

We present an optimized version of the symbolic shadowing algorithm for coarse graining a continuous state dynamical system, originally due to Hirata and co-workers [Phys. Rev. E 70, 016215 (2004)PLEEE81539-375510.1103/PhysRevE.70.016215]. We validate our algorithm by finding generating partitions presented previously in the literature. We show that, unlike the original, the optimized algorithm can approximate generating partitions for periodically driven continuous-time nonlinear oscillators. We recover known generating partitions for the driven Duffing oscillator and compute generating partitions for the driven van der Pol oscillator. We also examine the problem of how algorithms such as ours can be applied "objectively," that is, by starting from arbitrary initial partition guesses. By applying our algorithm to large ensembles of initial random partitions, we show that symbolic shadowing leads to a multiplicity of candidate generating partitions that localize points in phase space to a high degree, thus making it difficult to select the best choice(s). We thus propose using the Lempel-Ziv complexity to identify partitions from this set of candidates that are, in a specific sense, "minimal," i.e., those with contiguous cells, fewer cell boundaries, and a smaller number of cells compared to their rivals. We also show how our methods can be used to indicate the appropriate number of symbols needed to approximate a generating partition.