RESUMEN
A low-dimensional dynamical system is observed in an experiment as a high-dimensional signal, for example, a video of a chaotic pendulums system. Assuming that we know the dynamical model up to some unknown parameters, can we estimate the underlying system's parameters by measuring its time-evolution only once? The key information for performing this estimation lies in the temporal inter-dependencies between the signal and the model. We propose a kernel-based score to compare these dependencies. Our score generalizes a maximum likelihood estimator for a linear model to a general nonlinear setting in an unknown feature space. We estimate the system's underlying parameters by maximizing the proposed score. We demonstrate the accuracy and efficiency of the method using two chaotic dynamical systems-the double pendulum and the Lorenz '63 model.
RESUMEN
Traditionally, interactions between laser beams or filaments were considered to be deterministic. We show, however, that in most physical settings, these interactions ultimately become stochastic. Specifically, we show that in the nonlinear propagation of laser beams, the shot-to-shot variation of the nonlinear phase shift increases with distance, and ultimately becomes uniformly distributed in [0, 2π]. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in- or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the outcome of the interaction become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of these stochastic interactions are universal and fully predictable. These statistics can be efficiently computed using a novel universal model for stochastic interactions, even when the noise distribution is unknown.