General approach for dealing with dynamical systems with spatiotemporal periodicities

TL;DR

This paper introduces a universal theoretical framework for analyzing dynamical systems with periodicities, applicable across classical, quantum, stochastic, and nonlinear systems, enabling efficient characterization through symmetry considerations.

Contribution

The authors develop a general formalism that links symmetry properties to the parameter dependence of dynamical systems with periodicities, demonstrated on a Bose-Einstein condensate example.

Findings

Symmetry considerations largely determine parameter dependence.

Few measurements suffice to characterize system behavior.

Framework applicable to diverse types of dynamical systems.

Abstract

Dynamical systems often contain oscillatory forces or depend on periodic potentials. Time or space periodicity is reflected in the properties of these systems through a dependence on the parameters of their periodic terms. In this paper we provide a general theoretical framework for dealing with these kinds of systems, regardless of whether they are classical or quantum, stochastic or deterministic, dissipative or nondissipative, linear or nonlinear, etc. In particular, we are able to show that simple symmetry considerations determine, to a large extent, how their properties depend functionally on some of the parameters of the periodic terms. For the sake of illustration, we apply this formalism to find the functional dependence of the expectation value of the momentum of a Bose-Einstein condensate, described by the Gross-Pitaewskii equation, when it is exposed to a sawtooth potential whose amplitude is periodically modulated in time. We show that, by using this formalism, a small set of measurements is enough to obtain the functional form for a wide range of parameters. This can be very helpful when characterizing experimentally the response of systems for which performing measurements is costly or difficult.