Data-Driven Stabilization of Periodic Orbits

Abstract

Periodic orbits are among the simplest non-equilibrium solutions to dynamical systems, and they play a significant role in our modern understanding of the rich structures observed in many systems. For example, it is known that embedded within any chaotic attractor are infinitely many unstable periodic orbits (UPOs) and so a chaotic trajectory can be thought of as 'jumping' from one UPO to another in a seemingly unpredictable manner. A number of studies have sought to exploit the existence of these UPOs to control a chaotic system. These methods rely on introducing small, precise parameter manipulations each time the trajectory crosses a transverse section to the flow. Typically these methods suffer from the fact that they require a precise description of the Poincaré mapping for the flow, which is a difficult task since there is no systematic way of producing such a mapping associated to a given system. Here we employ recent model discovery methods for producing accurate and parsimonious parameter-dependent Poincaré mappings to stabilize UPOs in nonlinear dynamical systems. Specifically, we use the sparse identification of nonlinear dynamics (SINDy) method to frame model discovery as a sparse regression problem, which can be implemented in a computationally efficient manner. This approach provides an explicit Poincaré mapping that faithfully describes the dynamics of the flow in the Poincaré section and can be used to identify UPOs. For each UPO, we then determine the parameter manipulations that stabilize this orbit. The utility of these methods are demonstrated on a variety of differential equations, including the Rössler system in a chaotic parameter regime.

A toolkit for data-driven discovery of governing equations in high-noise regimes

“…The true system has ODEs: Three training trajectories had initial conditions [2, 0], [4,1], and [7,1]; and two holdout trajectories had initial conditions [3,2] and [6,3]. The initial conditions of the first training trajectory were from [10]; all others were selected at random.…”

“…Since its introduction, SINDy has been applied to a wide range of systems, including for reduced-order models of fluid dynamics [37,38,36,27,24,14,12] and plasma dynamics [19,35], turbulence closures [3,4,49], nonlinear optics [51], numerical integration schemes [53], discrepancy modeling [30,21], boundary value problems [50], multiscale dynamics [18], identifying dynamics on Poincare maps [6,7], tensor formulations [26], and systems with stochastic dynamics [5,13]. It can also be used to jointly discovery coordinates and dynamics simultaneously [15,34].…”

Deep Learning of Conjugate Mappings

“…With such C k the UPO is now stable. Note that the choice of C k can be automated using linear matrix inequalities [71].…”

“…The inability for polynomial mappings to faithfully describe the chaotic return map dynamics thus limits our ability to forecast the chaotic system, understand the statistics of the attractor, and pull out periodic orbits [71].…”

“…We further highlight the utility of the conjugate mapping in the context of controlling chaos. In a related work it was shown that the SINDy models in chaotic regions are able to locate period one and two UPOs [71], but attempts to located higher period UPOs was prevented due to numerical inaccuracy. To demonstrate the utility of our results, we report here that using the latent mapping g we are able to locate and stabilize periodic orbits at c = 11 up to at least period six.…”