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 Poincar\'e 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 Poincar\'e 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 Poincar\'e mapping that faithfully describes the dynamics of the flow in the Poincar\'e 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 R\"ossler system in a chaotic parameter regime.