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.
Highlights
Since their inception by Henri Poincaré at the turn of the twentieth century, return maps, or Poincaré maps as they are commonly referred, have significantly influenced our understanding of recurrent and chaotic dynamical systems
Periodic/cyclic orbits of this mapping manifest themselves as periodic orbits of the continuous-time dynamical system, and Poincaré mappings provide an accessible method of understanding the flow on and near periodic orbits in phase space, many of which are unstable and responsible for the rich, dynamical structures manifest in many systems
We demonstrate that a recent method for data-driven discovery of Poincaré maps [28] can be used to overcome this barrier to stabilizing unstable periodic orbits (UPOs) of nonlinear dynamical systems
Summary
Since their inception by Henri Poincaré at the turn of the twentieth century, return maps, or Poincaré maps as they are commonly referred, have significantly influenced our understanding of recurrent and chaotic dynamical systems. Since xis taken to correspond to a UPO, it follows that at least one of the eigenvalues of A lies outside the unit circle in the complex plane, implying that the fixed point xis linearly unstable To stabilize this orbit, we apply the method of [36] by introducing small parameter manipulations at each intersection of the continuous-time trajectory with the Poincaré section. It can be a difficult task choosing the appropriate functions to include in the library of candidate functions and at the time of writing this manuscript there does not appear to be any prescriptions that are broadly applicable This is a potential limitation of the method, we will see in Section IV that for the examples considered using a simple library of monomials can reliably capture the qualitative dynamics of a desired Poincaré mapping. We refer the reader to [26], [36] for a more complete discussion of the choice of the threshold parameter as it relates to the control matrix K
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
