Finding unstable periodic orbits: a hybrid approach with polynomial optimization

TL;DR

This paper introduces a hybrid polynomial optimization method to find and stabilize unstable periodic orbits in polynomial ODE systems, improving convergence and enabling the computation of optimal orbits in chaotic dynamics.

Contribution

A novel hybrid approach combining polynomial optimization and control strategies to compute and stabilize unstable periodic orbits in polynomial ODE systems.

Findings

Successfully applied to three low-dimensional chaotic systems.

Enhanced convergence of periodic orbit computations.

Controlled systems exhibit reduced instability or stabilization.

Abstract

We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems parameterized by a scalar $k$ such that the original ODE system is recovered for $k = 0$, and such that the optimal orbit is less unstable, or even stabilized, for $k>0$. Periodic orbits for the controlled system can be more easily converged with traditional methods and numerical continuation in $k$ allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics.