Finding unstable periodic orbits: A hybrid approach with polynomial optimization

PHYSICA D-NONLINEAR PHENOMENA

卷:427

Article

Lakshmi, Mayur V.¹  Fantuzzi, Giovanni¹  Chernyshenko, Sergei I.¹  lasagna, DaviDe² 

[1] Imperial Coll London, Dept Aeronaut, London SW7 2AZ, England

[2] Univ Southampton, Fac Engn & Environm, Southampton SO17 1BJ, Hants, England

关键词: Periodic orbits;    Nonlinear dynamics;    Polynomial optimization;    Auxiliary functions;    Differential equations;   

DOI  :  10.1016/j.physd.2021.133009

来源: Elsevier

【 摘 要 】

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 often 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. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

Free