Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization

TL;DR

This paper introduces sum-of-squares optimization methods to compute tight bounds on long-term averages in polynomial deterministic and stochastic dynamical systems, addressing challenges related to local attractors and basin of attraction issues.

Contribution

It develops computer-assisted sum-of-squares techniques for bounding long-term averages in polynomial systems, including deterministic and stochastic cases, with practical bounds demonstrated on the van der Pol oscillator.

Findings

Achieved within 1% bounds on the van der Pol oscillator's limit cycle averages.

Derived tight bounds on stochastic expectations across various noise levels.

Addressed basin of attraction limitations in bounding methods.

Abstract

We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.