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We describe methods for proving upper and lower bounds on infinite-timeaverages in deterministic dynamical systems and on stationary expectations instochastic systems. The dynamics and the quantities to be bounded are assumedto be polynomial functions of the state variables. The methods arecomputer-assisted, using sum-of-squares polynomials to formulate sufficientconditions that can be checked by semidefinite programming. In thedeterministic case, we seek tight bounds that apply to particular localattractors. An obstacle to proving such bounds is that they do not holdglobally; they are generally violated by trajectories starting outside thelocal basin of attraction. We describe two closely related ways past thisobstacle: one that requires knowing a subset of the basin of attraction, andanother that considers the zero-noise limit of the corresponding stochasticsystem. The bounding methods are illustrated using the van der Pol oscillator.We bound deterministic averages on the attracting limit cycle above and belowto within 1%, which requires a lower bound that does not hold for the unstablefixed point at the origin. We obtain similarly tight upper and lower bounds onstochastic expectations for a range of noise amplitudes. Limitations of ourmethods for certain types of deterministic systems are discussed, along withprospects for improvement.

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