# Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

## TL;DR

This paper establishes a duality framework for bounding long-time averages in nonlinear dynamical systems, showing auxiliary functions can provide arbitrarily sharp bounds and identify regions containing extremal trajectories.

## Contribution

It proves the strong duality between extremal trajectory problems and auxiliary function optimization, enabling precise bounds and trajectory localization in nonlinear systems.

## Key findings

- Auxiliary functions yield arbitrarily sharp upper bounds on time averages.
- Nearly minimal auxiliary functions identify phase space regions containing extremal trajectories.
- Semidefinite programming can construct auxiliary functions for polynomial systems like the Lorenz system.

## Abstract

For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxiliary function provides phase space volumes in which all nearly maximal trajectories are guaranteed to lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz system.