Sensitivity Analysis of Chaotic Systems using Unstable Periodic Orbits

TL;DR

This paper introduces a new adjoint sensitivity method for chaotic systems that uses unstable periodic orbits, allowing accurate sensitivity calculations without exponential divergence, demonstrated on Lorenz and Kuramoto-Sivashinsky systems.

Contribution

The paper develops a bounded adjoint sensitivity technique based on unstable periodic orbits, improving sensitivity analysis in chaotic dynamical systems.

Findings

Most orbits predict similar sensitivities.

Sensitivity calculations are stable over long periods.

Method effectively decouples stability from sensitivity computations.

Abstract

A well-behaved adjoint sensitivity technique for chaotic dynamical systems is presented. The method arises from the specialisation of established variational techniques to the unstable periodic orbits of the system. On such trajectories, the adjoint problem becomes a time periodic boundary value problem. The adjoint solution remains bounded in time and does not exhibit the typical unbounded exponential growth observed using traditional methods over unstable non-periodic trajectories (Lea et al., Tellus 52 (2000)). This enables the sensitivity of period averaged quantities to be calculated exactly, regardless of the orbit period, because the stability of the tangent dynamics is decoupled effectively from the sensitivity calculations. We demonstrate the method on two prototypical systems, the Lorenz equations at standard parameters and the Kuramoto-Sivashinky equation, a one-dimensional partial differential equation with chaotic behaviour. We report a statistical analysis of the sensitivity of these two systems based on databases of unstable periodic orbits of size 10^5 and 4x10^4, respectively. The empirical observation is that most orbits predict approximately the same sensitivity. The effects of symmetries, bifurcations and intermittency are discussed and future work is outlined in the conclusions.