Sensitivity Analysis of Chaotic Systems using Unstable Periodic Orbits

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.

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• A Preconditioned Multiple Shooting Shadowing Algorithm for the Sensitivity Analysis of Chaotic Systems
• Decomposing the Dynamics of the Lorenz 1963 model using Unstable Periodic Orbits: Averages, Transitions, and Quasi-Invariant Sets
• Sensitivity of long periodic orbits of chaotic systems
• Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach
• Ghost states underlying spatial and temporal patterns: how non-existing invariant solutions control nonlinear dynamics
• Data-driven guessing and gluing of unstable periodic orbits
• Adjoint-based variational method for constructing periodic orbits of high-dimensional chaotic systems
• Adjoint chaos via cumulant truncation