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Sensitivity analysis of chaotic systems using unstable periodic orbits

Sensitivity analysis of chaotic systems using unstable periodic orbits
Sensitivity analysis of chaotic systems using unstable periodic orbits
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 ∼ 10and ∼ 4 × 104, 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.
547-580
Lasagna, Davide
0340a87f-f323-40fb-be9f-6de101486b24
Lasagna, Davide
0340a87f-f323-40fb-be9f-6de101486b24

Lasagna, Davide (2018) Sensitivity analysis of chaotic systems using unstable periodic orbits. SIAM Journal on Applied Dynamical Systems, 17 (1), 547-580. (doi:10.1137/17M114354X).

Record type: Article

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 ∼ 10and ∼ 4 × 104, 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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Sensitivity Analysis of Chaotic Systems using Unstable Periodic Orbits - Accepted Manuscript
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More information

Accepted/In Press date: 18 November 2017
e-pub ahead of print date: 15 February 2018

Identifiers

Local EPrints ID: 415777
URI: http://eprints.soton.ac.uk/id/eprint/415777
PURE UUID: 0390d701-b1ae-49c6-ad3b-2281221dd87b
ORCID for Davide Lasagna: ORCID iD orcid.org/0000-0002-6501-6041

Catalogue record

Date deposited: 23 Nov 2017 17:30
Last modified: 26 Nov 2021 07:04

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