Sensitivity Analysis of Chaotic Systems Using Unstable Periodic Orbits

Lasagna, Davide

doi:10.1137/17m114354x

Cited by 28 publications

(60 citation statements)

References 38 publications

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“…will be mostly dominated by the first term and not by the details of the shadowing sensitivity algorithm. Note that when using a periodic trajectory of period T for sensitivity analysis [22], the error E T 1 px 0 , pq is identically zero, since the shadowing direction is periodic and is found from the solution of the tangent problem (15). Only the random error E T 0 px 0 , pq affects the sensitivity results.…”

Section: Error Analysismentioning

confidence: 99%

“…where the definition (22) has been used. Note that the bound on C T 2 px 0 , pq shows that the term of the left hand side of (45) decays to zero as T Ñ 0.…”

Section: 2mentioning

confidence: 99%

“…We compare periodic shadowing sensitivities with a) sensitivity calculations on unstable periodic orbits (UPOs) following Ref. [22], b) finite-difference gradients (with ∆ρ " 0.5), and c) Least-Squares Shadowing gradient calculations. The LSS data is obtained from digitizing data points reported in figure 6 of Ref.…”

Section: Sensitivity With Respect To Perturbations To ρmentioning

confidence: 99%

“…The sensitivity calculations on UPOs were performed using the method discussed in Ref. [22]. Since periodic orbits have a fixed period, statistics are calculated using orbits with period variation not greater than one time unit.…”

Section: Sensitivity With Respect To Perturbations To ρmentioning

confidence: 99%

“…Here, the topology of the problem can be introduced to derive periodic boundary conditions in time for the sensitivity equations, as recently shown in Ref. [22]. For chaotic trajectories, however, it currently appears unlikely that an efficient strategy exists that can be used to provide exact initial/boundary conditions and ones needs to rely on approximations.…”

Section: Introductionmentioning

confidence: 99%

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Periodic shadowing sensitivity analysis of chaotic systems

Lasagna

Sharma

Meyers

2019

Journal of Computational Physics

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The sensitivity of long-time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction, the uniformly-bounded-in-time solution of the sensitivity equations. Although its existence is formally guaranteed for certain systems, methods to determine it are hardly available. One practical approach is the Least-Squares Shadowing (LSS) algorithm (Q Wang, SIAM J Numer Anal 52, 156, 2014), whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm. Here, we present an alternative, potentially simpler shadowing-based algorithm, termed periodic shadowing. The key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time. We show that this is not only justifiable when the reference trajectory is itself periodic, but also possible and effective for chaotic trajectories. Our error analysis shows that periodic shadowing has the same convergence rates as LSS when the time span T is increased: the sensitivity error first decays as 1{T and then, asymptotically as 1{?T . We demonstrate the approach on the Lorenz equations, and also show that, as T tends to infinity, periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits (D Lasagna, SIAM J Appl Dyn Syst 17, 1, 2018) for which there is no shadowing error. Finally, finite-difference approximations of the sensitivity are also examined, and we show that subtle non-hyperbolicity features of the Lorenz system introduce a small, yet systematic, bias.

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Section: Error Analysismentioning

confidence: 99%

“…where the definition (22) has been used. Note that the bound on C T 2 px 0 , pq shows that the term of the left hand side of (45) decays to zero as T Ñ 0.…”

Section: 2mentioning

confidence: 99%

Section: Sensitivity With Respect To Perturbations To ρmentioning

confidence: 99%

Section: Sensitivity With Respect To Perturbations To ρmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Periodic shadowing sensitivity analysis of chaotic systems

Lasagna

Sharma

Meyers

2019

Journal of Computational Physics

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Adjoint-based limit cycle oscillation instability sensitivity and suppression

Jónsson

Martins

2022

Nonlinear Dyn

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Limit cycle oscillations (LCO) can be supercritical or subcritical. The supercritical response is benign, while the subcritical response can be destructive. It is desirable to avoid subcritical responses by enforcing LCO stability through design optimization constraints. However, there is a need for a computational approach that can model this instability and can handle hundreds or thousands of design variables. We propose a simple metric to determine the LCO stability using a fitted bifurcation diagram slope to address this need. We develop an adjoint-based formula to compute the stability derivative efficiently with respect to many design variables. To evaluate the stability derivative, we only need to compute the time-spectral adjoint equation three times no matter the number of design variables in the optimization problem. The proposed adjoint method is verified with finite differences, achieving a five-digit agreement between the two approaches. We consider a stability-constrained LCO parameter optimization problem and demonstrate that the optimizer can suppress the instability. Finally, we consider a more realistic LCO speed and stability-constrained mass minimization problem for an airfoil. The proposed method can be extended to optimization problems with a PDE-based model, opening the door to applications where high-fidelity models are needed.

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Adjoint sensitivity analysis of chaotic systems using cumulant truncation

Craske

2019

Chaos, Solitons & Fractals

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We describe a simple and systematic method for obtaining approximate sensitivity information from a chaotic dynamical system using a hierarchy of cumulant equations. The resulting forward and adjoint systems yield information about gradients of functionals of the system and do not suffer from the convergence issues that are associated with the tangent linear representation of the original chaotic system. The functionals on which we focus are ensemble-averaged quantities, whose dynamics are not necessarily chaotic; hence we analyse the system's statistical state dynamics, rather than individual trajectories. The approach is designed for extracting parameter sensitivity information from the detailed statistics that can be obtained from direct numerical simulation or experiments. We advocate a data-driven approach that incorporates observations of a system's cumulants to determine an optimal closure for a hierarchy of cumulants that does not require the specification of model parameters. Whilst the sensitivity information from the resulting surrogate model is approximate, the approach is designed to be used in the analysis of turbulence, whose degrees of freedom and complexity currently prohibits the use of more accurate techniques. Here we apply the method to obtain functional gradients from low-dimensional representations of Rayleigh-Bénard convection.

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Sensitivity Analysis of Chaotic Systems Using Unstable Periodic Orbits

Cited by 28 publications

References 38 publications

Periodic shadowing sensitivity analysis of chaotic systems

Periodic shadowing sensitivity analysis of chaotic systems

Adjoint-based limit cycle oscillation instability sensitivity and suppression

Adjoint sensitivity analysis of chaotic systems using cumulant truncation

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