Periodic shadowing sensitivity analysis of chaotic systems

Abstract

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.

Sensitivity of long periodic orbits of chaotic systems

“…where standard parameters σ = 10, β = 8/3 and ρ = 28 are used throughout. As in other sensitivity studies on the Lorenz equations [10,15,39,43,55,56], we consider the sensitivity of the period average of the observable J(t) = u 3 (t) with respect to perturbations of ρ. Numerical integration of chaotic trajectories is performed using a classical fourth-order Runge-Kutta method with ∆t = 0.005.…”

“…Linearisation of (16) as reported in Ref. [39] shows that the gradient of the period average is given by the inner product…”

“…The functional form (39) is sufficient to explain the structure of the tails in figure 10. To this end, assume that periodic orbits appear in bifurcations at critical values ν b as ν is increased, as expressed by (39). Focusing at a givenT , assume also that the number of periodic orbits is large, so that J (ν) can be thought of as a random variable, with the coefficients C 0 , C 1 , C 2 and the bifurcation point ν b being random variables with values differing from orbit to orbit.…”

“…An important remark is that sensitivity computations using periodic orbits do not suffer from shadowing errors displayed by shadowing methods applied to chaotic trajectories, e.g. the Least-Squares Shadowing [67] and Periodic Shadowing algorithms [39]. For such algorithms, convergence proofs have been offered that suggest that the standard deviation of sensitivity calculations on hyperbolic systems should first decay asT −1 as a result of the approximations of the exact shadowing direction involved in these algorithms.…”

“…Carefully conducted numerical approximations of the gradient using a finite-difference formula (see Ref. [39] for details) show that the response of the average of u 3 to perturbation of ρ in the Lorenz equations is approximately J ν 1.002, well below the asymptotic value from long periodic orbits. We remark that this difference is likely not a bias arising from using periodic orbits.…”