2 research outputs found
Periodic Shadowing Sensitivity Analysis of Chaotic Systems
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 is increased: the sensitivity error first decays as and then,
asymptotically as . We demonstrate the approach on the Lorenz
equations, and also show that, as 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