| dc.description.abstract | How does long-term chaotic behavior respond to small parameter perturbations? Using detailed models, chaotic systems are frequently simulated across disciplines – from climate science to astrophysics. But, an efficient computation of parametric derivatives of their statistics or long-term averages, also known as linear response, is an open problem. The difficulty is due to an inherent feature of chaos: an exponential growth over time of infinitesimal perturbations, which renders conventional methods for sensitivity computation inapplicable. More sophisticated recent approaches, including ensemble-based and shadowing-based methods are either computationally impractical or lack convergence guarantees. We propose a novel alternative known as space-split sensitivity or S3, which evaluates linear response as an efficiently computable, provably convergent ergodic average. The main contribution of this thesis is the development of the S3 algorithm for uniformly hyperbolic systems – the simplest setting in which chaotic attractors occur – with one-dimensional unstable manifolds. S3 can enable applications of the computed sensitivities to optimization, control theory and uncertainty quantification, in the realm of chaotic dynamics, wherein these applications remain nascent.
We propose a transformation of Ruelle’s rigorous linear response formula, which is ill-conditioned in its original form, into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle’s formula, called the S3 decomposition, that is differentiable on the unstable manifold. The S3 decomposition ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similar to in a non-chaotic system. The remainder, known as the unstable contribution, is regularized and converted into a computable ergodic average. The S3 algorithm presented here can be naturally extended to systems with higher-dimensional unstable manifolds.
The secondary contributions of this thesis are analysis and applications of existing methods, including those shadowing-based and ensemble-based, to compute linear response. A feasibility analysis of ensemble sensitivity calculation, which is a direct evaluation of Ruelle’s formula, reveals a problem-dependent, typically poor rate of convergence, rendering it computationally impractical. Shadowing-based sensitivity computation is not guaranteed to converge because of atypicality of shadowing orbits. This atypicality also implies that small parameter perturbations can lead, contrary to popular belief, to a large change in the statistics of a chaotic system, a consequence being that numerical simulations of chaotic systems may not reproduce their true long-term behaviors. | |
