Shadowing Pseudo-Orbits and Gradient Descent Noise Reduction

Shadowing trajectories are one of the most powerful ideas of modern dynamical systems theory, providing a tool for proving some central theorems and a means to assess the relevance of models and numerically computed trajectories of chaotic systems. Shadowing has also been seen to have a role in state estimation and forecasting of nonlinear systems. Shadowing trajectories are guaranteed to exist in hyperbolic systems, but this is not true of non-hyperbolic systems, indeed it can be shown there are systems that cannot have long shadowing trajectories. In this paper we consider what might be called shadowing pseudo-orbits. These are pseudo-orbits that remain close to a given pseudo-orbit, but have smaller mismatches between forecast state and verifying state. Shadowing pseudo-orbits play a useful role in the understanding and analysis of gradient descent noise reduction, state estimation and forecasting nonlinear systems, because their existence can be ensured for a wide class of non-hyperbolic systems. New theoretical results are presented that that extend classical shadowing theorems to shadowing pseudo-orbits. These new results provide some insight into the convergence behaviour of gradient descent noise reduction methods. The paper also discusses, in the light of the new results, some recent numerical results for an operational weather forecasting model when gradient descent noise reduction was employed.