A PDE Approach to Finite Time Indicators in Ergodic Theory

Abstract

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators of the dynamics. On the one hand, they provide the same results on applications than other popular indicators; on the other hand, their PDE based definition -that we show robust under suitable perturbations -allows one to study them using the traditional tools of PDE environment. In particular, we formulate this approximating device in the Lyapunov exponents framework and we compare the operative use of them to the common use of the Fast Lyapunov Indicators to detect the phase space structure of quasi-integrable systems.

Convergence to the Time Average by Stochastic Regularization

“…We now consider the regularization of G T based on a vanishing stochastic perturbation, previously introduced in [4] by following a technique described in [11]. First, we consider the intermediate function…”

“…Our starting point is a class of globally defined approximate first integrals previously introduced in [4], which generalize the usual time average of any phase-space function.…”

“…In this paper, we obtain regularization by further averaging with respect to a stochastic perturbation (see [11], and also [4]) in order to deal with smooth functions. To use such regularized functions in any perturbation framework, we need estimates which include also the derivatives.…”

“…We now consider the regularization of G T based on a vanishing stochastic perturbation, previously introduced in [2] by following a technique described in [5]. More precisely, let (Ω, F , P ) be a probability space and w t : Ω → R n a n-dimensional Wiener process.…”

“…Then, we obtain a stochastic differential equation by perturbing the Hamilton equations with a white noise İt = 0 φt = g(I) + 2ν ẇt (7) whose flow is Φ t ν (I, ϕ, ω) = (I, ϕ + g(I)t + 2νw t (ω)). As in [2], for µ, ν > 0 we introduce…”