Mapping densities in a noisy state space

Abstract

Weak noise smooths out fractals in a chaotic state space and introduces a maximum attainable resolution to its structure. The balance of noise and deterministic stretching/contraction in each neighborhood introduces local invariants of the dynamics that can be used to partition the state space. We study the local discrete-time evolution of a density in a two-dimensional hyperbolic state space, and use the asymptotic eigenfunctions for the noisy dynamics to formulate a new state space partition algorithm. 1. Motivation and outline Chaotic systems’ main feature is their high sensitivity to initial conditions. That makes direct numerical integration of the equations difficult and often calls for alternative methods for the evaluation of long-time averages of observables, such as decay of correlations, diffusion coefficients, energy spectra, or escape rates [7]. To properly weigh these averages, one needs to understand which regions of the state space are more or less relevant for the dynamics, in other words make a partition [3]. Invariants of the dynamics such as unstable periodic oribts have been successfully used to partition the state space [1]. However, noise, modelled by stochastic variables, erases periodic orbits. One has to look for new invariants. For that reason, we previously have studied [4, 2] the evolution of densities of trajectories and determined eigenfunctions of the local Fokker-Planck operator in the vicinity of the deterministic periodic orbits. The eigenfunctions are then used to partition the state space. All that was done in discrete time in one dimension. In order to develop a similar algorithm in higher dimensions, the first step is again to study the evolution of densities in the neighborhood of the periodic points of the deterministic system. In the present contribution we focus on the asymptotic evolution in two dimensions, forward and backward in time of a noiseless hyperbolic map (sect. 2), to which we successively add weak, uncorrelated, isotropic noise (sect. 3). In both cases the densities asymptotically align with the unstable (stable) direction of the monodromy matrix when iterated forward (backward) in time. We finally use our results to propose a definition of a neighborhood for an optimal partition of the state space. 2. Deterministic evolution We start by reviewing the deterministic evolution of densities and observables in the neighborhood of a fixed point x0 of the two-dimensional map x′ = f(x). We assume that the fixed point is hyperbolic, i.e., that the Jacobian matrix evaluated at the fixed point, Mi j(x0) = ∂ fi(x) ∂x j ∣∣∣∣∣ x=x0 , (1) has eigenvalues |Λs| 1. Consider the simplest example, a map f(x) = (Λsx,Λuy) (2) that is contracting along the x-axis and expanding along the y-axis. Now consider a density of trajectories ρ(x), for instance a Gaussian placed around the fixed point of f(x), and apply the Perron-Frobenius operator [1] to it: L ρ(x) = ∫ dz δ (x − f(z)) ρ(z) = 1 |ΛuΛs| ρ ( x Λs , y Λu ) , (3) so that, after n iterations, Lnρ(x) = 1 |Λu| ρ( x Λ n s , y Λ n u ) |Λs| . (4) One can see this as a density, which is losing mass by a factor of |Λu| at each iteration. This expression can be renormalized by a factor of |Λu| , when taking the limit n → ∞. If the initial density is a normalized Gaussian ρ(x) ∝ exp [ −(x2 + y2)/2σ2 ] , we obtain lim n→∞ |Λu|L = lim n→∞ exp [ − x2 2(σΛs )2 ] √ 2πσ2Λ s = δ(x) (5) meaning the limiting density is supported on the y-axis, the unstable manifold of the fixed point of the map. Of course this was the simplest possible example, given that the contracting and expanding directions of the fixed point are already separated by the coordinates. This is not the case in general, and one needs to do something different from what we just described. We will follow Rugh’s formalism [8] for a general two-dimensional map f(x) with a hyperbolic fixed point x0: the equation fy(xi, yi) = y f has ar X iv :1 30 3. 09 51 v1 [ nl in .C D ] 5 M ar 2 01 3 a unique solution, which we can call φs(xi, y f ), which is analytic and a contraction. On the other hand, one can define φu(xi, y f ) = fx ( xi, φs(xi, y f ) ) (6)