Introduction

Periodic orbit theory is a remarkable tool in classical [1,2,3] and quantum [4,5] chaotic systems, permitting the evaluation of long time properties, such as escape rates, dynamical averages and energy levels in terms of short unstable recurrent motions, that is, periodic orbits or ``cycles''. In classical hyperbolic systems with known topology, such as the repeller of the map 5x(1-x) discussed below, convergence can be impressive: the escape rate is computed to 9 digits from the 8 cycles of period 4 or less. The lowest energy levels of helium [6] are computed to an accuracy far better than would be expected from a semiclassical approximation. Even when there is intermittency, hence hyperbolicity is lost, it is possible to get sensible results using special techniques in both the classical [7,8] and quantum [9] cases.

Recently the theory has been extended to classical systems with weak additive noise, using Feynmann diagrams [10] or smooth conjugacy techniques [11]. There are a number of motivations for such extensions: noise at some level is present in all physical systems; it regularizes the theory, replacing Dirac $\delta$-functions by smooth kernels (see below) and fractal distributions by smooth functions; there is also some hope that the noise may effectively truncate the theory, rendering irrelevant contributions from periodic orbits longer than the finite memory of the system.

The result of these investigations is a weak noise perturbation theory, representing the trace of the evolution operator and derived quantities as a power series expansion in $\sigma$, the noise level. The coefficients are combinations of higher derivatives of the map evaluated at the periodic orbits of the deterministic unperturbed system. Numerically, the coefficients themselves converge at a similar rate to the classical periodic orbit theory, but the power series in $\sigma$ is useful only for weak noise, say $\sigma<0.03$, suggesting the following question, the subject of this paper:

To what extent does periodic orbit theory survive strong noise, and how fast does it converge?

Strong noise differs qualitatively from weak noise in a number of respects: The stochastic dynamics is equally close to many slightly different deterministic dynamical systems, so the concept of a unique perturbation theory becomes less defined, in addition to the lack of convergence of such a theory. Also, Gaussian noise has no preferred status; for weakly stochastic systems, all types of noise distributions with a given variance $\sigma^2$ are identical to order $\sigma^2$.

The approach taken here is that the relevant quantity, the trace of an evolution operator, is evaluated numerically, using very little detailed information about the dynamics, in particular without reference to periodic orbits. The method is general enough to include any type of dynamics (hyperbolic, intermittent, attracting) and uncorrelated noise, subject to smoothness of both dynamics and the noise distribution, with the latter decaying exponentially at large distances. Here, as in Refs. [10,11] the noise is additive, but this is not a necessary condition.

From the trace, it is straightforward to construct the spectral determinant, and hence highly convergent expansions for escape rates and dynamical averages, in the spirit of cumulant expansions, as in standard periodic orbit theory. This has some similarities to Ref. [12], where various approximations to the quantum trace are compared.

Section II outlines the formalism required for the calculation, in particular casting the trace as a multidimensional integral. Section III discusses numerical approaches for evaluating this integral. Finally, the results and their ramifications are discussed in Section IV.