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# Formalism

The goal is to determine the long time properties of stochastic dynamical systems, here one dimensional maps with additive noise:

 (1)

where f(x) is a known function, for example the logistic map

 (2)

is a measure of the strength of the noise, and are independent identically distributed random variables with unit variance,

 (3)

such as a normalized Gaussian distribution. The methods used here are equally applicable to and that depend on x, and non-Gaussian noise distributions.

Instead of the Langevin form (1) it is more convenient to consider the discrete Fokker-Planck equation for a probability distribution transported by the dynamics and diffusing due to the noise:

 (4)

where is the noise kernel, for example

 (5)

reducing to a Dirac in the deterministic limit.

Long time properties of the dynamics are obtained from the leading eigenvalue(s) of the linear evolution operator , which are (the inverses of) solutions of the characteristic equation

 (6)

For example, the probability of a point initially in an open system remaining there after n iterations is typically proportional to where the escape rate is related to the leading zero z0 by

 (7)

Dynamical averages and diffusion coefficients can be obtained from the leading zero of appropriately weighted evolution operators [1,2,3].

The spectral determinant (6) of an infinite dimensional operator may be defined by its cumulant expansion in powers of z, using the matrix relation and Taylor expanding the logarithm:

 = = (8)

Cn may be obtained from all the traces with , and an approximation for z0 is obtained by numerical root finding on the n'th degree polynomial given by the truncation of the determinant. The above expression for Cn quickly gets complicated; it is easier to expand the derivative

 (9)

which leads to the recursive equation

 (10)

The trace is straightforward to write down as an n-dimensional integral, a discrete periodic chain reminiscent of a path integral, obtained in Ref. [13],

 (11)

where the index j is cyclic, so xn=x0. In the noiseless () limit, the integrand is a product of Dirac -functions, and the trace is given by a sum over the fixed points of fn, that is, the n-cycles of f. In Refs. [10,11] the weak noise limit is obtained by a saddlepoint expansion of the integral around these cycles. Here, the integral is performed numerically, up to n=5, as described in the following section.

Next: Numerical methods Up: Traces and determinants of Previous: Introduction
Carl Philip Dettmann
1998-05-15