Formalism

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

$$x_{n+1}=f(x_n)+\sigma\xi_n\;\;,$$ (1)

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

$$f(x)=\lambda x(1-x)\;\;,$$ (2)

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

$$\langle \xi_m\xi_n \rangle=\delta_{mn}\;\;,$$ (3)

such as a normalized Gaussian distribution. The methods used here are equally applicable to $\sigma$ and $\xi$ 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 $\rho(x)$ transported by the dynamics and diffusing due to the noise:

$$\rho_{n+1}(x)\equiv{\cal L}[\rho_n](x) =\int \delta_\sigma(x-f(x'))\rho_n(x')dx'$$ (4)

where $\delta_\sigma(y)$ is the noise kernel, for example

$$\delta_\sigma(y)=\frac{e^{-y^2/(2\sigma^2)}}{\sigma\sqrt{2\pi}}\;\;,$$ (5)

reducing to a Dirac $\delta$ in the deterministic $\sigma=0$ limit.

Long time properties of the dynamics are obtained from the leading eigenvalue(s) of the linear evolution operator ${\cal L}$, which are (the inverses of) solutions of the characteristic equation

$$\det(1-z{\cal L})=0\;\;.$$ (6)

For example, the probability of a point initially in an open system remaining there after n iterations is typically proportional to $e^{-\gamma n}$ where the escape rate $\gamma$ is related to the leading zero z₀ by

$$\gamma=-\ln z_0\;\;.$$ (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 $\ln\det={\rm tr}\ln$ and Taylor expanding the logarithm:

 

$$\det(1-z{\cal L}) \exp\left(-\sum_{n=1}^{\infty}\frac{z^n}{n}{\rm tr}{\cal L}^n\right)$$ =

$$1-z{\rm tr}{\cal L}+\frac{z^2}{2}\left[({\rm tr}{\cal L})^2-{\rm tr}{\cal L}^2\right] -\ldots$$ =

$$\equiv \sum_{n=0}^{\infty}C_nz^n$$ (8)

C_n may be obtained from all the traces ${\rm tr}{\cal L}^m$ with $m\leq n$, 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 C_n quickly gets complicated; it is easier to expand the derivative

$$-z\frac{d}{dz}\det(1-z{\cal L}) =\det(1-z{\cal L})\sum_{n=1}^{\infty}z^n{\rm tr}{\cal L}^n\;\;,$$ (9)

which leads to the recursive equation

$$C_n=\frac{1}{n}\left({\rm tr}{\cal L}^n- \sum_{m=1}^{n-1}C_m{\rm tr}{\cal L}^{n-m}\right)\;\;.$$ (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],

$${\rm tr}{\cal L}^n=\int\prod_{j=0}^{n-1}dx_j \prod_{j=0}^{n-1}\delta_\sigma(x_{j+1}-f(x_j))$$ (11)

where the index j is cyclic, so x_n=x₀. In the noiseless ($\sigma=0$) limit, the integrand is a product of Dirac $\delta$-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.