HIGHER CORRELATION-FUNCTIONS OF CHAOTIC DYNAMIC-SYSTEMS - A GRAPH THEORETICAL APPROACH

For dynamical systems conjugated to the Bernoulli shift and their higher-dimensional extensions of Kaplan-Yorke type we calculate higher-order correlation functions by means of a graph theoretical method. The graphs relevant to this problem are forests of incomplete double binary trees. Our method has similarities with the Feynman graph approach in quantum field theory: The 'free field' corresponds to a Gaussian random dynamics, the 'interacting field' to a chaotic process with non-trivial higher-order correlations. The 'coupling constant' tau-1/2 is a time scale parameter that measures how much the chaotic process differs from a Gaussian process. We develop a pertubation theory around the Gaussian limit case for sums of iterates of the fully developed logistic map with arbitrary coefficients. The dynamical systems considered have a physical meaning in the sense that they describe the movement of a damped particle under the influence of a chaotic kick force. For tau --> 0 a Langevin process is generated, for tau > 0 there is a complicated chaotic process with higher-order correlation functions represented by double binary forests. The discussion can be generalized to complex mappings describing the movement of a charged particle in a chaotic electric and constant magnetic field.