Closing the hierarchy of moment equations in nonlinear dynamical systems

The moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems. In the present paper a systematic approach for closing this hierarchy is proposed, based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale. The method is applied to a one-dimensional vector field in the presence of noise, and to two prototypes of chaotic behavior. Excellent agreement with numerical results is obtained. Special emphasis is placed on the role of symmetries, and on the origin of the composite oscillations found for certain types of moments in the chaotic systems. [S1063-651X(98)01310-5].