The reduction of complex dynamical systems using principal interaction patterns

Abstract

A method of constructing low-dimensional nonlinear models capturing the main features of complex dynamical systems with many degrees of freedom is described. The system is projected onto a linear subspace spanned by only a few characteristic spatial structures called Principal Interaction Patterns (PIPs). The expansion coefficients are assumed to be governed by a nonlinear dynamical system. The optimal low-dimensional model is determined by identifying spatial modes and interaction coefficients describing their time evolution simultaneously according to a nonlinear variational principle. The algorithm is applied to a two-dimensional geophysical fluid system on the sphere. The models based on Principal Interaction Patterns are compared to models using Empirical Orthogonal Functions (EOFs) as basis functions. A PIP-model using 12 patterns is capable of capturing the long-term behaviour of the complete system monitored by second-order statistics, while in the case of EOFs 17 modes are necessary.