Calculated Attractor Dimensions for Low-Order Spectral Models

Abstract

Advancing knowledge about the phase space topologies of nonlinear hydrodynamic or dynamical systems has raised the question of whether the structure of the attractors in which the solutions are eventually confined can be characterized rigorously and economically. It is shown by applying the Lyapunov exponents, Lyapunov dimension, and correlation dimension to several low-order truncated spectral models that these quantities give useful information about the phase space structure and predictability characteristics of such attractors. The Lyapunov exponents measure the average exponential rate of convergence or divergence of nearby solution trajectories in an appropriate phase space. The Lyapunov dimension dL incorporates the dynamical information of the Lyapunov exponents to give an estimate of the dimension of the system attractor, while the correlation dimension v is a more geometrically motivated measure that is simple to compute and related to more classical dimensions. The Lyapunov exponents detect bifurcations between solution regimes and also subtle predictability differences between attractors. As measures of chaotic attractor dimension, v>dL in all cases, and the ratio v/dL is smallest at values of the forcing just above the transition to chaos. Changes in the Lyapunov dimension are concentrated in a small range of forcing values, while the correlation dimension varies more uniformly. The value of dL is tied closely to the number of positive Lyapunov exponents, while v is more sensitive to the magnitude of the chaotic component of the system. Variations in these measures for a hierarchy of convection models support the idea that the appearance of strong chaos in two-dimensional models is truncation-related, and can be delayed to arbitrarily large forcing if enough modes are included.