Predicting chaotic statistics with unstable invariant tori

TL;DR

This paper demonstrates that unstable invariant tori embedded in chaotic PDEs can be used to predict statistical properties of hyperchaotic systems, extending periodic orbit theory to higher-dimensional invariant sets.

Contribution

Introduces a novel numerical method to identify unstable invariant 2-tori in a chaotic PDE and shows their weighted averages approximate chaotic statistics.

Findings

Unstable invariant 2-tori can be numerically converged in a hyperchaotic PDE.

Weighted sums over these tori approximate the system's statistical properties.

The approach extends periodic orbit theory to higher-dimensional invariant sets.

Abstract

It has recently been speculated that statistical properties of chaos may be captured by weighted sums over unstable invariant tori embedded in the chaotic attractor of hyperchaotic dissipative systems; analogous to sums over periodic orbits formalized within periodic orbit theory. Using a novel numerical method for converging unstable invariant 2-tori in a chaotic PDE, we identify many quasiperiodic, unstable, invariant 2-torus solutions of a modified Kuramoto-Sivashinsky equation exhibiting hyperchaotic dynamics with two positive Lyapunov exponents. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori -- with weights computed based on their respective stability eigenvalues -- approximate statistics for the chaotic dynamics. These results are a step towards including higher-dimensional invariant sets in a generalized periodic orbit theory for hyperchaotic systems including spatio-temporally chaotic PDEs.