Correlation functions in chaotic systems from periodic orbits

Linear response for macroscopic observables in high-dimensional systems

The long-term average response of observables of chaotic systems to dynamical perturbations can often be predicted using linear response theory, but not all chaotic systems possess a linear response. Macroscopic observables of complex dissipative chaotic systems, however, are widely assumed to have a linear response even if the microscopic variables do not, but the mechanism for this is not well-understood. We present a comprehensive picture for the linear response of macroscopic observables in high-dimensional coupled deterministic dynamical systems, where the coupling is via a mean field and the microscopic subsystems may or may not obey linear response theory. We derive stochastic reductions of the dynamics of these observables from statistics of the microscopic system and provide conditions for linear response theory to hold in finite dimensional systems and in the thermodynamic limit. In particular, we show that for large systems of finite size, the linear response is induced via self-generated noise. We present examples in the thermodynamic limit where the macroscopic observable satisfies linear response theory (LRT), although the microscopic subsystems individually violate LRT, as well as a converse example where the macroscopic observable does not satisfy LRT despite all microscopic subsystems satisfying LRT when uncoupled. This latter, maybe surprising, example is associated with emergent nontrivial dynamics of the macroscopic observable. We provide numerical evidence for our results on linear response as well as some analytical intuition.

Abnormal mixing of passive scalars in chaotic flows

We study the relaxation of a passive scalar towards the uniform equilibrium distribution in an advection-diffusion problem where the phase space for the pure advection problem is a mixture of chaotic domains and elliptic islands. Since the advection-diffusion problem is linear, the relaxation can be characterized by the eigenvalues and eigenmodes of the evolution operator. Almost degenerate eigenvalues then give rise to deviations from simple exponential decay behavior. We show by example that the corresponding eigenmodes can be supported by islands or weakly connected chaotic domains. These theoretical considerations are related to some experimental observations in two-dimensional flows.