Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

Abstract

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria.We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure.We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenzlike) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz 1430028-1 Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by THE UNIVERSITY OF WAIKATO on 11/16/14. For personal use only. V. Araujo et al.attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

Exact Recovery of Chaotic Systems from Highly Corrupted Data

“…For the Lorenz equations (3.3), and more generally, for any so-called geometric Lorenz-like system (see [AGP14] for definition and properties) of a flow on a three-dimensional manifold, the ergodic basin B(µ) covers a full Lebesgue measure subset of the topological basin of attraction Λ. Property (3.4) of a physical measure shows that asymptotically, the time average of a continuous observable of the flow equals its space average.…”

“…Explicitly when the underlying attractor of the flow has Hausdorff dimension greater than two, the flow satisfies some mixing properties and the governing equation vector f has a sparse representations in the space of multivariable polynomials, then the polynomial coefficients of f as well as the outlier vectors can be exactly recovered as the unique solution to a partial 1 -minimization problem with high probability (depends on the number of measurements), as long as the number of measurements is big enough and sparse level is low enough. We prove theoretical reconstruction guarantees by combining the partial sparse recovery results [BSV11] with statistical behavior of the Lorenzlike systems [AGP14,AMV15]. It is based on the observation that although individual trajectories are highly unpredictable due to the sensitivity property to initial conditions of chaotic dynamic systems, their statistical behavior is understandable and share many of the same properties of random sequences.…”

Optimisation of chaotically perturbed acoustic limit cycles

“…It is chaotic, i.e. it produces unpredictable dynamics; and for the range of parameters we use, it is quasi-hyperbolic [65,66], which means that it behaves like a hyperbolic system for the purpose of this analysis [53,67]. These two conditions fulfil the chaotic hypothesis 2 , which holds in high-dimensional fluids systems.…”

On the topology of the Lorenz system

“…The singularities prevent the attractor from being hyperbolic [3,21], and are the main reason for the instability of the system [21]. The dynamics slow down near the singularity and this is the main difficulty in numerical analysis of the flow [2].…”