相关论文: Statistical properties of Lorenz like flows, recen…

We describe some recent results on the dynamics of singular-hyperbolic (or Lorenz-like) attractors: attractors in this class are expansive and so sensitive with respect to initial data; they admit a unique physical measure whose support is…

A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a…

After reviewing known results on sensitiveness and also on robustness of attractors together with observations on their proofs, we show that for attractors of three-dimensional flows, robust chaotic behavior meaning sensitiveness to initial…

For geometric Lorenz attractors (including the classical Lorenz attractor) we obtain a greatly simplified proof of the central limit theorem which applies also to the more general class of codimension two singular hyperbolic attractors. We…

Over the last 10 years or so, advanced statistical properties, including exponential decay of correlations, have been established for certain classes of singular hyperbolic flows in three dimensions. The results apply in particular to the…

We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two strong different senses. Firstly, the flow is expansive: if two points remain close for all times, possibly with time reparametrization, then their…

We present criteria for statistical stability of attracting sets for vector fields using dynamical conditions on the corresponding generated flows. These conditions are easily verified for all singular-hyperbolic attracting sets of $C^2$…

We study geometrical and dynamical properties of the so-called discrete Lorenz-like attractors, that can be observed in three-dimensional diffeomorphisms. We propose new phenomenological scenarios of their appearance in one parameter…

Dynamics arising persistently in smooth dynamical systems ranges from regular dynamics (periodic, quasiperiodic) to strongly chaotic dynamics (Anosov, uniformly hyperbolic, nonuniformly hyperbolic modelled by Young towers). The latter…

The Lorenz attractor is one of the best known examples of applied mathematics. However, much of what is known about it is a result of numerical calculations and not of mathematical analysis. As a step toward mathematical analysis, we allow…

We provide conditions on the coupling function such that a system of 4 globally coupled identical oscillators has chaotic attractors, a pair of Lorenz attractors or a 4-winged analogue of the Lorenz attractor. The attractors emerge near the…

It is known that sectional-hyperbolic attracting sets, for a $C^2$ flow on a finite dimensional compact manifold, have at most finitely many ergodic physical invariant probability measures. We prove an upper bound for the number of distinct…

We consider the robust family of Geometric Lorenz attractors. These attractors are chaotic in the sense that they are transitive and have sensitive dependence on the initial conditions. Moreover, they support SRB measures whose ergodic…

An {\em attractor} is a transitive set of a flow to which all positive orbit close to it converges. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central…

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures…

We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been…

We prove the results in [1] using Theorem 1 of the recent paper [2] by Crovisier and Yang. References: [1] Arbieto, A., Rojas, C., Santiago, B., Existence of attractors, homoclinic tangencies and singular-hyperbolicity for flows,…

The Lorenz '96 model is an adjustable dimension system of ODEs exhibiting chaotic behavior representative of dynamics observed in the Earth's atmosphere. In the present study, we characterize statistical properties of the chaotic dynamics…

We prove that a partially hyperbolic attracting set for a C2 vector field, having slow recurrence to equilibria, supports an ergodic physical/SRB measure if, and only if, the trapping region admits non-uniform sectional expansion on a…

Lorenz attractors are important objects in the modern theory of chaos. The reason from one side is that they are met in various natural applications (fluid dynamics, mechanics, laser dynamics, etc.). At the same time, Lorenz attractors are…