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 basins cover a full Lebesgue measure subset of points in the topological basin of attraction. Here we prove that the SRB measures depend continuously on the dynamics in the weak$^*$ topology.
We present an example of a new strange attractor which, as we show, belongs to a class of wild pseudohyperbolic spiral attractors. We find this attractor in a four-dimensional system of differential equations which can be represented as an extension of the Lorenz system.
Cascades of Lorenz attractors in the Shimizu-Morioka model
National Research University Higher School of Economics（国立高等教育经济学院）；Lobachevsky State University of Nizhny Novgorod（下诺夫哥罗德国立罗巴切夫斯基大学）；Georgia State University（佐治亚州立大学）
In this note, we consider the thermodynamic formalism for Lorenz attractors of flows in any dimension. Under a mild condition on the Hölder continuous potential function $ϕ$, we prove that for an open and dense subset of $C^1$ vector fields, every Lorenz attractor supports a unique equilibrium state. In particular, we obtain the uniqueness for the measure of maximal entropy.
Lorenz attractors play an important role in the modern theory of dynamical systems. The reason is that they are robust, i.e. preserve their chaotic properties under various kinds of perturbations. This means that such attractors can exist in applied models and be observed in experiments. It is known that discrete Lorenz attractors can appear in local and global bifurcations of multidimensional diffeomorphisms. However, to date, only partial cases were investigated. In this paper bifurcations of homoclinic and heteroclinic cycles with quadratic tangencies of invariant manifolds are studied. A f…
Homoclinic tangencies to resonant saddles and discrete Lorenz attractors
研究了一类二次同宿切线到鞍不动点的三参数一般开折中周期轨道的分支。我们应用重新标度技术的第一次返回（庞加莱）的地图，并表明，重新标度的地图可以带来一个地图渐近接近的三维Henon地图$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$，如已知的，表现出野生双曲洛伦兹吸引子在一些开放的参数域。在此基础上，我们证明了类Lorenz吸引子的无穷级联的存在性。
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 robust, in the sense that they are generally not destroyed by small perturbations (autonomous, non-autonomous, stochastic). This allows us to be sure that the observed in the experiment object is exactly the chaotic attractor, rather than a long-time periodic orbit. Discrete-time analogs of the Lorenz attractor possess even more complicated structure -- they…
We define an extension of the geometric Lorenz model, defined on the three sphere. This geometric model has an invariant one dimensional trefoil knot, a union of invariant manifolds of the singularities. It is similar to the invariant trefoil knot arising in the classical Lorenz flow near the classical parameters. We prove that this geometric model is topologically equivalent to the geodesic flow on the modular surface, once compactifying the latter.
In this article we consider the ergodic optimization for hyperbolic flows and Lorenz attractors with respect to both continuous and Holder continuous observables. In the context of hyperbolic flows we prove that a Baire generic subset of continuous observables have a unique maximizing measure, with full support and zero entropy, and that a Baire generic subset of Holder continuous observables admit a unique and periodic maximizing measure. These results rely on a relation between ergodic optimization for suspension semiflows and ergodic optimization for the Poincaré map with respect to induced…
We prove that heterodimensional cycles can be created by unfolding a pair of homoclinic tangencies in a certain class of C-infinity diffeomorphisms. This implies the existence of a C2- open domain in the space of dynamical systems with a certain type of symmetry where systems with heterodimensional cycles are dense in C-infinity. In particular, we describe a class of three-dimensional flows with a Lorenz-like attractor such that an arbitrarily small perturbation of any such flow can belong to this domain - in this case the corresponding heterodimensional cycles belong to a chain-transitive att…
In this article we construct the parameter region where the existence of a homoclinic orbit to a zero equilibrium state of saddle type in the Lorenz-like system will be analytically proved in the case of a nonnegative saddle value. Then, for a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the detected parameter region is carried out to discover several new interesting bifurcation scenarios.
Unidirectionally coupled Lorenz systems in which the drive possesses a chaotic attractor and the response admits two stable equilibria in the absence of the driving is under investigation. It is found that double chaotic attractors coexist in the dynamics. The approach is applicable for chains of coupled Lorenz systems. The existence of four chaotic attractors in three coupled Lorenz systems is also demonstrated.
Computing geometric Lorenz attractors with arbitrary precision
Lorenz吸引子是由E. N. Lorenz是\emph{strange attractors}的第一个例子。然而，洛伦兹的研究主要是基于（非严格）数值模拟，直到最近，证明存在的洛伦兹吸引仍然难以捉摸。为了解决这个问题，一些作者引入了几何洛伦兹模型，并证明了几何洛伦兹模型有一个奇怪的吸引子。在2002年，原始的洛伦兹模型表现得像一个几何洛伦兹模型，因此有一个奇怪的吸引子。在本文中，我们表明几何洛伦兹吸引子是可计算的，以及它们的物理措施。
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 families of such maps. We pay especially our attention to such a scenario that can lead to period-2 Lorenz-like attractors. These attractors have very interesting dynamical properties and we show that their crises can lead, in turn, to the emergence of pseudohyperbolic discrete Lorenz shape attractors of new types. We also show examples of all these attractors in …
We consider one parameter families of vector fields introduced by Rovella, obtained through modifying the eigenvalues of the geometric Lorenz attractor, replacing the expanding condition on the eigenvalues of the singularity by a contracting one. We show that there is no statistical stability within the set of parameters for which there is a physical measure supported on the attractor. This is achieved obtaining a similar conclusion at the level of the corresponding one-dimensional contracting Lorenz maps.
It is known that nonuniformly hyperbolic maps admitting singularities have at most countably many ergodic Sinai-Ruelle-Bowen (SRB) measures. These maps include the Belykh attractor, the geometric Lorenz attractor, and more general Lorenz-type systems. In this paper, we establish easily verifiable sufficient conditions guaranteeing that the number of ergodic SRB measures is at most finite, and provide examples and nonexamples showing that the conditions are necessary in general.
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 $C^2$ vector fields using known results, providing robust examples of statistically stable singular attracting sets (encompassing in particular the Lorenz and geometrical Lorenz attractors). These conditions are shown to hold also on the persistent but non-robust family of contracting Lorenz flows (also known as Rovella attractors), providing examples of statistical…
In hyperbolic dynamics, a well-known result is that every hyperbolic attracting set, have a finite pseudo-orbit tracing property (FPOTP). It's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics; Komuro in [Lorenz attractors do not have the pseudo-orbit tracing property], provides a negative answer for this question, by proving that the geometric Lorenz Attractor doesn't have a FPOTP. In this paper, we generalized the result of Komuro, we prove that every codimension one sectional-hyperbolic attractor set with a unique singularity Lorenz-like, which is of bounda…
Singular hyperbolicity is a weakened form of hyperbolicity that has been introduced for vector fields in order to allow non-isolated singularities inside the non-wandering set. A typical example of a singular hyperbolic set is the Lorenz attractor. However, in contrast to uniform hyperbolicity, singular hyperbolicity does not immediately imply robust topological properties, such as the transitivity. In this paper, we prove that open and densely inside the space of $C^1$ vector fields of a compact manifold, any singular hyperbolic attractors is robustly transitive.
Statistical properties of Lorenz like flows, recent developments and perspectives
我们评论的统计行为的洛伦兹方程及其吸引子的数学结果，更一般的奇异双曲系统的类。这种系统的数学理论原来是令人惊讶的困难。值得注意的是，洛伦兹吸引子存在的严格证明仅在2000年左右提出，计算机辅助证明以及扩展的双曲理论发展到包括吸引子鲁棒包含平衡。我们提出了一些主要的结果，对这样的系统的pacitcal行为。我们发现，对于三维流的吸引子，鲁棒混沌行为等价于某些双曲结构的存在，称为奇异双曲性。这些结构，反过来，与物理措施的存在：\emph{in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure}。然后，我们给出了更多的细节奇异双曲（洛伦兹）吸引子的动力学最近的结果。
