The Lorenz attractor was introduced in 1963 by E. N. Lorenz as one of the first examples of \emph{strange attractors}. However Lorenz' research was mainly based on (non-rigourous) numerical simulations and, until recently, the proof of the existence of the Lorenz attractor remained elusive. To address that problem some authors introduced geometric Lorenz models and proved that geometric Lorenz models have a strange attractor. In 2002 it was shown that the original Lorenz model behaves like a geometric Lorenz model and thus has a strange attractor. In this paper we show that geometric Lorenz attractors are computable, as well as their physical measures.
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.
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 …
This study introduces a modified quadratic Lorenz attractor. The properties of this new chaotic system are analysed and discussed in detail, by determining the equilibria points, the eigenvalues of the Jacobian, and the Lyapunov exponents. The numerical simulations, the time series analysis, and the projections to the $xy$-plane, $xz$-plane, and $yz$-plane are conducted to highlight the chaotic behaviour. The multiplicative form of the new system is also presented and the simulations are conducted using multiplicative Runge-Kutta methods.
In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using t…
We give an analytic (free of computer assistance) proof of the existence of a classical Lorenz attractor for an open set of parameter values of the Lorenz model in the form of Yudovich-Morioka-Shimizu. The proof is based on detection of a homoclinic butterfly with a zero saddle value and rigorous verification of one of the Shilnikov criteria for the birth of the Lorenz attractor; we also supply a proof for this criterion. The results are applied in order to give an analytic proof of the existence of a robust, pseudohyperbolic strange attractor (the so-called discrete Lorenz attractor) for an o…
We prove, by employing rigorous numerics, that Shimizu-Morioka system has a Lorenz attractor for an open set of parameter values. Using this result, we prove the existence of a discrete version of the Lorenz attractor for a class of three-dimensional polynomial diffeomorphisms.
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.
We study chaotic dynamics in nonholonomic model of Celtic stone. We show that, for certain values of parameters characterizing geometrical and physical properties of the stone, a strange Lorenz-like attractor is observed in the model. We study also bifurcation scenarios for appearance and break-down of this attractor.
In this paper, we show that geometric Lorenz attractors have Hausdorff dimension strictly greater than $2$. We use this result to show that for a "large" set of real functions the Lagrange and Markov Dynamical spectrum associated to these attractors has persistently non-empty interior.
We prove exponential decay of correlations for a class of $C^{1+α}$ uniformly hyperbolic skew product flows, subject to a uniform nonintegrability condition. In particular, this establishes exponential decay of correlations for an open set of geometric Lorenz attractors. As a special case, we show that the classical Lorenz attractor is robustly exponentially mixing.
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.
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 also obtain the functional central limit theorem and moment estimates, as well as iterated versions of these results. A consequence is deterministic homogenisation (convergence to a stochastic differential equation) for fast-slow dynamical systems whenever the fast dynamics is singularly hyperbolic of codimension two.
We study the non-wandering set of contracting Lorenz maps. We show that if such a map $f$ doesn't have any attracting periodic orbit, then there is a unique topological attractor. Precisely, there is a compact set $Λ$ such that $ω_f(x)=Λ$ for a residual set of points $x \in [0,1]$. Furthermore, we classify the possible kinds of attractors that may occur.
We study a class of geometric Lorenz flows, introduced independently by Afraimovic, Bykov & Sil'nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.
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 the whole attractor, which is hyperbolic and the equilibrium state with respect to the center-unstable Jacobian; the hitting time associated to a geometric Lorenz attractor satisfies a logarithm law; the rate of large deviations for the physical measure on the ergodic basin of a geometric Lorenz attractor is exponential.
In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.
We prove that every geometric Lorenz attractor has superpolynomial decay of correlations with respect to the unique SRB measure. Moreover, we prove the Central Limit Theorem and Almost Sure Invariance Principle for the time-1 map of the flow of geometric Lorenz attractors. In particular, our results apply to the classical Lorenz attractor.
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…
We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map $\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$ which, as known, exhibits wild hyperbolic Lorenz-like attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of Lorenz-like attractors.
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…
The exact Lyapunov dimension formula for the Lorenz system has been analytically obtained first due to G.A.Leonov in 2002 under certain restrictions on parameters, permitting classical values. He used the construction technique of special Lyapunov-type functions developed by him in 1991 year. Later it was shown that the consideration of larger class of Lyapunov-type functions permits proving the validity of this formula for all parameters of the system such that all the equilibria of the system are hyperbolically unstable. In the present work it is proved the validity of the formula for Lyapun…
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…
We give the first examples of flows which exhibit robust singular attractors containing a wild hyperbolic set (in the sense of Newhouse). A hyperbolic set is said to be wild, if it has tangencies between its stable and unstable manifolds, in a robust way. The only restriction on the ambient manifold is that its dimension should be at least 5.
It was established in 2006 that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is non-simple.
This paper reports the finding of a simple one-parameter family of three-dimensional quadratic autonomous chaotic systems. By tuning the only parameter, this system can continuously generate a variety of cascading Lorenz-like attractors, which appears to be richer than the unified chaotic system that contains the Lorenz and the Chen systems as its two extremes. Although this new family of chaotic systems has very rich and complex dynamics, it has a very simple algebraic structure with only two quadratic terms (same as the Lorenz and the Chen systems) and all nonzero coefficients in the linear …
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.
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…
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 the time variable in the three dimensional Lorenz system to be complex, hoping that solutions that have resisted analysis on the real line will give up their secrets in the complex plane. Knowledge of singularities being fundamental to any investigation in the complex plane, we build upon earlier work and give a complete and consistent formal development of …
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.
This study covers an analytical approach to calculate positively invariant sets of dynamical systems. Using Lyapunov techniques and quantifier elimination methods, an automatic procedure for determining bounds in the state space as an enclosure of attractors is proposed. The available software tools permit an algorithmizable process, which normally requires a good insight into the systems dynamics and experience. As a result we get an estimation of the attractor, whose conservatism only results from the initial choice of the Lyapunov candidate function. The proposed approach is illustrated on …
We comment on mathematical results about the statistical behavior of Lorenz equations an its attractor, and more generally to 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 statisitcal behavior of such sy…
A novel cryptography method based on the Lorenz's attractor chaotic system is presented. The proposed algorithm is secure and fast, making it practical for general use. We introduce the chaotic operation mode, which provides an interaction among the password, message and a chaotic system. It ensures that the algorithm yields a secure codification, even if the nature of the chaotic system is known. The algorithm has been implemented in two versions: one sequential and slow and the other, parallel and fast. Our algorithm assures the integrity of the ciphertext (we know if it has been altered, wh…
The classical Lorenz system is considered. For many years, this system has been the subject of study by numerous authors. However, until now the structure of the Lorenz attractor is not clear completely yet, and the most important question at present is to understand the bifurcation scenario of chaos transition in this system. Using some numerical results and our bifurcational geometric approach, we present a new scenario of chaos transition in the classical Lorenz system.
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…
This paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems, such as the Lorenz attractor. The algorithms compute the derivative of long time averaged "statistical" quantities to infinitesimal perturbations of the system parameters. The algorithms are demonstrated on the Lorenz attractor. We show that sensitivity derivatives of statistical quantities can be accurately estimated using a single, short trajectory (over a time interval of 20) on the Lorenz attractor.
In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has a positive maximal Lyapunov exponents and this property is robust, i.e. it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, su…
A symmetric Lorenz map is obtain by ``flipping'' one of the two branches of a symmetric unimodal map. We use this to derive a Sharkovsky-like theorem for symmetric Lorenz maps, and also to find cases where the unimodal map restricted to the critical omega-limit set is conjugate to a Sturmian shift. This has connections with properties of unimodal inverse limit spaces embedded as attractors of some planar homeomorphisms.
