On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent α > 1 of certain monotone combinatorial types. We prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated two-dimensional strong unstable manifold. For monotone families of Lorenz maps we prove that each infinitely renormalizable combinatorial type has a unique representative within the family. We also prove that each infinitely renormalizable map has no wandering intervals, is ergodic, and has a uniquely ergodic minimal Cantor attractor of measure zero.     

Bounds for the chaotic region in the Lorenz model

In a previous paper, the authors made an extensive numerical study of the Lorenz model, changing all three parameters of the system. We conjectured that the region of parameters where the Lorenz model is chaotic is bounded for fixed r. In this paper, we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using Fenichel theory. The theoretical bounds are complemented with numerical studies performed using the Maximum Lyapunov Exponent and OFLI2 techniques, and a comparison of both sets of results is shown. Finally, we provide a complete three-dimensional model of the chaotic regime depending on the three parameters.     

Statistical dynamics of the Lorenz model

The dynamics of the Lorenz model in the turbulent regime (r>r _T is investigated by applying methods for treating many-body systems. Symmetry properties are used to derive relations between correlation functions. The basic ones are evaluated numerically and discussed for several values of the parameterr. A theory for the spectra of the two independent relaxation functions is presented using a dispersion relation representation in terms of relaxation kernels and characteristic frequencies. Their role in the dynamics of the system is discussed and it is shown that their numerical values increase in proportion to r. The approximation of the relaxation kernels that represent nonlinear coupling between the variables by a relaxation time expression and a simple mode coupling approximation, respectively, is shown to explain the two different fluctuation spectra. The coupling strength for the modes is determined by a Kubo relation imposing selfconsistency. Comparison with the experimental spectra is made for three values ofr.     

On the statistical dynamics of the Lorenz model

We use the theory of stochastic differential equations with rapidly fluctuating coefficients to study the statistical dynamics of the Lorenz model in the turbulent region. On the assumption that the system is ergodic we are able to calculate self-consistently several basic statistical quantities in terms of the parameters of the model. Our results are in good agreement with numerical computations.Supported financially by the Summer Study Program in Geophysical Fluid Dynamics at the Woods Hole Oceanographic Institution.     

Ergodicity of randomly perturbed Lorenz model

We show, by using the Liapunov method, that the Lorenz model perturbed by Gaussian white noise is ergodic for any Rayleigh number. Our theory confirms two properties which have been found by numerical calculation. We also discuss the ergodicity of some other randomly perturbed dissipative systems, a one-dimensional laser, and a homopolar disk dynamo model of the geomagnetism.     

Chaos and limit cycles in the Lorenz model

The Lorenz model is interpreted as a damping motion under a time-dependent force. The range of the Rayleigh number r in which limit cycles exist is studied by numerical simulation. The shape of the limit cycle is given.     

Hyperbolicity and Astigmatism

We study the mechanism of hyperbolicity in high-dimensional Hamiltonian systems. Especially we consider ergodic billiards with focusing components in dimensions d3. In this case astigmatism serves as an obstacle to hyperbolicity in billiards with large focusing components. The notion of absolutely focusing mirrors is extended to the dimensions d3 and the first classes of ergodic billiards with both focusing and dispersing components are constructed in d3.     

Transient behavior in periodic regions of the Lorenz model

It is shown by numerical simulation that in a transient stage trajectories in phase space are unstable also in the range of r in which a stable limit cycle exists.     

A Feigenbaum sequence of bifurcations in the Lorenz model

For some high values of the Rayleigh numberr, the Lorenz model exhibits laminar behavior due to the presence of a stable periodic orbit. A detailed numerical study shows that, forr decreasing, the turbulent behavior is reached via an infinite sequence of bifurcations, whereas forr increasing, this is due to a collapse of the stable orbit to a hyperbolic one. The infinite sequence of bifurcations is found to be compatible with Feigenbaum's conjecture.     

The Lorenz model and the method of Carleman embedding

The Lorenz model is investigated by means of the Carleman embedding. Attention is focused on the analytical properties of the Laplace transform of a trajectory. It is shown that beyond the turbulent threshold finite linear embedding violates basic analytical requirements.     

Probability distribution for the number of cycles between successive regime transitions for the Lorenz model

The Lorenz model has been widely used for exploring many real world problems. In this paper we obtain, with the help of an invariant manifold technique, the return map for the maximum value of the variable x of the model and use this return map to derive the simple, empirically obtained, regime transition rules for forecasting regime changes and length in the new regime for the model. The probability distribution for number of cycles between successive regime transitions of the Lorenz model may be of interest in many disciplines. We apply the Perron-Frobenius algorithm over the return map to estimate the probability distribution for the number of cycles between successive regime transitions. These probabilities are also estimated for the forced Lorenz model, which is a conceptual model to explore the effects of sea surface temperature on seasonal rainfall.     

Analytical investigation of the hopf bifurcation in the Lorenz model

It is shown analytically that the bifurcation is subcritical always in the parameter range b > 0 and σ > b + 1, where b denotes the aspect ratio and σ the Prandtl number. The calculation is based on the centre manifold concept and makes use of the averaging method.     

Transition between turbulent and periodic states in the Lorenz model

The difference among turbulent states, which appear alternating with periodic states as the Rayleigh number r increases, is investigated by numerical simulation. The transition between turbulent and periodic states is also studied.     

外强迫对Lorenz系统初值可预报性的影响

利用强迫Lorenz模型, 研究了外强迫对Lorenz系统混沌性质、 映射结构及初值可预报性的影响, 并以海表温度为大气运动的外强迫, 用实际大气海洋资料分析了外强迫对大气可预报性的影响. 结果发现, 系统混沌现象的出现与外强迫有关, 外强迫改变了Lorenz系统的运动规律, 使围绕两奇怪吸引子运动的随机性减少. 考虑外强迫后, 系统运动轨迹的概率密度函数呈不对称的双峰结构, 且Lorenz映射由无外强迫时的一个尖点分离为两个尖点, 尖点的偏离方向和偏离位置分别与外强迫的正负和大小有关. 外强迫可减小Lorenz系统对初值的敏感性, 提高系统的初值可预报性, 尤其是外强迫越大, 可预报性提高的幅度也越大. 这些结果在不同强度海表温度强迫下的实际大气可预报性分析中得到了证实, 即海温异常越大, 实际大气变量的可预报性也越大.     

Semi-Focusing Billiards: Hyperbolicity

In this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of ℝ³. We also prove that a related class of semi-focusing billiards has mixed dynamics, i.e., their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other. These billiards are the first rigorous examples of billiards in domains of ℝ³ with divided phase space. The first author was partially supported by the NSF grant #0140165 and the Humboldt Foundation. The second author was partially supported by the FCT (Portugal) through the Program POCTI/FEDER.     

Analytic form of the simplest limit cycle in the Lorenz model

A method to obtain the analytic form of limit cycles is presented from a viewpoint of constants of motion in dissipative systems. The method is applied to the Lorenz equation in the high-Rayleigh-number limit. The analytic form of the simplest limit cycle is obtained, which is in a good agreement with the result of computer simulation.     

Structure-Preserving Algorithms for the Lorenz System

Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville＇s formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge-Kutta （RK4） method and the fifth-order Runge-Kutta-Fehlberg （RKF45） method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space axe less than those of the Runge-Kutta methods.     

A computational strategy for multiscale systems with applications to Lorenz 96 model

Numerical schemes for systems with multiple spatio-temporal scales are investigated. The multiscale schemes use asymptotic results for this type of systems which guarantee the existence of an effective dynamics for some suitably defined modes varying slowly on the largest scales. The multiscale schemes are analyzed in general, then illustrated on a specific example of a moderately large deterministic system displaying chaotic behavior due to Lorenz. Issues like consistency, accuracy, and efficiency are discussed in detail. The role of possible hidden slow variables as well as additional effects arising on the diffusive time-scale are also investigated. As a byproduct we obtain a rather complete characterization of the effective dynamics in Lorenz model.