基于Lorenz系统的数值天气转折期预报理论探索

以Lorenz系统为研究对象,对数值天气转折期预报中的动力学特征进行了理论研究,通过对Lorenz系统平衡点稳定性的讨论,得到了区分准稳定区域和准不稳定区域的分界曲面,由此标定出准稳定区域和准不稳定区域.在准稳定区域,Lorenz曲线保持相对稳定,能够在该平衡点周围周期运动;在准不稳定区域,Lorenz曲线可能会从这个平衡点周围跃过分界曲面而进入另外一个平衡点周围,即发生突变,这是Lorenz系统的一个重要动力学特征;对数值天气转折期预报与气候突变检测、预测给出一种新理论和新方法.     

New interpretation and size of strange attractor of the Lorenz model of turbulence

Using the equivalence of the Lorenz equations with laser equations we show how the irregular jumping from one segment of the variable space to the other one can be understood. We also estimate the size of the Lorenz attractor.     

The real and complex Lorenz equations in rotating fluids and lasers

The Lorenz equations are derived systematically from amplitude equations of weakly nonlinear dispersively unstable physical systems near criticality when weak dissipation is added. This derivation is only valid if the undamped neutral curve is not destabilised by the addition of weak dissipation. The addition of extra weak dispersive effects make some of the coefficients complex and yields a complex set of Lorenz equations. Both sets of equations are derived in examples in laser optics and baroclinic instability.     

Studying the Lorenz equations with one-dimensional maps from successive local maxima inz

The Lorenz equations is studied with one-dimensional maps from successive local maxima inz. It is found that the periodic windows embedded in chaotic region of the Lorenz equations could be ordered in a systematic way according to symbolic dynamics of three letters.     

Experimental observation of Lorenz chaos in the Quincke rotor dynamics

In this paper, we report experimental evidence of Lorenz chaos for the Quincke rotor dynamics. We study the angular motion of an insulating cylinder immersed in slightly conducting oil and submitted to a direct current electric field. The simple equations which describe the dynamics of the rotor are shown to be equivalent to the Lorenz equations. In particular, we observe two bifurcations in our experimental system. Above a critical value of the electric field, the cylinder rotates at a constant rate. At a second bifurcation, the system becomes chaotic. The characteristic shape of the experimental first return map provides strong evidence for Lorenz-type chaos.     

Ludvig Lorenz and His Non-Maxwellian Electrical Theory of Light

Maxwell’s celebrated electromagnetic theory of light dates from 1865. Two years later, without appealing to the ether as a carrier of light waves, the Danish physicist Ludvig Lorenz (1829–1891) independently published another electrical theory of light based on optical equations and the novel idea of retarded potentials. In spite of resting on a very different conceptual foundation, Lorenz’s theory led to almost the same results as Maxwell’s. But whereas Maxwell’s field theory heralded a revolution in physics, Lorenz’s alternative was largely forgotten and soon relegated to a footnote in the history of physics. In part based on archival material and other sources in Danish, this paper offers a detailed contextual account of Lorentz’s theory and its reception in the physics community. Moreover, it includes a brief introduction to other of Lorenz’s scientific contributions and discusses the reasons why his electrical theory of light failed to attract serious interest.     

在非线性三波耦合的不稳定饱和过程中出现的Lorenz型混沌

我们证明了在等离子体中，三波耦合方程同构于Ｌｏｒｅｎｚ型方程，预言了在此过程中会出现Ｌｏｒｅｎｚ型混沌     

Chaos in the segmented disc dynamo

It is shown that the equations describing the segmented disc dynamo in the presence of friction are identical to the Lorenz equations.     

Lorenz-type chaotic pulsing of a 15NH3-laser

The dynamics of the ¹⁵NH₃ laser operating on the v₂ = 1; aR (4, 4) rotational transition at 153 μm is investigated experimentally. Self pulsing with period doubling and chaos is observed along with periodic windows. For resonant tuning, pulsing as expected for the Lorenz case is found. Phase space projections of the Lorenz type pulsing are given. All observations agree with the predictions of the Lorenz equations extended to allow for detuning.     

Octonionic Lorenz-like condition

In this study, the octonion algebra and its general properties are defined by the Cayley–Dickson’s multiplication rules for octonion units. The field equations, potential equations and Maxwell equations for electromagnetism are investigated with the octonionic equations and these equations can be compared with their vectorial representations. The potential and wave equations for fields with sources are also provided. By using Maxwell equations, a Lorenz-like condition is newly suggested for electromagnetism. The existing equations including the photon mass provide the most acknowledged Lorenz condition for the magnetic monopole and the source.     

Riddling and chaotic synchronization of coupled piecewise-linear Lorenz maps

We investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale, which has serious consequences on the predictability of the final state for such a coupled system. We found that there are wide parameter intervals for which two piecewise-linear Lorenz maps exhibit riddled basins (globally or locally), which indicates that there are riddled basins in coupled Lorenz equations, as previously suggested by numerical experiments. The use of piecewise-linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins.     

Hybrid TS fuzzy modelling and simulation for chaotic Lorenz system

The projection of the chaotic attractor observed from the Lorenz system in the $X$--$Z$ plane is like a butterfly, hence the classical Lorenz system is widely known as the butterfly attractor, and has served as a prototype model for studying chaotic behaviour since it was coined.In this work we take one step further to investigate some fundamental dynamic behaviours of a novel hybrid Takagi--Sugeno (TS) fuzzy Lorenz-type system, which is essentially derived from the delta-operator-based TS fuzzy modelling for complex nonlinear systems, and contains the original Lorenz system of continuous-time TS fuzzy form as a special case. By simply and appropriately tuning the additional parametric perturbations in the two-rule hybrid TS fuzzy Lorenz-type system, complex (two-wing) butterfly attractors observed from this system in the three dimensional (3D) $X$--$Y$--$Z$ space are created, which have not yet been reported in the literature, and the forming mechanism of the compound structures have been numerically investigated.     

Amplification of intrinsic fluctuations by the Lorenz equations

Macroscopic systems (e.g., hydrodynamics, chemical reactions, electrical circuits, etc.) manifest intrinsic fluctuations of molecular and thermal origin. When the macroscopic dynamics is deterministically chaotic, the intrinsic fluctuations may become amplified by several orders of magnitude. Numerical studies of this phenomenon are presented in detail for the Lorenz model. Amplification to macroscopic scales is exhibited, and quantitative methods (binning and a difference-norm) are presented for measuring macroscopically subliminal amplification effects. In order to test the quality of the numerical results, noise induced chaos is studied around a deterministically nonchaotic state, where the scaling law relating the Lyapunov exponent to noise strength obtained for maps is confirmed for the Lorenz model, a system of ordinary differential equations.     

Sequences of gluing bifurcations in an analog electronic circuit

We report on the experimental investigation of gluing bifurcations in the analog electronic circuit which models a dynamical system of the third order: Lorenz equations with an additional quadratic nonlinearity. Variation of one of the resistances in the circuit changes the coefficient at this nonlinearity and replaces the Lorenz route to chaos by a different scenario which leads, through the sequence of homoclinic bifurcations, from periodic oscillations of the voltage to the irregular ones. Every single bifurcation “glues” in the phase space two stable periodic orbits and creates a new one, with the doubled length: a sequence of such bifurcations results in the birth of the chaotic attractor.     

A simple Lorenz circuit and its radio frequency implementation

A remarkably simple electronic circuit design based on the chaotic Lorenz system is described. The circuit consists of just two active nonlinear elements (high-speed analog multipliers) and a few passive linear elements. Experimental implementations of the circuit exhibit the classic butterfly attractor and the hysteretic transition from steady state to chaos observed in the Lorenz equations. The simplicity of the circuit makes it suitable for radio frequency applications. The power spectrum of the observed oscillations displays a peak frequency as high as 930 kHz and significant power beyond 1 MHz.     

Limit Cycles near Stationary Points in the Lorenz System

The limit cycles in the Lorenz system near the stationary points are analysed numerically. A plane in phase space of the linear Lorenz system is used to locate suitable initial points of trajectories near the limit cycles. The numerical results show a stable and an unstable limit cycle near the stationary point. The stable limit cycle is smaller than the unstable one and has not been previously reported in the literature. In addition, all the limit cycles in the Lorenz system are theoreticallv Proven not to be planar.     

螺旋线行波管注波互作用时域理论

研究了螺旋线行波管中电子注与高频场互作用的时域理论.电子对场的作用由高频场方程和空间电荷场方程模拟,场对电子注的作用由运动方程模拟.在螺旋导电面模型下利用安培环路定理和法拉第电磁感应定律得到了时域高频场方程.利用空间电荷波模型处理空间电荷场,得到了空间电荷场方程.将高频场和空间电荷场代入洛伦兹力方程,得到了运动方程.利用耦合阻抗处理高频场方程的激励源,使得高频场方程的求解能够借助诸如HFSS或HFCS等高频模拟软件来实现,增强了时域理论的灵活性.基于上述理论,编写软件数值模拟某螺旋线行波管,验证了时域理论的可行性.     

A derivation of the Lorenz equations for some unstable dispersive physical systems

The amplitude equations which are valid in the neighbourhood of the bifurcation point of a class of dispersively unstable physical systems when small dissipation is included are shown to be transformable to the Lorenz equations. There is a strong connection with systems which yield equations solvable by the inverse scattering transform when damping is excluded and spatial variation included. Two examples are given in very brief detail: (1) a two-layer model for baroclinic instability, (2) the laser equations giving rise to the SIT equations.     

Circuit bounds on stochastic transport in the Lorenz equations

In turbulent Rayleigh–Bénard convection one seeks the relationship between the heat transport, captured by the Nusselt number, and the temperature drop across the convecting layer, captured by the Rayleigh number. In experiments, one measures the Nusselt number for a given Rayleigh number, and the question of how close that value is to the maximal transport is a key prediction of variational fluid mechanics in the form of an upper bound. The Lorenz equations have traditionally been studied as a simplified model of turbulent Rayleigh–Bénard convection, and hence it is natural to investigate their upper bounds, which has previously been done numerically and analytically, but they are not as easily accessible in an experimental context. Here we describe a specially built circuit that is the experimental analogue of the Lorenz equations and compare its output to the recently determined upper bounds of the stochastic Lorenz equations [1]. The circuit is substantially more efficient than computational solutions, and hence we can more easily examine the system. Because of offsets that appear naturally in the circuit, we are motivated to study unique bifurcation phenomena that arise as a result. Namely, for a given Rayleigh number, we find a reentrant behavior of the transport on noise amplitude and this varies with Rayleigh number passing from the homoclinic to the Hopf bifurcation.     

Stability analysis of two coupled Lorenz lasers and the coupling-induced periodic →chaotic transition

We have performed a linear stability analysis of two Lorenz lasers coupled by their electric fields and have shown that the bad cavity condition becomes a function of coupling and that a good cavity instability may occur if the injected fields are inverted before injection. In addition, we show that the symmetrically coupled Lorenz system is isomorphic to the original Lorenz system with new parameters. The stability analysis also predicts a lowering of the second laser threshold with coupling for both the chaotic and self-pulsing regimes. Numerical integration of the equations is in agreement with these predictions and has revealed a coupling induced transition from self-pulsing to chaotic behavior. The classification of the behavior of the coupled system in the parameter space of the coupling constants has been investigated and shows that the results of symmetric coupling allow enough of a margin for an experimental test of the theory. This would allow experimentalists to observe the actual Lorenz instability at excitations as low as 4–5 times above threshold.