Lorenz方程中两种尺度的相互作用

在Lorenz方程中的混沌区域，存在着两种尺度的相互作用.其中代表大尺度运动的Rayleigh数r决定空气中小尺度的对流和湍流运动，而对流和湍流运动又对大尺度运动具有反馈作用，两者的相互作用最终使r值减小，空气运动进入一种定常状态.在反馈过程中，垂直热通量xy及垂直温度梯度z的变化是决定r变化的主要原因，并且反馈的最终结果表现出突变的特征.     

Operator Derivation of the Gauge-Invariant Proca and Lehnert Equations; Elimination of the Lorenz Condition

Using covariant derivatives and the operator definitions of quantum mechanics, gauge invariant Proca and Lehnert equations are derived and the Lorenz condition is eliminated in U(1) invariant electrodynamics. It is shown that the structure of the gauge invariant Lehnert equation is the same in an O(3) invariant theory of electrodynamics.     

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.     

随机脉冲微分方程的p阶矩稳定性和参激白噪声作用下Lorenz系统的脉冲同步

考察了随机脉冲微分系统的p阶矩稳定性问题,在更符合脉冲系统一般假设的情况下,建立了条件更弱的随机脉冲微分系统p阶矩稳定性判定定理.并应用该判定定理,考察了参激白噪声作用下Lorenz系统的脉冲同步问题,证明了同步误差系统的p阶矩稳定性,从而说明在p阶矩的意义下,两个系统是可以用脉冲方法实现同步的.数值模拟验证了随机Lorenz系统脉冲同步的可行性. 关键词： 随机脉冲微分方程 p阶矩稳定性')" href="#">p阶矩稳定性 脉冲 同步     

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.     

Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model

The system of equations introduced by Lorenz to model turbulent convective flow is studied here for Rayleigh numbersr somewhat smaller than the critical value required for sustained chaotic behavior. In this regime the system is found to exhibit transient chaotic behavior. Some statistical properties of this transient chaos are examined numerically. A mean decay time from chaos to steady flow is found and its dependence uponr is studied both numerically and (very close to the criticalr) analytically.This work was supported in part by NASA grant NSG 5209; partial support of computer costs was provided by the University of Maryland-Baltimore County Computer Center.     

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.     

Oscillatory convection and chaos in a Lorenz-type model of a rotating fluid

A four-mode model of convection in a rotating fluid layer is studied. The model is an extension of the Lorenz model of Rayleigh-Bénard convection, the extra mode accounting for the regeneration of vorticity by rotation. Perturbation theory is applied to show that the Hopf bifurcations from conductive and steady convective solutions can be either supercritical or subcritical. Perturbation theory is also used at large Rayleigh numbersr to predict novel behavior. Supercritical oscillatory convection of finite amplitude is found by numerical integration of the governing equations. The general picture is of a series of oscillatory solutions stable over larger intervals, interspersed by short bursts of chaos.     

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.     

Karhunen–Loève decomposition of peripheral blood flow signal

The Karhunen–Loève expansion is applied to scalar signals and the effect of window length (t_w), time lag (τ) and embedding dimension (d) is analysed for periodic signals and for signals modeled by the Lorenz equations. For τ≠k/2f_i (f_i are characteristic frequencies of the signal, k is positive integer), we obtain 2m modes from an m-periodic signal. For a large set of parameters a finite number of modes was not obtained from the Lorenz system. It is further shown that, on the time scale of a minute, the peripheral blood flow signal contains oscillatory modes that occur in pairs thereby confirming that the blood flow through the cardiovascular system is oscillatory. Some of the difficulties of applying Karhunen–Loève expansion to scalar signals are pointed out.     

Spectral Resolution and Inversion of the Bloch Equations with Relaxation

It is demonstrated that the linear Bloch equations, describing near-resonant excitation of two-level media with relaxation, can be resolved into a 3n-dimensional nonlinear system associated with a special spectral problem, generalizing the classical Zakharov–Shabat spectral problem. Remarkably, for n = 1 it is the well-known Lorenz system, and for n > 1 several such systems coupled with each other in a manner dependant on the excitation pulse. The unstable manifold of a saddle equilibrium point in this ensemble characterizes possible excitations of the spins from the initial equilibrium state. This enables us to get a straightforward geometric extension of the inverse scattering method to the damped Bloch equations and hence invert them, i.e., design frequency selective pulses automatically compensated for the effect of relaxation. The latter are essential, for example, in nuclear magnetic resonance and extreme nonlinear optics.      

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.     

Analogy between the Lorenz strange attractor and a bistable stochastic oscillator

The relation between the aperiodic solution of the Lorenz model and that of a stochastic anharmonic oscillator is explored. The stochastic oscillator is constructed by replacing (t) in the Lorenz model by a stochastic variable(t) of specified statistics. The resulting system is of course not isomorphic to the Lorenz model, but does share with it a number of statistical properties. Thus, within the confines of these measures the two systems are physically very similar.     

T-points: A codimension two heteroclinic bifurcation

The local bifurcation structure of a heteroclinic bifurcation which has been observed in the Lorenz equations is analyzed. The existence of a particular heteroclinic loop at one point in a two-dimensional parameter space (a T point) implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits, as well as countably many other topologically more complicatedT points in a small neighborhood in parameter space.     

Why Does the Geometric Product Simplify the Equations of Physics?

In the last decades it was observed that Clifford algebras and geometric product provide a model for different physical phenomena. We propose an explanation of this observation based on the theory of bounded symmetric domains and the algebraic structure associated with them. The invariance of physical laws is a result of symmetry of the physical world that is often expressed by the symmetry of the state space for the system implying that this state space is a symmetric domain. For example, the ball of all possible velocities is a bounded symmetric domain. The symmetry on this ball follow from the symmetry of the space-time transformations between two inertial systems, which fixes the so-called symmetric velocity between them. The Lorenz transformations acts on the ball Sof symmetric velocities by conformal transformations. The ball Sis a spin ball (type IV in Cartan's classification). The Lie algebra of this ball is defined a triple product that is closely related to geometric product. The relativistic dynamic equations in mechanics and for the Lorenz force is described by this Lie algebra and the triple product.     

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.     

Hysteresis in the Lorenz equations

The phenomenon of hysteresis, associated with multiple stable solution behaviours, has been predicted and observed in the Lorenz equations when r and σ are large.     

Master-slave synchronization and the Lorenz equations

Since the seminal remark by Pecora and Carroll [Phys. Rev. Lett. 64, 821 (1990)] that one can synchronize chaotic systems, the main example in the related literature has been the Lorenz equations. Yet this literature contains a mixture of true and false, and of justified and unsubstantiated claims about the synchronization properties of the Lorenz equations. In this note we clarify some of the confusion. (c) 1997 American Institute of Physics.     

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.