Lorenz model

Figure 2.. Polarized reflectivity spectra of p-BEDO-TTF)5[CsHg(SCN)4]2 for E 1 L and E L at 300, 200, 100 and 10 K. (L is BEDO-TTF stack direction). The fit with Drude-Lorenz model for T=10 K is shown by thin solid line.

This result may be expressed in the more customary units of cubic meters or cubic nanometers by dividing by dTiSp. Thus, Op is equal to 1.472 x 10 nm at 25°C. When the calculation is repeated at 50°C the result is p = 1.471 x 10 nm. One expects the polarizability to be independent of temperature in a range where the electrons in the molecule remain in the same molecular orbitals. The small change in the polarizability reflects the weakness of the Lorentz-Lorenz model, which is based on continuum concepts. However, the estimated change is small, so that one may assume that the model is reasonably good. 

In his paper on "Deterministic Nonperiodic Flow" [9], F.N. Lorenz studies the behavior of the groimd atmosphere. The paper did not attract much attention at the time of its publication. Lorenz models the atmospheric behavior via nonlinear equations using three variables ... 

Guidi, G.M., J. Halloy A. Goldbeter. 1995. Chaos suppression by periodic forcing Insights from Dictyostelium cells, from a multiply regulated biochemical system, and from the Lorenz model. In Chaos and Complexity. J. Trfin Thanh Vin et al, eds. Editions Frontieres, Gif-sur-Yvette, France, pp. 135-46. 

CMLL Clansins-Mossotti/Lorentz-Lorenz (model)... 

Although the Lorenz model is not a model of chemical kinetics, there is some similarity in both model types the right-hand side is of the polynomial type with first- and second-order terms. In this chapter, we will present results of the analysis of a nonlinear model—also with three variables— the catalytic oscillator model. 

Yorke, J. A., Yorke, E. D. (1979) Metastable chaos The transition to sustained chaotic behavior in the Lorenz model. J. Stat. Phys. 21, 263... 

Fig. 8.5. The product of the total rate of dissipation times temperature (solid line) in Js and the time derivative of excess work (dashed line) vs. time in the following processes for the Lorenz model (a) Gravity is initially set in the direction along which the temperature decreases, and the system is at a stable motionless conductive stationary state at t = 0, invert the direction of gravity the motionless conductive state becomes unstable and the system approaches the convective stationary state, (b) The reverse process. The temperature difference is AT = 4K for both cases...

The theoretical results, based on the Lorenz model, agree with experiments qualitatively in that the total excess work change is of the same order of magnitude as the heat release measured in the experiments, and this is a major confirmation of our theory. 

There may be several reasons for a lack of quantitative agreement. First, the convective stationary state in the theory is a focus, not a node. (A node is approached with an eigenvalue that is real and negative and hence provides for a damped monotonic approach, whereas a focus is approached with a complex eigenvalue with the real part negative, that is a damped oscillatory approach.) In the experiments, however, the convective stationary state is a node due to the rigid boundaries. Second, because of the truncation to the first order in Fourier modes in the Lorenz model, this model can be a good approximation... 

The Lorentz-Lorenz model is obtained when the host material is chosen as air or vacuum (eh=l). The EMA equation reduces to ... 

In the Lorenz model, the saddle value is positive for the parameter values corresponding to the homoclinic butterfly. Therefore, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely the unstable manifold of the other one, and vice versa. The occurrence of such an intersection leads, in turn, to the existence of a hyperbolic limit set containing transverse homoclinic orbits, infinitely many saddle periodic orbits and so on [1]. In the case of a homoclinic butterfly without symmetry there is also a region in the parameter space for which such a rough limit set exists [1, 141, 149]. However, since this limit set is unstable, it cannot be directly associated with the strange attractor — a mathematical image of dynamical chaos in the Lorenz equation. 

Case (a) corresponds to a codimension-three bifurcation, while Cases (b) and (c) are of codimension four. However, if the system exhibits some symmetry, then all of the above three bifurcations reduce to codimension two. It was established in [126, 127, 129] that a symmetric homoclinic butterfly with either a = 0 or A = 0 appears in the so-called extended Lorenz model, and in the Shimizu-Morioka system, as well as in some cases of local bifurcations of codimension three in the presence of certain discrete symmetries [129]. 

Fig. C.2.2. The Andronov-Hopf bifurcation curve AH and a pitch-fork curve r = 1 in the (r, (r)-plane of the Lorenz model at 6 = 8/3.

Fig. C.2.3. A partial bifurcation diagram for the asymmetric Lorenz model. The point CP is a cusp, at BT the system has a double-degenerate equilibrium state with two zero characteristic exponents (see Sec. 13.2).

We have seen that homoclinic bifurcations in symmetric systems have much in common. Let us describe next the universal scenario of the formation of a homoclinic loop to a saddle-focus in a typical system. In particular, this mechanism works adequately in the Rossler model, in the new Lorenz models, in the normal form (C.2.27), and many others. 

Let us visualize these steps using the example of the new Lorenz model [128]... 

The new Lorenz model is very rich in the sense of bifurcations. One of them is a non-transverse homoclinic saddle-node bifurcation. In Sec. C.2, we have... 

Roschin, N. V. [1978] Unsafe stability boundaries of the Lorenz model , J. AppL Math. Mech. 42(5), 1038-1041. 

---