Lorenz system

How does this relate to fluid turbulence The idea is that there exists a critical value of the Reynolds number, TZe, such that intermittent turbulent behavior can appear in the system for TZ > TZe- Moreover, if the behavior of the Lorenz system correctly identifies the underlying mechanism, it may be predicted that, as TZ changes, (1) the duration of the intermittently turbulent behavior will be random, and (2) the mean duration of the laminar phases in between will vary as... 

In the absence of noise, the system [179] describes the generation of a singlemode laser field interacting with a homogeneously broadened two-level medium [180]. The variables and parameters of the Lorenz system can be interpreted in terms of a laser system as q is the normalized electric field amplitude, the normalized polarization, q3 the normalized inversion, a = fe/y1 r = A+l, b = 72/71, with k the decay rate of the field in the cavity, yj and y2 the relaxation constants of the inversion and polarization, and A the pump parameter. Far-infrared lasers have been proposed as an example of a realization of the Lorenz system [162]. A detailed comparison of the dynamics of the system (42) and a far-infrared laser, plus a discussing the validity of the Lorenz system as laser model, can be found in Ref. 163. 

Figure 20. The bifurcation diagram of the Lorenz system for fixed a = 10, b =. The unstable and stable sets are shown by dashed and solid lines, respectively.

Figure 21. Structure of the phase space of the Lorenz system. An escape trajectory measured by numerical simulation is indicated by the filled circles. The trajectory of the Lorenz attractor is shown by a thin line the separatrixes T and T2 by dashed lines [182].

The Lorenz system, partially demonstrating chemical reactions, for creating esthetic patterns... 

As seen, two equations, may demonstrate reversible reactions. However, the Lorenz system is in fact a model of thermal convection, which includes not only a description of the motion of some viscous fluid or atmosphere, but also the information about distribution of heat, the driving force of thermal convection. The above set can be described by the following matrix, which, however, does not have any probabilistic significance ... 

Fig.3.14-3. Ci versus t for the Lorenz system demonstrating extreme sensitivity to initial conditions in case a and chaos in case h...

The Lorenz system is dissipative volumes in phase space contract under the flow. To see this, we must first ask how do volumes evolve ... 

Show that it is impossible for the Lorenz system to have either repelling fixed points or repelling closed orbits. (By repelling, we mean that all trajectories starting near the fixed point or closed orbit are driven away from it.)... 

Like the waterwheel, the Lorenz system (1) has two types of fixed points. The origin (x, y, z ) = (0,0,0) is a fixed point for all values of the parameters. It is like the motionless state of the waterwheel. For r > 1, there is also a symmetric pair of fixed points x = y = b(r -1), z = r -1. Lorenz called them C and C. They represent left- or right-turning convection rolls (analogous to the steady rotations of the waterwheel). As r —> 1, C and coalesce with the origin in a pitchfork bifurcation. 

Exponential divergence) Using numerical integration of two nearby trajectories, estimate the largest Liapunov exponent for the Lorenz system, assuming that the parameters have their standard values r = 28, a=lQ, b = 8/3. 

Transient chaos) Example 9.5.1 shows that the Lorenz system can exhibit transient chaos for r = 21, <7 = 10, b = j. However, not all trajectories behave this way. Using numerical integration, find three different initial conditions for which there is transient chaos, and three others for which there isn t. Give a rule of thumb which predicts whether an initial condition will lead to transient chaos or not. 

According to Gleick (1987, p. 149), Henon became interested in the problem after hearing a lecture by the physicist Yves Pomeau, in which Pomeau described the numerical difficulties he had encountered in trying to resolve the tightly packed sheets of the Lorenz attractor. The difficulties stem from the rapid volume contraction in the Lorenz system after one circuit around the attractor, a volume in phase space is typically squashed by a factor of about 14,000 (Lorenz 1963). 

Henon had a clever idea. Instead of tackling the Lorenz system directly, he sought a mapping that captured its essential features but which also had an adjustable amount of dissipation. Henon chose to study mappings rather than differential equations because maps are faster to simulate and their solutions can be followed more accurately and for a longer time. 

As desired, the Henon map captures several essential properties of the Lorenz system. (These properties will be verified in the examples below and in the exercises.)... 

The Henon map is invertible. This property is the counterpart of the fact that in the Lorenz system, there is a unique trajectory through each point in phase space. In particular, each point has a unique past. In this respect the Henon map is superior to the logistic map, its one-dimensional analog. The logistic map stretches and folds the unit interval, but it is not invertible since all points (except the maximum) come from two pre-images. 

The Henon map is dissipative. It contracts areas, and does so at the same rate everywhere in phase space. This property is the analog of constant negative divergence in the Lorenz system. 

The next property highlights an important difference between the Henon map and the Lorenz system. 

Some trajectories ofthe Henon map escape to infinity. In contrast, all trajectories of the Lorenz system are bounded they all eventually enter and stay inside a certain large ellipsoid (Exercise 9.2.2). But it is not surprising that the Henon map has some unbounded trajectories far from the origin, the quadratic term in (1) dominates and repels orbits to infinity. Similar behavior occurs in the logistic map—recall that orbits starting outside the unit interval eventually become unbounded. 

The Rossler system has only one nonlinear term, yet it is much harder to analyze than the Lorenz system, which has two. What makes the Rossler system less tractable ... 

The Rossler system lacks the sym metry of the Lorenz system. 

Robbins, K. A. (1979) Periodic solutions and bifurcation structure at high r in the Lorenz system. SIAM J. Appl. Math. 36,457. 

In a case of autonomous systems depending on three or more variables there exist more types of limit sets which, in some cases, may be extraordinarily complex. We shall discuss below an important case of a very complex limit set, which may occur in a dynamical system defined in the three-dimensional phase space. The following system of equations, known as the Lorenz system, in which a, r, b are certain control parameters,... 

When the control parameters a, b have fixed values, for example a = 10, b = 8/3, and the parameter r increases from the value r = 0, the nature of phase trajectories of the system (5.14) changes qualitatively. We shall describe properties of the Lorenz system only for the values of parameter r larger than r = 24.74 and smaller than a certain value rx (Lorenz in his work has studied the case a = 10, b = 8/3, r = 28 < rOT). When 24.74 < < r < rx, all three stationary points are unstable. However, trajectories do not escape to infinity — all the trajectories are attracted by a certain region of the phase space (attractor), containing the stationary points and which is approximately a two-dimensional surface. 

As we have concluded in Chapter 5, in the case of the Lorenz system a trajectory always remains within a confined region of the phase space, being non-periodical. For t - oo, the trajectory approaches a certain limit set the Lorenz attractor. It follows from the Liouville theorem that the Lorenz attractor has a zero volume (since divF < 0). This implies that, apparently, the Lorenz attractor is a point (dimension zero), a line (dimension one), or a plane (dimension two). Then, however, the trajectory for t -> oo would have remained within a confined region on the plane and, by virtue of the Poincare-Bendixon theorem. Hence, a conclusion follows that the Lorenz attractor has a fractional dimension, larger than two. 

Fundamentals of the theory of ordinary differential equations are given in books by Arnol d (1978), as well as Arrowsmith and Place. One may get acquainted with the Lorenz system in Gilmore s book. A book by Arnol d contains more advanced information on ordinary differential equations. 

One example of such a system is the Lorenz system [3] with n = 3. The system can be written as... 

A Poincare map is established by cutting across the trajectories in a certain region in the phase space, say with dimension n, with a surface that is one dimension less than the dimension of the phase space, n — 1. One such cut is also shown in Fig. 1. The equation that produces the return to the crossing the next time is a discrete evolution equation and is called the Poincare map. The dynamics of the continuous system that creates the Poincare map can be analyzed by the discrete equation. Therefore, chaotic behavior of the Poincare map can be used to identify chaos in the continuous system. For example, for certain parameters, the Henon discrete evolution equation is the Poincare map for the Lorenz systems. 

The minus sign before Aq is simply to give a better correspondence to the Lorenz system, i.e., a celebrated three-variable chaos-producing dynamical system (Lorenz, 1963). 

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