Lorenz attractor

The values could be almost repeating but not quite so. In chaos theory, these are called Lorenz attractor systems. 

We have encountered oscillating and random behavior in the convergence of open-shell transition metal compounds, but have never tried to determine if the random values were bounded. A Lorenz attractor behavior has been observed in a hypervalent system. Which type of nonlinear behavior is observed depends on several factors the SCF equations themselves, the constants in those equations, and the initial guess. 

The short program Lorenz.m calculates the concentrations for A, B and C for the initial conditions. c0=[l l 20]. Figure 3-37 displays the trajectories in a fashion that is not common in chemical kinetics. It is a plot of the time evolution of the values of A vs. B vs. C (see also Figure 3-35). Most readers will recognise the characteristic butterfly shape of the trajectory. The important aspect is that, in contrast to Figure 3-35, the trajectory is different each time. This time, it is not the effect of numerical errors but an essential aspect of the outcome. Even if the starting values for A, B and C are away from the butterfly , the trajectory moves quickly into it it is attracted by it and thus the name, Lorenz attractor. 

We now consider, for comparison, fluctuational escape from the Lorenz attractor, which, for a certain range of parameters, is a quasihyperbolic attractor consisting of unstable sets only [161] ... 

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

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].

Figure 22. The averaged escape trajectory (solid line) and the averaged fluctuational force (dashed line) during escape from the Lorenz attractor [182],...

It is clear that all of the escape trajectory from the Lorenz attractor lies on the attractor itself. The role of the fluctuations is, first, to bring the trajectory to a seldom-visited area in the neighborhood of the saddle cycle L, and then to induce a crossing of the cycle L. So we may conclude that the role of the fluctuations is different in this case, and the possibility of applying the Hamiltonian formalism will require a more detailed analysis of the crossing process. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

C. Sparrow, The Lorenz Equations Bifurcations, Chaos and Strange Attractors, Springer, New-York, 1982. 

Phase plane plots are not the best means of investigating these complex dynamics, for in such cases (which are at least 3-dimensional) the three (or more) dimensional phase planes can be quite complex as shown in Figures 16 and 17 (A-2) for two of the best known attractors, the Lorenz strange attractor [80] and the Rossler strange attractor [82, 83]. Instead stroboscopic maps for forced systems (nonautonomous) and Poincare maps for autonomous systems are better suited for investigating these types of complex dynamic behavior. 

The Lorenz and Rossler models are deterministic models and their strange attractors are therefore called deterministic chaos to emphasize the fact that this is not a random or stochastic behavior. 

Fig. 1. Examples of w-limit sets, (a) Rest point (b) limit cycle (c) Lorenz attractor [projection on the (Cj, c3) plane, a = 10, r = 30, b = 8/3].

Example 13.1 Lorenz equations The strange attractor The Lorenz equations (published in 1963 by Edward N. Lorenz a meteorologist and mathematician) are derived to model some of the unpredictable behavior of weather. The Lorenz equations represent the convective motion of fluid cell that is warmed from below and cooled from above. Later, the Lorenz equations were used in studies of lasers and batteries. For certain settings and initial conditions, Lorenz found that the trajectories of such a system never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity. Attractors in these systems are well-known strange attractors. 

Figure 23. El and E2 plots (a) Lorenz attractor, (b) Time series produced by a (pseudo) random number generator.

Lorenz Strange attractor in simple model of convection... 

In this chapter we ll follow the beautiful chain of reasoning that led Lorenz to his discoveries. Our goal is to get a feel for his strange attractor and the chaotic motion that occurs on it. 

In this section we ll follow in Lorenz s footsteps. He took the analysis as far as possible using standard techniques, but at a certain stage he found himself confronted with what seemed like a paradox. One by one he had eliminated all the known possibilities for the long-term behavior of his system he showed that in a certain range of parameters, there could be no stable fixed points and no stable limit cycles, yet he also proved that all trajectories remain confined to a bounded region and are eventually attracted to a set of zero volume. What could that set be And how do the trajectories move on it As we ll see in the next section, that set is the strange attractor, and the motion on it is chaotic. 

Now watch how 5(z) grows. In numerical studies of the Lorenz attractor, one finds that... 

The same could be true for the Lorenz equations. Although all trajectories are attracted to a bounded set of zero volume, that set is not necessarily an attractor, since it might not be minimal. To this day, no one has managed to prove that the Lorenz attractor seen in computer experiments is truly an attractor in this technical sense. But everyone believes it is, except for a few purists. 

By this ingenious trick, Lorenz was able to extract order from chaos. The function = /(z ) shown in Figure 9.4.3 is now called the Lorenz map. It tells us a lot about the dynamics on the attractor given we can predict z, by z, = /(Zq), and then use that information to predict Zj = /(zi), and so on, bootstrapping our way forward in time by iteration. The analysis of this iterated map is going to lead us to a striking conclusion, but first we should make a few clarifications. 

Second, the Lorenz map may remind you of a Poincare map (Section 8.7). In both cases we re trying to simplify the analysis of a differential equation by reducing it to an iterated map of some kind. But there s an important distinction To construct a Poincare map for a three-dimensional flow, we compute a trajectory s successive intersections with a two-dimensional surface. The Poincare map takes a point on that surface, specified by two coordinates, and then tells us how those two coordinates change after the first return to the surface. The Lorenz map is different because it characterizes the trajectory by only one number, not two. This simpler approach works only if the attractor is very flat, i.e., close to two-dimensional, as the Lorenz attractor is. 

So far we have concentrated on the particular parameter values <7 = 10, b =, r = 28, as in Lorenz (1963). What happens if we change the parameters It s like a walk through the jungle—one can find exotic limit cycles tied in knots, pairs of limit cycles linked through each other, intermittent chaos, noisy periodicity, as well as strange attractors (Sparrow 1982, Jackson 1990). You should do some exploring on your own, perhaps starting with some of the exercises. 

The voltages u,v,w at three different points in the circuit are proportional to Lorenz s x, y,z. Thus the circuit acts like an analog computer for the Lorenz equations. Oscilloscope traces of u t) vs. w t), for example, confirmed that the circuit was following the familiar Lorenz attractor. Then, by hooking up the circuit to a loudspeaker, Cuomo enabled us to hear the chaos—it sounds like static on the radio. 

Hysteresis between a fixed point and a strange attractor) Consider the Lorenz equations with <7 = 10 and b = 8/3. Suppose that we slowly turn the r knob up and down. Specifically, let r = 24.4 -h sin or, where ft) is small compared to typical orbital frequencies on the attractor. Numerically integrate the equations, and plot the solutions in whatever way seems most revealing. You should see a striking hysteresis effect between an equilibrium and a chaotic state. 

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