Lorenz map

Lorenz (1963) found a beautiful way to analyze the dynamics on his strange attrac- 

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, no one has been able to refute this argument in a rigorous sense. But by using his map, Lorenz was able to give a plausible counterargument that stable limit cycles do not, in fact, occur for the parameter values he studied. 

To show that this closed orbit is unstable, consider a slightly perturbed trajectory that has =r + t , where r , is small. After linearization as usual, we find 

Hence the deviation grows with each iteration, and so the original closed orbit is unstable. 

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G. Brix, M.E. Bellemann, U. Haberkorn, L. Gerlach, P. Bachert, W.J. Lorenz, Mapping the blodlstrlbutlon and catabolism of 5-fluorouracll In tumor-bearing rats by chemical-shift selective F-19 MR-lmagIng, Magn. Reson. Med. 34 (1995) 302-307. 

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. 

Given the Lorenz map approximation = /(z ), with / (z)l> 1 for all z, show that all closed orbits are unstable. 

As tools for analyzing differential equations. We have already encountered maps in this role. For instance, Poincare maps allowed us to prove the existence of a periodic solution for the driven pendulum and Josephson junction (Section 8.5), and to analyze the stability of periodic solutions in general (Section 8.7). The Lorenz map (Section 9.4) provided strong evidence that the Lorenz attractor is truly strange, and is not just a long-period limit cycle. 

To compare these results to those obtained for one-dimensional maps, we use Lorenz s trick for obtaining a map from a flow (Section 9.4). For a given value of c, we record the successive local maxima of x(r) for a trajectory on the strange attractor. Then we plot x,, vs. x , where denotes the th local maximum. This Lorenz map for c = 5 is shown in Figure 10.6.7. The data points fall very nearly on a one-dimensional curve. Note the uncanny resemblance to the logistic map ... 

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. 

Intermittency is not just a curiosity of the logistic map. It arises commonly in systems where the transition from periodic to chaotic behavior takes place by a saddle-node bifurcation of cycles. For instance. Exercise 10.4.8 shows that intermittency can occur in the Lorenz equations. (In fact, it was discovered there see Pomeau and Manneville 1980). 

Area contraction is the analog of the volume contraction that we found for the Lorenz equations in Section 9.2. As in that case, it yields several conclusions. For instance, the attractor A for the baker s map must have zero area. Also, the baker s map cannot have any repelling fixed points, since such points would expand area elements in their neighborhood. 

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. 

In theculinary spirit ofthe pastry map and the baker s map, Otto Rossler (1976) found inspiration in a taffy-pulling machine. By pondering its action, he was led to a system of three differential equations with a simpler strange attractor than Lorenz s. The Rossler system has only one quadratic nonlinearity xz ... 

This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors. 

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. 

Deursen R, Blum Lorenz CB, Reymond JL (2010) A searchable map of PubChem. J Chem Inf Model 50(11) 1924-1934 Chemscreener unpublished results... 

Following the method of Lorenz, one may obtain a quasi-one-dimensional map from successive local maxima Z (w = 1,2,...) of Z t), The result is a uni-modal map as shown in Fig. 7.17. Unlike the classical Lorenz chaos, the map seems to have a smooth maximum instead of a cusp structure. The route to chaos, if seen on the map, shows no difference from the usual period-doubling type except that the present map may not have a quadratic maximum the splitting of Ml into L/+1 and L/+i appears on the map as the bifurcation of 2-point cycles from a 2 -point cycle, and the mutual contact of L/4.1 and Z/+i at the saddle... 

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