The strange attractor

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. 

The Lorenz equations may produce deterministic chaos because we know how it will instantaneously change. However, for high enough Rayleigh numbers, the system becomes chaotic. Small changes in the initial conditions can lead to very different behavior after long time interval, since the small differences grow nonlinearly with feedback over time (known as the Butterfly effect). These equations are fairly well behaved and the overall patterns repeat in a quasi-periodic fashion. 

Let us solve the Lorenz equations below with MATLAB between t = 0 and 20 and prepare plots of v, versus t, and a state-space representation ofty versus yu andy3 versus y2 by using two different sets of initial conditions Ti(0) =y2(0) =y3(0) = 5.0 andyffO) =y2(0) = y3(0) = 5.0. 

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At the th period doubling the period of the oscillation is In the limit —> co we arrive at the strange attractor where the time variation of the concentrations is no longer periodic. This is the period-doubling route to chaos. 

Figure 4.12 shows sample a vs y plots obtained in this manner for a few elementary CA rules. Note that the patterns for nonlinear rules such as R18, R22, and 122 appear to possess a characteristic fractal-like structure reminiscent of the strange attractors appearing in continuous systems shown earlier. We will comment on the nature of this similarity a bit later on in this chapter. 

The orbits are dense in a state space region i.e. the orbits fills the phase space zone of the strange attractor fl. 

Figure 13. Phase diagram of the system (35) on the (to, H) plane obtained numerically for the parameter values T = 0.025, coo = 0.597, P = 1, y = 1. See text for a description of the symbols the various lines are guide to the eye. The working point P, with ay = 0.95, h = 0.13, shown by a thick plus, was chosen to lie in the region of coexistence of the period 1 stable limit cycle and of the strange attractor [168],...

In principle, it is possible to find the optimal path by direct solution of the Pontryagin Hamiltonian (37), with appropriate boundary conditions. We must stress that even for this relatively simple system, the solution is a formidable, and almost impossible, task. First of all, in general one has no insight into the appropriate boundary conditions, in particular into those at the starting time (which belong to the strange attractor). But even if the boundaries were known, in practice the determination of the optimal path is impossible the functional R of Eq. (36) has so many local minima, that it proved impractical to attempt a (general) search for the optimal path. 

On the theoretical physics side, the Kolmogorov-Arnold-Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the... 

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. 

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 ... 

Back in Chapter 9, we found that the solutions of the Lorenz equations settle down to a complicated set in phase space. This set is the strange attractor. As Lorenz (1963) realized, the geometry of this set must be very peculiar, something like an infinite complex of surfaces. In this chapter we develop the ideas needed to describe such strange sets more precisely. The tools come from fractal geometry. 

Hobson (1993) has recently developed a method for computing this unstable manifold to very high accuracy. As expected, it is indistinguishable from the strange attractor. Hobson also presents some enlargements of less familiar parts of the Henon attractor, one of which looks like Saturn s rings (Figure 12.2.4). 

In effect, the flow is acting like the pastry transformation, and the phase space is acting like the dough Ultimately the flow generates an infinite complex of tightly packed surfaces the strange attractor. 

Solution To find suitable initial conditions, we could use trial and error, or we could guess that transient chaos might occur near the ghost of the strange attractor of Figure 12.5.6. For instance, the point (Xq, )= (0.2,0.1) leads to the time series shown in Figure 12.5.8a. 

In the phase space, the trajectory followed by the system never passes again through the same point, but remains confined to a finite portion of this space (fig. 4.10) the system evolves towards a strange attractor (Ruelle, 1989). The unpredictability of the time evolution in the chaotic regime is associated with the sensitivity to initial conditions two points, initially close to each other on the strange attractor, will diverge exponentially in the course of time. 

In the phase plane (p, a, 13), the projection of the trajectory followed by the seven-variable system (6.2) takes the form of a strange attractor of which fom successive states are shown in fig. 6.12. The system remains confined within a portion of the phase space, but the curve it follows in the course of oscillations never passes twice through any given point. This is one way by which this behaviour differs from complex periodic oscillations. Some cycles on the strange attractor are... 

Fig. 6.13. Strange attractor corresponding to the chaotic behaviour of fig. 6.11. The curve is a projection in the space (a, )3, F) where Y represents the total fraction of receptor bound to extracellular cAMP. The range of variation of ATP, intracellular cAMP and fraction Y extends from 0 to 15, 1.545 to 1.585, and 0.15 to 0.9, respectively. The window shows a histogram of cycle lengths for 500 successive cycles on the strange attractor. The time interval between two successive peaks of cAMP is measured, emd the percentage of cycle lengths within 2 min intervals is plotted (Martiel Goldbeter, 1985a).

In the specific case considered in figs. 6.21 and 6.22, the mixing of 5% of periodic cells with 95% of chaotic cells results in periodic oscillations of cAMP (lower part of figures) the strange attractor associated with chaos thus transforms into a limit cycle. The periodic oscillations in the mixed suspension correspond to the behaviour predicted by the bifurcation diagram of fig. 6.3 for the effective value of parameter given by eqn (6.10). Qualitatively similar results are also obtained by numerical simulations when the two populations differ only by their intracellular supply of ATP, namely, VjV2-... 

Fig. 6.24. Suppression of chaos by a small-amplitude, periodic input of cAMP. The chaotic oscillations of cAMP (b) are the same as those considered in fig. 6.21 (top, left part). The system is subjected to a sinusoidal input of cAMP (a), as described by eqn (6.3d). Such forcing of the strange attractor leads to periodic oscillations of cAMP (c) the latter are obtained by numerical integration of the first two equations of system (6.3) and eqn (6.12) for A = 0.025 and r = 6 min (Li et ai, 1992b).

The phase space representation of trajectories computed numerically, as described above, has been introduced in another chapter of this volume. TTie systems considered there are Hamiltonian systems which arise in chemistry in the context of molecular dynamics problems, for example. The difference between Hamiltonian systems and the dissipative ones we are considering in this chapter is that, in the former, a constant of the motion (namely the energy) characterizes the system. A dissipative system, in contrast, is characterized by processes that dissipate rather than conserve energy, pulling the trajectory in toward an attractor (where in refers to the direction in phase space toward the center of the attractor). We have already seen two examples of attractors, the steady state attractor and the limit cycle attractor. These attractors, as well as the strange attractors that arise in the study of chaotic systems, are most easily defined in the context of the phase space in which they exist. 

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

An example of a calculation of the Lyapunov exponents and dimension, for a simple four-variable model of the peroxidase-oxidase reaction will help to clarify these general definitions. The following material is adapted from the presentation in Ref. 94. As described earlier, the Lyapunov dimension and the correlation dimension, D, serve as upper and lower bounds, respectively, to the fractal dimension of the strange attractor. The simple four-variable model is similar to the Degn—Olsen-Ferram (DOP) model discussed in a previous section but was suggested by L. F. Olsen a few years after the DOP model was introduced. It remains the simplest model the peroxidase-oxidase reaction which is consistent with the most experimental observations about this reaction. The rate equations for this model are ... 

Whereas attractor in the steady state has zero dimensions and Euclidian dimension of the limit cycle is two, it is not possible to define the Euclidian dimension of the strange attractor. However, using the concept of fractal geometry, it is possible to define the dimension of such an attractor, which is not an integer. 

The phase-plane plot in three dimensions is further shown in Fig. (12.8). The phase-plane plot in three dimensions has fractal geometry and the attractor is called the strange attractor. 

In the case of non-deterministic chaos (noisy) oscillations, the attractor is found to be as shown in Fig. 12.9(b) corresponding to the time series generated from thiophenol + bromate oscillator [12], which is quite different as compared to the strange attractor. The phase-plane plots of thio-phenol oscillator based on experimental results Fig. 12.9(a) displays an unusual attractor since noise in oscillations is produced by the heterogeneous reaction. 

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