A strange attractor

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

Figure 4 shows a pattern of the concentration when the chaotic motion is established as well as the evolution of the deviation from two very close initial conditions. Note that nowadays it is very difficult to prove rigorously that a strange attractor is chaotic. In accordance with [35], a nonlinear system has chaotic dynamics if ... 

According to the Shilnikov s theorem, the reactor presents a chaotic behavior. In order to test the presence of a strange attractor, it is necessary to raise the value of xe ax to introduce a perturbation in the vector field around the homoclinic orbit. Taking xemax = 5, the results of the simulation are shown in Figure 18, where the sensitive dependence on initial conditions has been corroborated. 

The profile and phase plots show oscillatory behavior of Figure 4.56, but no pattern or periods can be seen therein. This is called a strange attractor in modern nonlinear dynamics theory. A strange attractor can be chaotic or nonchaotic (high-dimensional torus). Differentiating between chaotic and nonchaotic strange attractors is beyond the scope of this undergraduate book. 

From the viewpoint of experimental workers, slow relaxations are abnormally (i.e. unexpected) slow transition processes. The time of a transition process is determined as that of the transition from the initial state to the limit (t -> oo) regime. The limit regime itself can be a steady state, a limit cycle (a self-oscillation process), a strange attractor (stochastic self-oscillation), etc. 

Since (11.12) has an infinite number of degrees of freedom [523], we constructed a pseudophase space [4,32] for the system of (11.12) and (11.13) using the model variables c(t), c(t + t°/2), c(t + t°), Figure 11.11. The use of three dimensions is in accordance with the embedding dimension that Ilias et al. [514] have found. The attractor of our system is quite complicated geometrically, i.e., it is a strange attractor. The real phase space is of infinite dimension. However, trajectories may be considered to lie in a low-dimensional space (attractor). The model parameters take the same values as in Figure 11.10 and time runs for 10 days. 

Hence, if we start with a enormous solid blob of initial conditions, it eventually shrinks to a limiting set of zero volume, like a balloon with the air being sucked out of it. All trajectories starting in the blob end up somewhere in this limiting set later we ll see it consists of fixed points, limit cycles, or for some parameter values, a strange attractor. 

Finally, we define a strange attractor to be an attractor that exhibits sensitive dependence on initial conditions. Strange attractors were originally called strange because they are often fractal sets. Nowadays this geometric property is regarded as less important than the dynamical property of sensitive dependence on initial conditions. The terms chaotic attractor and fractal attractor are used when one wishes to emphasize one or the other of those aspects. 

At first the trajectory seems to be tracing out a strange attractor, but eventually it stays on the right and spirals down toward the stable fixed point (Recall that both C and C are still stable at r = 21. jThetimeseriesof y vs. t shows the same result an initially erratic solution ultimately damps down to equilibrium (Figure 9.5.3). 

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. 

Now it s time to return to dynamics. Suppose that we re studying a chaotic system that settles down to a strange attractor in phase space. Given that strange attractors typically have fractal microstructure (as we ll see in Chapter 12), how could we estimate the fractal dimension ... 

A strange attractor typically arises when the flow contracts the blob in some directions (reflecting the dissipation in the system) and stretches it in others (leading to sensitive dependence on initial conditions). The stretching cannot go on forever— the distorted blob must be folded back on itself to remain in the bounded region. 

The transformation shown in Figure 12.1.3 is normally called a horseshoe map, but we have avoided that name because it encourages confusion with another horseshoe map (the Smale horseshoe), which has very different properties. In particular, Smale s horseshoe map does not have a strange attractor its invariant set is more like a strange saddle. The Smale horseshoe is fundamental to rigorous discussions of chaos, but its analysis and significance are best deferred to a more advanced course. See Exercise 12.1.7 for an introduction, and Gucken-heimer and Holmes (1983) or Arrowsmith and Place (1990) for detailed treatments., ... 

In this section we discuss another two-dimensional map with a strange attractor. It was devised by the theoretical astronomer Michel Henon (1976) to illuminate the microstructure of strange attractors. 

In a striking series of plots, Henon provided the first direct visualization of the fractal structure of a strange attractor. He set a = 1.4, b = 0.3 and generated the at-... 

Numerical integration shows that y this system has a strange attractor for... 

Figure 12.4.1 shows a time series measured by Roux ct al. (1983). At first glance the behavior looks periodic, but it really isn t—the amplitude is erratic. Roux et al. (1983) argued that this aperiodicity corresponds to chaotic motion on a strange attractor, and is not merely random behavior caused by imperfect experimental control. 

The method is based on time delays. For instance, define a two-dimensional vector x(r) = (B(t). B(t -I- t)) for some delay T > 0. Then the time series B t) generates a trajectory x(r) in a two-dimensional phase space. Figure 12.4.2 shows the result of this procedure when applied to the data of Figure 12.4.1, using t = 8.8 seconds. The experimental data trace out a strange attractor that looks remarkably like the Rdssler attractor ... 

Now the tangle resolves itself—the points fall on a fractal set, which we interpret as a cross section of a strange attractor for (1). The successive points (x(t),y(ty) are found to hop erratically over the attractor, and the system exhibits sensitive dependence on initial conditions, just as we d expect. 

Ueda attractor) Consider the system x + kx + % = Bcos i, with k = 0.1, B = 12. Show numerically that the system has a strange attractor, and plot its Poincare section. 

Henon, M. (1976) A two-dimensional mapping with a strange attractor. Cotntnun. 

Holmes, P. (1979) A nonlinear oscillator with a strange attractor. Phil. Trans. Roy. Soc. A 292,419. 

On the other hand, the abundance of experimental material stimulates an evolution of the theories explaining non-linear phenomena. For example, as shown above, the transition in a chemical reaction from the stationary state to the state of periodical oscillations, the so-called Hopf bifurcation, is a certain elementary catastrophe. The transition in a chemical reaction to the chaotic state may be explained in terms of catastrophes associated with a loss of stability of a certain iterative process or by using the notion of a strange attractor (anyway, it turns out that both the systems are closely related). The equations of a chemical reaction with diffusion have been extensively studied lately. Based on the progress being made in this area, further interesting achievements in theory may be anticipated, particularly for the phenomena associated with catastrophes — the loss of stability by a non-linear system. 

A strange attractor resulting from such a deterministic model is called deterministic chaos to emphasize the fact that it is not a random or stochastic variation. 

Back to chemistry. We have already shown that the Brusselator leads to the existence of a limit cycle. In more elaborate models, this cycle can be subdivided into various periodicities to eventually give rise to a strange attractor. 

Several groups have shown the existence of a strange attractor in the BZ reaction. The representation below was taken from the work of Gyorgi, Rempe and Field [14]. 

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. 

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