Attractors strange

There are many examples in nature where a system is not in equilibrium and is evolving in time towards a thennodynamic equilibrium state. (There are also instances where non-equilibrium and time variation appear to be a persistent feature. These include chaos, oscillations and strange attractors. Such phenomena are not considered here.)... 

Shaw R 1981 Strange attractors, chaotic behaviour, and information flow Z. Naturf. a 36 80-112... 

We shall now analyse the stmcture of a chemical strange attractor and describe why the dynamics may be classified as chaotic. 

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. 

Several important topics have been omitted in this survey. We have described only a few of the routes by which chaos can arise in chemical systems and have made no attempt to describe in detail the features of the different kinds of chemical strange attractor seen in experiments. A wide variety of chemical patterns have been observed and while the many aspects of the mechanisms for their appearance are understood, some features like nonlinear... 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

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 contrast to dissipative dynamical systems, conservative systems preserve phase-space volumes and hence cannot display any attracting regions in phase space there can be no fixed points, no limit cycles and no strange attractors. There can nonetheless be chaotic motion in the sense that points along particular trajectories may show sensitivity to initial conditions. A familiar example of a conservative system from classical mechanics is that of a Hamiltonian system. 

Figure 4.8 shows four snapshot views of the structure of the Henon strange attractor for a = 1.4 and p = 0.3. In the figure, plots (b), (c) and (d) provide enlargements of the small window regions shown in plots (a), (b) and (c), respectively. The attractor possesses two noteworthy properties ... 

Fig. 4.9 Snapshots of strange attractors defined by the expressions appearing in equation 4,32.

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. 

Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

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 ... 

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

It is important to remark that this behavior is similar to that previously considered by Eqs.(9), when two external periodic disturbances are applied. Nevertheless, this behavior can be very difficult to obtain, because the lobe of Figure 8 is small. Figures 10 and 11 shows chaotic oscillations and a new strange attractor. By simulation it is possible to obtain plots similar to those in Figures 2, 4, 5 and 6. 

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. 

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