Chaos strange attractor

Grebogi, C., Ott, E, and Yorke, J. A. (1987) Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics. Science 238, 632. 

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.)... 

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... 

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

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. 

Today this infinite complex of surfaces would be called a fractal. It is a set of points with zero volume but infinite surface area. In fact, numerical experiments suggest that it has a dimension of about 2.05 (See Example 11.5.1.) The amazing geometric properties of fractals and strange attractors will be discussed in detail in Chapters 11 and 12. But first we want to examine chaos a bit more closely. 

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 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 describe some beautiful experiments on the Belousov-Zhabotin-sky chemical reaction. The results show that strange attractors really do occur in nature, not just in mathematics. For more about chemical chaos, see Argoul et al. (1987). 

Even when (1) has no strange attractors, it can still exhibit complicated dynamics (Moon and Li 1985). For i nstance, consider a regime in which two or more stable limit cycles coexist. Then, as shown in the next example, there can be transient chaos before the system settles down. Furthermore the choice of final state depends sensitively on initial conditions (Grebogi et al. 1983b). 

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. 

Coexisting chaos and limit cycle) Consider the double-well oscillator (12.5,1) with parameters S = 0.15, F =0.3, and (O — 1. Show numerically that the system has at least two coexisting attractors a large limit cycle and a smaller strange attractor. Plot both in a Poincare section,... 

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. 

Ruel D. Ruelle, Turbulence, Strange Attractors, and Chaos, World Scientific, Singapore, 1995. 

R13) 1979-2 Rossler, O. E. Chaos and Strange Attractors in Chemical Kinetics, Springer Series in Synergetics, vol. 3, 107-113... 

Classical mechanics provides the least ambiguous statement of the nature of chaotic motion, with chaos also defined through a heirarchy of ideal model systems. We note, at the outset, that isolated molecule dynamics relates to chaotic motion in conservative Hamiltonian systems. This is distinct from chaotic motion in dissipative systems where considerable simplifications result from the reduction in degrees of freedom during evolution10 and where objects such as strange attractors and fractal dimensions play an important role. 

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