Fractal attractor

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Dinicola S, D Anselmi F, Pasqualato A et al (2011) A systems biology approach to cancer fractals, attractors, and nonlinear dynamics. OMICS 15 93-104... 

For such an iteration function scheme, a fractal attractor exists. The best known example for obtaining a fractal set is the square representation in the complex plane... 

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. 

Our work in the previous three chapters has revealed quite a bit about chaotic systems, but something important is missing intuition. We know what happens but not why it happens. For instance, we don t know what causes sensitive dependence on initial conditions, nor how a differential equation can generate a fractal attractor. Our first goal is to understand such things in a simple, geometric way. 

The dough is rolled out and flattened, then folded over, then rolled out again, and so on. After many repetitions, the end product is a flaky, layered structure—the culinary analog of a fractal attractor. 

Show that for a < +, the baker s map has a fractal attractor A that attracts all orbits. More precisely, show that there is a set A such that for any initial condition (. o > o) > the distance from 6"(Xq,> o) to A converges to zero as n —>. ... 

Grassberger, P. (1981) On the Hausdorff dimension of fractal attractors. J. Stat. Phys. 26, 173. 

This sequence appears when you, for example, repeatedly press the cosine button on a pocket calculator and it converges to 0.739 if x is an arbitrary number in radians. Other iterative processes do not converge to a specific number, but produce what seems like a random set of numbers. These numbers may be such that they always are close to a certain set, which may be a. fractal. This set is called a fractal attractor or strange attractor. 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

Note that while the attractor for a o is a fractal, it is not a chaotic attractor, since initially neighboring points do not undergo the same kind of exponential divergence that we observed earlier in the Bernoulli map. 

Fig. 4.5 Sets of stable attractors for the first six critical values of a. Note the self-similarity between the boxed subpattern for oe and the entire pattern for 04 appearing two lines above. A Cantor-set-like fractal pattern appears in the limit an-+oo-...

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. 

The fractal dimension was introduced earlier in section 2.1.1. If the minimum number of d-dimensional boxes of side e needed to eover the attractor A, N e), scales as... 

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be... 

Measure Entropy In the same way as the information dimension, Dp generalizes the fractal dimension. Dp, of an attractor. 4, by taking into account the relative frequency with which the individual e-boxes of a partition are visited by points on the attractor, so too the finite set entropy generalizes to a finite measure entropy,... 

Actively working groups are sure to include physical chemists (experimental and theoretical) and mathematicians (pure and applied). "Graphs theory , "dynamics , "non-linear oscillations , "chaos , "attractor , "synergetics , "catastrophes and finally "fractals these are the key words of modern kinetics. 

Although the detailed features of the interactions involved in cortisol secretion are still unknown, some observations indicate that the irregular behavior of cortisol levels originates from the underlying dynamics of the hypothalamic-pituitary-adrenal process. Indeed, Ilias et al. [514], using time series analysis, have shown that the reconstructed phase space of cortisol concentrations of healthy individuals has an attractor of fractal dimension dj = 2.65 0.03. This value indicates that at least three state variables control cortisol secretion [515]. A nonlinear model of cortisol secretion with three state variables that takes into account the simultaneous changes of adrenocorticotropic hormone and corticotropin-releasing hormone has been proposed [516]. 

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... 

The process of obtaining fractal sets at the transition from order to chaos can be regarded as an example of the change of boundaries between different regions which possess gravity centers (attractors) influencing the distribution of points in the region. Now the boundary constitutes a kind of order-disorder phase transition [20, 21]. 

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. 

---