Attractors, three-dimensional

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],... 

The definition of an atom and its surface are made both qualitatively and quantitatively apparent in terms of the patterns of trajectories traced out by the gradient vectors of the density, vectors that point in the direction of increasing p. Trajectory maps, complementary to the displays of the density, are given in Fig. 7.1c and d. Because p has a maximum at each nucleus in any plane that contains the nucleus (the nucleus acts as a global attractor), the three-dimensional space of the molecule is divided into atomic basins, each basin being defined by the set of trajectories that terminate at a given nucleus. An atom is defined as the union of a nucleus and its associated basin. The saddle-like minimum that occurs in the planar displays of the density between the maxima for a pair of neighboring nuclei is a consequence of a particular kind of critical point (CP), a point where all three derivatives of p vanish, that... 

Fig. 15. Three-dimensional attractors constructed by the time-delay method from several of the experimental time series reproduced in Fig. 13. (From Ref. 71.)...

A major difference between competitive and cooperative systems is that cycles may occur as attractors in competitive systems. However, three-dimensional systems behave like two-dimensional general autonomous equations in that the possible omega limit sets are similarly restricted. Two important results are given next. These allow the Poincare-Bendix-son conclusions to be used in determining asymptotic behavior of three-dimensional competitive systems in the same manner used previously for two-dimensional autonomous systems. The following theorem of Hirsch is our Theorem C.7 (see Appendix C, where it is stated for cooperative systems). 

The ELF is a scalar function of three variables, and in order to obtain more information from it, it is necessary to use a mathematical approach called differential topology analysis. This was first done by Silvi and Savin,11 and later on extended by them and co-workers.45,46 Unfortunately, one cannot visualize in a global way a three-dimensional function. Usually, one resorts to isosurfaces like the ones in Figure 1, or to contour maps. A three-dimensional function has a richer structure than a one-dimensional function, and their mathematical characterization introduces some new words which are necessary to understand in order to go further. It is the purpose of this section to explain this new terminology in a manner as simpler as possible. Let us begin with a one-dimensional (ID) example, a function f(x) like the one in Figure 3. The function has three maxima and two minima characterized by the sign of the second derivative. In three dimensions (3D) there are more possibilities, for there are nine second derivatives. Hence, one does not talk about maxima but about attractors. In ID, the attractors are points, in... 

Second, the Lorenz map may remind you of a Poincare map (Section 8.7). In both cases we re trying to simplify the analysis of a differential equation by reducing it to an iterated map of some kind. But there s an important distinction To construct a Poincare map for a three-dimensional flow, we compute a trajectory s successive intersections with a two-dimensional surface. The Poincare map takes a point on that surface, specified by two coordinates, and then tells us how those two coordinates change after the first return to the surface. The Lorenz map is different because it characterizes the trajectory by only one number, not two. This simpler approach works only if the attractor is very flat, i.e., close to two-dimensional, as the Lorenz attractor is. 

Figure 12.3.5 shows a Poincare section of the attractor. We slice the attractor with a plane, thereby exposing its cross section. (In the same way, biologists examine complex three-dimensional structures by slicing them and preparing slides.) If we take a further one-dimensional slice or Lorenz section through the Poincare section, we find an infinite set of points separated by gaps of various sizes. 

Roux et al. (1983) also considered the attractor in three dimensions, by defining the three-dimensional vector x(r) = (B(r),B(r+T),B(r+2T)). To obtain a Poincare section of the attractor, they computed the intersections of the orbits x(r) with a fixed plane approximately normal to the orbits (shown in projection as a dashed line in Figure 12.4.2). Within the experimental resolution, the data fall on a one-dimensional curve. Hence the chaotic trajectories are confined to an approximately two-dimensional sheet. 

For certain parameter values this chemical system can exhibit fixed point, periodic or chaotic attractors in the three-dimensional concentration phase space. We consider the parameter set... 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the ehaos assoeiated with motion on the attractor and some geometrical measme of the structural 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 diseussion we consider three-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

In a three-dimensional differential equation the system is going to collapse on a structure that is strictly less than three in dimension, but strictly more than two. Such a strange observation leads us immediately to the concept of fractional dimensionality and the attractor associated with such a fractional dimensionality is called a strange attractor, due to its unusual nature. The fractional dimensionality is also desaibed as a fractal. ... 

When related calculations are carried out for molecules (and also solids) [107,108], one either plots color-coded two-dimensional ELF(r) slices to represent the local ELF values, or one draws three-dimensional isosurfaces referring to a chosen ELF value, usually around 0.8 or close to it. With the advent of powerful computer graphics, such representations produce aesthetically compelling color figures from which ELF "attractors" (bonded and nonbonded electron pairs as well as atomic shells) show up and reflect the underlying molecular symmetry. All-electron calculations are generally preferable for this purpose [106]. 

Chaos does not occur as long as the torus attraaor is stable. As a parameter of the system is varied, however, this attractor may go through a sequence of transformations that eventually render it unstable and lead to the possibility of chaotic behavior. An early suggestion for how this happens arose in the context of turbulent fluid flow and involved a cascade of Hopf bifurcations, each of which generate additional independent frequencies. Each additional frequency corresponds to an additional dimension in phase space the associated attractors are correspondingly higher dimensional tori so that, for example, two independent frequencies correspond to a two-dimensional torus (7 ), whereas three independent frequencies would correspond to a three-dimensional torus (T ). The Landau theory suggested that a cascade of Hopf bifurcations eventually accumulates at a particular value of the bifurcation parameter, at which point an infinity of modes becomes available to the system this would then correspond to chaos (i.e., turbulence). 

Aldiough direct measurements of variables characterizing (he individual flow and chemical transport processes under field condidons are not technically feasible, their cumulative effect can be characterized by the phase-space analysis of time-series data for the infiltration and outflow rates, capillary pressure, and dripping-water frequency. The tune-series of low-frequency fluctuadons (assumed to represent intrafracture flow) are described by three-dimensional attractors similar to those fi m die sohidon of the Kuramoto-Sivashinsky equadon. These attractoia demonstrate die stretching and folding of fluid elements, followed by diffusion. 

Two types of attractors were known since the times of Poincare points or closed curves (limit cycles). The third type was discovered in 1971 [60]. It is so-called strange attractor , which can exist in three- and more-dimensional systems. In accordance with these three known types of attractors, three different kind of system s behaviour are possible after bifurcation (1) transition into a new stable steady state (2) undamped self-oscillations, and (3) chaotic regime (turbulence). 

Arneodo, A., Coullet, P. Tresser, C. (1980). Occurrence of strange attractors in three-dimensional Volterra equations. Phys. Letters, 79A, 259-63. 

This delay method (Takens, 1981) enables us to construct, for example, two-and three-dimensional portraits of the attractor. A first test to which the attractor can be subjected is to observe whether the attractor has a clear structure. If the... 

The construction of two- and three-dimensional phase portraits from experimental data by use of the delay technique is now routine. Figure 8.18 shows two-dimensional phase portraits of periodic and chaotic attractors for the BZ reaction. 

Figures. 18 Two-dimensional phase portraits in the BZ reaction constructed by the delay technique, (a) A periodic state, (b) two-dimensional projection of a three-dimensional phase portrait of a chaotic attractor. (Adapted from Swinney, 1983.)...

HANUSSE - In fact the morphology analysis is performed on the complete trajectory given by the simulation of the full model. One interesting result is that the results do not seem to depend on the space in which you work (concentrations, reaction rates, combinations of them). You obtain the same essential dynamical features. This stability of the solution is to be compared to that of the reconstruction procedure of three dimensional attractors from a unique time series, in studies of chaotic behaviour. 

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