Attractor point

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 spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

Naturally, the state of an RBN will change with time. If there are no external influences, then it has been shown that the state of any RBN will settle into either a point attractor (i.e., steady state) or a cycle. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

Figures 4.44 and 4.45, best viewed in color, show a benign complication of the problem caused by the Lewis numbers. If, however, we reduce the Lewis number LeA further to 0.07, the system trajectories indicate periodic explosions of the underlying system throughout all time, and the trajectories do not converge to the steady state at all, even with what we thought to be proper feedback. The trajectory that these curves settle at is called a periodic attractor of the system in contradistinction to the earlier encountered point attractor of Figures 4.43 or 4.44, for example. A point attractor, or more accurately a fixed-point attractor, is a more commonly encountered steady state in chemical and biological engineering systems. It could be called a stationary nonequilibrium state to distinguish it from the stationary equilibrium states associated with closed or isolated batch processes.

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

A dynamic system may exhibit qualitatively different behavior for different values of its control parameters 0. Thus, a system that has a point attractor for some value of a parameter may oscillate (limit cycle) for some other value. The critical value where the behavior changes is called a bifurcation point, and the event a bifurcation [32]. More specifically, this kind of bifurcation, i.e., the transition from a point attractor to a limit cycle, is referred to as Hopf bifurcation. 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

Hopfield networks [18] are able to store patterns as point attractors in n dimensional binary space and recall them in response to partial or degraded versions of stored patterns. For this reason, they are known as content addressable memories where each memory is a point attractor for nearby, similar patterns. Traditionally, known patterns are loaded directly into the network (see the learning rule 10 below), but in this paper we investigate the use of a Hopfield network to discover point attractors by sampling from a fitness function. A Hopfield network is a neural network consisting of n simple connected processing units. The values the units take are represented by a vector, u ... 

Biological systems dominated by a point attractor may be disturbed by the action of a drug but they come to their original state as soon as the drug is eliminated or excreted from the body, or they reach a new steady state when the drug concentration is kept constant. 

Important note A stationary nonequilibrium state can also be called a point attractor. We will see later that other types of attractor are also possible. 

Detailed analysis of a number and types of critical points (attractors) of the ELF field enables a characterisation of quantitative changes of ELF, as well as the electronic structure of the N-Ol bond, within the catastrophe theory. For F01N02, three catastrophes observed during dissociation of the N-Ol bond, one cusp and two fold catastrophes, are shown in Fig. 19.2. The V(N,01) attractor is annihilated in the cusp catastrophe and the protocovalent bond is created by two attractors, V(N) and V(01). Subsequently the V(N) and V(01) attractors are annihilated in the fold catastrophes. 

For both independence and finite variance of the involved random variables, the central limit theorem holds a probability distribution gradually converges to the Gaussian shape. If the conditions of independence and finite variance of the random variables are not satisfied, other limit theorems must be considered. The study of limit theorems uses the concept of the basin of attraction of a probability distribution. All the probability density functions define a functional space. The Gaussian probability function is a fixed point attractor of stochastic processes in that functional space. The set of probability density functions that fulfill the requirements of the central limit theorem with independence and finite variance of random variables constitutes the basin of attraction of the Gaussian distribution. The Gaussian attractor is the most important attractor in the functional space, but other attractors also exist. 

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