Dimensional Discrete Maps

The case of multidimensional discrete-time mappings of the form 

Equation 4.65 takes a particularly simple form for one-dimensional maps, Xj+i 

The criterion for one-dimensional chaos is then s])ecified succinctly by  

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The time evolution of the discrete-valued CA rule, F —> F, is thus converted into a two-dimensional continuous-valued discrete-time map, 3 xt,yt) —> (a y+i, /y+i). This continuous form clearly facilitates comparisons between the long-time behaviors of CA and their two-dimensional discrete mapping counter-... 

In many ways, May s sentiment echoes the basic philosophy behind the study of CA, the elementary versions of which, as we have seen, are among the simplest conceivable dynamical systems. There are indeed many parallels and similarities between the behaviors of discrete-time dissipative dynamical systems and generic irreversible CA, not the least of which is the ability of both to give rise to enormously complicated behavior in an attractive fashion. In the subsections below, we introduce a variety of concepts and terminology in the context of two prototypical discrete-time mapping systems the one-dimensional Logistic map, and the two-dimensional Henon map. 

Consider, once again, a one-dimensional discrete-time map... 

Self-Organizing Maps (SOMs) or Kohonen maps are types of Artificial Neural Networks (ANNs) that are trained using supervised/unsupervised learning to produce a low-dimensional discretized representation (typically 2-dimensional) of an arbitrary dimension of input space of the training samples (Zhong et al. 2005). 

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

The predictive power of Feigenbaum s theory may strike you as mysterious. How can the theory work, given that it includes none of the physics of real systems like convecting fluids or electronic circuits And real systems often have tremendously many degrees of freedom—how can all that complexity be captured by a one-dimensional map Finally, real systems evolve in continuous time, so how can a theory based on discrete-time maps work so well ... 

In a lattice-gas the individual cells are the structural units of a D-dimensional regular lattice. Each cell is defined by its position vector r on the discrete space, a finite number of states s(r ) and a set of transition rules E that map the state of the cell at time t into the state at time t + 1. A finite number of particles reside in each cell. A discrete velocity c, with / = 1,..., k is associated to each particle. This velocity is chosen such that the particle can propagate to a neighboring cell in unit time. Each velocity direction is subject to an exclusion principle of utmost single occupancy. The combinations of occupancies define the set of possible states associated with each cell. The configuration of each cell is defined by the Boolean field... 

The Poincare map is a method to transform the continuous flow in n-dimensional phase space to an equivalent discrete flow (map) in a phase space of (n — l)-dimensions (or (n — 2)-dimensions for Hamiltonian flows). 

The intersections of the continuous Hamiltonian flow in the 2n-dimensional phase space, defined by Equation (71), with the surface of section defined by Equation (72), transforms the continuous flow to an equivalent discrete flow (map), on a (2n — 2)-dimensional surface of section (see Figure 12). 

By using the discrete variable method the 3-dimensional Schrodinger equation is mapped onto a system of ordinary differential equations in the radial coordinate. We solve this system by using the finite element method. 

The full diffusion-reaction problem is therefore mapped onto a discrete, one-dimensional process that evolves according to the following diffusion "rules" (Grebenkov et al., 2005). Molecules currently at site i move to the left (site f — 1) or right (site i + 1) with probability l/2d, or stay on the current site with probability (1 — cr) (1 — 1/d), wherein (7 = 1/ (1 + A/a) is the absorption probability of the molecule with the surface. Finally, the molecule crosses the surface with probability a (1 — 1/d). Collectively, these rules can be written as a finite difference equation (Grebenkov et al., 2005), which is a discretized version of Eqns. 4 to 6 ... 

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