Discrete dynamical system

We restrict our attention to symplectic one-step discretizations of (1), which leads to discrete dynamical systems of the form... 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

Apart from their pedagogical value, reversible rules may be used to explore possible relationships between discrete dynamical systems and the dynamics of real mechanical systems, for which the microscopic laws are known to be time-reversal invariant. What sets such systems apart from continuous idealizations is their exact reversibility, discreteness assures us that computer simulations run for arbitrarily long times will never suffer from roundoff or truncation errors. As Toffoli points out, ...the results that one obtains have thus the force of theorems [toff84a]. ... 

As should have been made plausible by the above discussion, symbolic dynamics provides an intuitive conceptual bridge between continuous and discrete dynamical systems. On the one hand, except for the fact that the coarse-grained discrete dynamics of symbolic systems is typically nonlocal in character (see the following section), symbolic dynamical systems are essentially CA-in-disguise. On the other hand, by exploiting the fundamental CA property of continuity in the Cantor-set topology under the metric defined by equation 4.49, real-valued representations of CA dynamics may be readily obtained. We discuss these two alternative points of... 

The introductory chapter of this book identified four basic motivations for studying CA. The subsequent chapters have discussed a wide variety of CA models predicated on the first three of these four motivations namely, using CA as... (1) as powerful computational engines, (2) as discrete dynamical system simulators, and (3) as conceptual vehicles for studying general pattern formation and complexity. However, we have not yet presented any concrete examples of CA models predicated on the fourth-and arguably the deepest-motivation for studying CA as fundamental models of nature. A discussion of this fourth class of CA models is taken up in earnest in this chapter, whose narrative is woven around a search for an answer to the beisic speculative question, Is nature, at its core, a CA "... 

DDLab is an interactive graphics program for studying many different kinds of discrete dynamical systems. Arbitrary architectures can be defined, ranging from Id, 2d or 3d CA to random Boolean networks. 

Auxiliary discrete dynamical systems and relaxation analysis 130... 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

The fast stage of relaxation of a complex reaction network could be described as mass transfer from nodes to correspondent attractors of auxiliary dynamical system and mass distribution in the attractors. After that, a slower process of mass redistribution between attractors should play a more important role. To study the next stage of relaxation, we should glue cycles of the first auxiliary system (each cycle transforms into a point), define constants of the first derivative network on this new set of nodes, construct for this new network an (first) auxiliary discrete dynamical system, etc. The process terminates when we get a discrete dynamical system with one attractor. Then the inverse process of cycle restoration and cutting starts. As a result, we create an explicit description of the relaxation process in the reaction network, find estimates of eigenvalues and eigenvectors for the kinetic equation, and provide full analysis of steady states for systems with well-separated constants. 

RELAXATION OF MULTISCALE NETWORKS AND HIERARCHY OF AUXILIARY DISCRETE DYNAMICAL SYSTEMS... 

We can understand better this asymptotics by using the Markov chain language. For nonseparated constants a particle in has nonzero probability to reach and nonzero probability to reach A, . The zero-one law in this simplest case means that the dynamics of the particle becomes deterministic with probability one it chooses to go to one of vertices A, A3 and to avoid another. Instead of branching, A2 A and A2 A3, we select only one way either A2 A] or A2 A3. Graphs without branching represent discrete dynamical systems. 

The map (p determines discrete dynamical system on a set of components V — A,. We call it the auxiliary discrete dynamical system for a given network of monomolecular reactions. Let us decompose this system and find the cycles Cy and their basins of attraction, Att(Cy). 

Notice that for the graph that represents a discrete dynamic system, attractors are ergodic components, whereas basins are connected components. 

An auxiliary reaction network is associated with the auxiliary discrete dynamical system. This is the set of reactions A, A q with kinetic constants k,. The correspondent kinetic equation is... 

For deriving of the auxiliary discrete dynamical system we do not need the values of rate constants. Only the ordering is important. Below we consider multiscale ensembles of kinetic systems with given ordering and with well-separated kinetic constants ( (i) k(,(2) > > for some permutation cr). 

First, let us find the eigenvectors for zero eigenvalue. Dimension of zero eigenspace is equal to the number of fixed points in the discrete dynamical system. If A,- is a fixed point then the correspondent eigenvalue is zero, and the right eigenvector r has only one nonzero coordinate, concentration of A, r = 5ij. 

In the simplest case, the auxiliary discrete dynamical system for the reaction network W is acyclic and has only one attractor, a fixed point. Let this point be A (n is the number of vertices). The correspondent eigenvectors for zero eigenvalue are r = S j and Z = 1. For such a system, it is easy to find explicit analytic solution of kinetic equation (32). 

The second simple particular case on the way to general case is a reaction network with components A, ..., A whose auxiliary discrete dynamical system has one attractor, a cycle with period t > 1 A +i A - +x. ., A ... 

After that, we create a new auxiliary discrete dynamical system for the new reaction network on the set A],... We can describe this new... 

Again we should analyze, whether this new cycle is a sink in the new reaction network, etc. Finally, after a chain of transformations, we should come to an auxiliary discrete dynamical system with one attractor, a cycle, that is the sink of the transformed whole reaction network. After that, we can find stationary distribution by restoring of glued cycles in auxiliary kinetic system and applying formulas (11)-(13) and (15) from Section 2. First, we find the stationary state of the cycle constructed on the last iteration, after that for each vertex Ay that is a glued cycle we know its concentration (the sum of all concentrations) and can find the stationary distribution, then if there remain some vertices that are glued cycles we find distribution of concentrations in these cycles, etc. At the end of this process we find all stationary concentrations with high accuracy, with probability close to one. 

Auxiliary discrete dynamical system for the network (48) includes the chain and one reaction ... 

For one catalytic cycle, relaxation in the subspace = 0 is approximated by relaxation of a chain that is produced from the cycle by cutting the limiting step (Section 2). For reaction networks under consideration (with one cyclic attractor in auxiliary discrete dynamical system) the direct generalization works for approximation of relaxation in the subspace = 0 it is sufficient to perform the following procedures ... 

The auxiliary discrete dynamical system for reaction network (50) is... 

The general case cycles surgery for auxiliary discrete dynamical system with arbitrary family of attractors... 

This dominant system graph is acyclic and, moreover, represents a discrete dynamical system, as it should be (not more than one outgoing reaction for any component). Therefore, we can estimate the eigenvalues and eigenvectors on the base of formulas (35) and (37). It is easy to determine the order of constants because fcjj = 41 32/ 21 this constant is the smallest nonzero constant in the obtained acyclic system. Finally, we have the following ordering of constants A3 —> Ai —> A2 —A4 and A5 —> A5 —> A4. 

The vertex A4 is the fixed point for the discrete dynamical system. There is no reaction A4 For convenience, we include the eigenvectors f" and for zero eigenvalue, K4 = 0. These vectors correspond to the steady state is the steady-state vector, and the functional f is the conservation law. 

Figure 4 Four possible auxiliary dynamical systems for the reversible triangle of reactions with k2T>kij for (/,/)y (2,l) (a) kn>ki2, k23>k i (b) kn>ki2, kn>k2i (c) ki2>ku, k2i>kn and (d) ki2>k- 2, kii >/c23- Foi" each vertex the outgoing reaction with the largest rate constant is represented by the solid bold arrow, and other reactions are represented by the dashed arrows. The digraphs formed by solid bold arrows are the auxiliary discrete dynamical systems. Attractors of these systems are isolated in frames.

Let us take a multiscale network and perform the iterative process of auxiliary dynamic systems construction and cycle gluing, as it is prescribed in Section 4.3. After the final step the algorithm gives the discrete dynamical system O " with fixed points A . 

The fixed points A of the discrete dynamical system O " are the glued ergodic components G C js/ of the initial network W. At the same time, these points are attractors of O ". Let us consider the correspondent decomposition of this... 

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