Lyapunov functional

Chapter 5 provides some examples of purely analyti( al tools useful for describing CA. It discusses methods of inferring cycle-state structure from global eigenvalue spectra, the enumeration of limit cycles, the use of shift transformations, local structure theory, and Lyapunov functions. Some preliminary research on linking CA behavior with the topological characteristics of the underlying lattice is also described. 

A slightly different Lyapunov function from the one defined above can be used to put bounds on the transient length. Let... 

Using the Lyapunov function Lanti-symmetric, given by... 

To see that this is a reasonable approach to take, we look more closely at equation 10.9. It is easy to show that the energy function is in fact a Lyapunov Function. In particular, as the neural net evolves according to the dynamics specified by equation 10.7, itself either remains constant or decreases. The attractors of the system therefore reside at the local minima of the energy surface. 

To show that f is a Lyapunov function, we assume that the net evolves asynchronously and calculate the explicit change in that is induced by a change in the neuron ... 

Having shown that the energy function (equation 10.9) is a Lyapunov function, let us go back to the main problem with which we started this section, namely to find an appropriate set of synaptic weights. Using the results of the above discussion, we know that we need to have the desired set of patterns occupy the minimum points on the energy surface. We also need to be careful not to destroy any previously stored patterns when we add new ones to our net. Our task is therefore to find a... 

It is clear from Eq.(7) that the real time t flows dynamically. In computing the trajectories with sufficiently small step sizes of r, the elapsing of real time t was followed by simultaneously integrating the relation (7). Here the real time t will be given in the relativistic system of units. The Lyapunov functions Ai(t) are in units of 1/t. 

In Figs. l(a)-l(c) the Lyapunov functions are shown for Z = 50, m = 1 and different scaled energies. Fig. 1(a) shows results for v = 0, p = 0, e = 10. Al( ) tends to zero indicating that this trajectory is regular. This figure has the same shape as that for the nonrelativistic hydrogen atom in a uniform magnetic field (Schweizer et.al., 1988). In Fig. 1(b) the Lyapunov function for v = 0, p = 0, e = 50 is shown. It tends to some positive value, which means that this trajectory is chaotic. While for v = 0, p = 0, e = 100 (Fig. 1(c)) we find that the trajectory is unstable. 

Proof. By analyzing the steady-state behavior of AD model (11), four possible equilibrium points are obtained (see Table 1). However, it is observed that only the point P4 has physical meaning under NOC. This means that under NOC, AD model (11) has a single equilibrium point P4, which depends on the process kinetics and the influent composition. Now, in order to evaluate the stability of the internal d mamics of AD model (11), the following candidate Lyapunov function (CLP) is proposed... 

Show that F(0) is a Lyapunov function ), which can be used to prove the global stability. 

A valuable byproduct of passive arithmetic is the suppression of limit cycles and overflow oscillations. Formally, the signal power of a conceptually infinite-precision implementation can be viewed as a Lyapunov function bounding the squared amplitude of the finite-precision implementation. 

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

From the various versions of this method we will choose only one. Let V < 0 and, only at the rest point under study c, V - 0. Then let Vhave its minimum, V(c) = at the point c and for some e > Vmin the set specified by the inequality V(c0) < e is finite. Therefor any initial conditions c0 from this set the solution of eqn. (73) is c(t, k, c0) - c at t - oo. V(c) is called a Lyapunov function. The arbitrary function whose derivative is negative because of the system is called a Chetaev or sometimes a dissipative function. Physical examples are free energy, negative entropy, mechanical energy in systems with friction, etc. Studies of the dissipative functions can often provide useful information about a given system. A modern representation for the second Lyapunov method, including a method of Lyapunov vector functions, can be found in ref. 20. 

In the previous section we introduced the Lyapunov functions for chemical kinetic equations that are the dissipative functions G. The function RTG is treated as free energy. Since G < 0 and the equality is obtained only at PDE, and for the construction of G it suffices to know only the position of equilibrium N, there exist limitations on the non-steady-state behaviour of a closed system that are independent of the reaction mechanism. If in the initial composition N = N, the other composition N can be realized during the reaction only in the case when... 

---