Lyapunov surfaces

Here, similar to the scalar fields, the velocity is a continuous function therefore there is a unique value of u in every node. Generally, this is not true for the traction vector. However, for a Lyapunov surface, where the normal is continuous, the tractions are also continuous. 

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. 

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... 

Figure 16. Scattering resonances of the full rotational-vibrational Hamiltonian describing the dissociation of CO2 on a LEPS surface obtained by equilibrium point quantization with (2.8). The resonances with 7 = 0,..., 10 are given by dots. Their close vicinity explains the formation of hyphens , i.e., unresolved sequences of dots. Note that rotation is very slightly destabilizing in the present model. The successive hyphens are the bending progressions with V2 = 0,. .. 5. The solid line is given by the Lyapunov exponent of the symmetric-stretch periodic orbit 0 expressed as an imaginary energy.

The first of the principal Horn and Jackson results is as follows. If the system obeys the law of mass action (or acting surfaces), then if it has a positive PCB it demonstrates a "quasi-thermodynamic behaviour, i.e. its positive steady state is unique and stable and a global Lyapunov function exists. 

Let V be a region in space bounded by a closed surface S (of Lyapunov-type [24, 50]), and f (x) be a vector field acting on this region. A Lyapunov-type surface is one that is smooth. The divergence (Gauss) theorem establishes that the total flux of the vector field across the closed surface must be equal to the volume integral of the divergence of the vector (see Theorem 10.1.1). 

The two first cases, although cumbersome, will close the system of equations. However, the third case will imply the use of an extra equation or the use of a discontinuous element. When equivalent integral equations to partial differential equations are developed, it is required that the surface is of a Lyapunov type [29, 40], For the purpose of this book, we will assume that this type of surfaces have the condition of having a continuous normal vector. The integral formulation also can be generated for Kellog type surfaces, which allow the existence of corners that are not too sharp. To avoid complications, we can assume that even for very sharp corners the normal vector is continuous, as depicted in Fig. 10.9. 

The Lyapunov function O in the form of type (4.71) definite quadratic expression can be constructed for many other simple schemes of catalytic transformations, too, to allow the conclusion about stability of the catalyst in these systems. In particular, this conclusion is true in the case of any intermediate linear transformations—that is, one free of interactions between active centers of the catalyst. The conclusion also is vahd for the cases of more complex schemes that imply possibilities of the forma tion and coexistence of intermediates of the stepwise transformations, which escape the catalyst surface for the gas (liquid) phase provided that the intermediate catalytic complexes do not interact with one another. 

A two-dimensional surface of section of phase space with the same parameters as Fig. 4a is shown in Fig. 4b. We measure maximal Lyapunov exponents averaged over a finite time interval and plot each of them to the initial point set on the section of phase space (q = qi = 0). The brighter region represents the more unstable region, and two vertical lines in the center of Figure 4b... 

As noted earlier, a necessary and sufficient condition for TST to be exact at a dividing surface is that any classical trajectory crossing the surface will never recross it. It is of practical interest to determine when TST must fail. Usually, the complete potential energy surface is not available so that one would want to develop local criteria for the failure of TST. Here we will shoxv that the stability properties of periodic orbits can be used to answer this question. Loosely speaking, a periodic orbit is defined as stable in the sense of Lyapunov if every trajectory originating at t=0 close enough to the periodic orbit remains close to the orbit for all time t. Obviously, TST cannot be exact if the pods is stable, since there are, by definition, trajectories crossing the pods that stay in its vicinity forever. For these trajectories and so TST is not exact. 

To prove Lyapunov stability let us surround the point O by a sphere 5 of radius e. Let > 0 be the minimum of the function V (x) on the surface of the sphere (it is strictly positive because all points of the sphere lie at a finite distance from the origin). Since V is continuous and V (O) = 0, it follows that for any point xq chosen sufficiently close to O the value of the function V (x) is strictly less than V. ... 

Fig. 9.1.1. Geometrical interpretation of a Lyapunov function. The surface V x) = constant has no contact with a vector field, i.e. the tangent at any point of the surface is transverse to the vector field so that every trajectory goes inwards the sphere.

For cases having an extra degeneracy (for example an equilibrium state with zero characteristic exponent and zero first Lyapunov value) the boundary of the stability region may lose smoothness at the point There may also exist situations where the boimdary is smooth but bifurcations in different nearby one-parameter families are different (i.e. there does not exist a versal one-parameter family, for example, such as the case of an equilibrium state with a pair of purely imaginary exponents and zero first Lyapunov value). In such cases the procedure is as follows. Consider a surface 971 of a smaller dimension (less than (p — 1)) which passes through the point and is a part of the stability boundary, selected by some additional conditions in the above examples the condition is that the first Lyapunov value be zero. If (fc — 1) additional conditions are imposed, then the surface 971 will be (P fc)-dimensional and it is defined by a system of the form... 

Consider first the case where the first Lyapunov value I2 is non-zero. Following the scheme outlined in the preceding section, we first derive the equation of the boundary of the stability region near e = 0. Next we will find the conditions under which is a smooth surface of codimension one. Finally, we will select the governing parameter and investigate the transverse families. 

So, we can now simply apply the results of the previous section. Thus, if the family (11.3.4) is in a general position (i.e. the rank of the matrix (11.2.18) is maximal if I2 0, this condition reduces to the inequality (11.2.9)), then the set of parameter values which corresponds to the existence of a fixed point with a imit multiplier and zero Lyapunov values Z2> , Zjb i, forms a smooth surface of codimension (fc — 1) that passes through e = 0, The families of maps transverse to SDt can be recast into the form... 

Then, in the case of general position the surface (which corresponds to equilibrium states with a pair of purely imaginary eigenvalues and with the first k — 1) zero Lyapunov values equal to zero) is a -smooth surface of codimension k passing through the point e = 0 in the parameter space. All transverse families in this case depend on k governing parameters /io,..., /ifc-i and may be written in polar coordinates as follows ... 

Remark. This statement remains valid (with obvious modifications) also in the case of the on-edge homoclinic loop to a degenerate saddle-node. In this case, /i is a vector of parameters (of dimension equal to the number of zero Lyapunov values plus one), and an additional bifurcation parameter e is introduced as before. A stable periodic orbit exists when the saddle-node disappears (the region /j> Dq m our notations), or when e > hhomi/j) fjL Do. Here, the surface e = hhomifJ ) corresponds to the homoclinic loop of the border saddle equilibrium Oi, as illustrated in Fig. 12.1.7. 

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