Lyapunov functions problem

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... 

Stationary domain walls. For the sake of simplicity, let us consider a plane domain wall perpendicular to the axis X, which separates two semi-infinite roll systems with wave vectors ki and lc2 [51], as shown in Fig. 15a. Because the Lyapunov functional densities of both roll patterns are equal, there is no reason for a motion of the domain wall, hence it is motionless [52]. The problem is governed by the following system of ordinary differential equations ... 

The second and, apparently, the more general idea is the idea of formation of a new scientific discipline - "Model Engineering" (Gorban and Karlin, 2005 Gorban et al., 2007). The subject of the discipline is the choice of an outset statement of the solved problem which is the most suitable (optimal) both for conceptual analysis and for computations. The transfer of kinetic description into the space of thermodynamic variables became a main method for this discipline. In the method the solved problem can be represented as one-criterion problem of search for extremum of the function that has the properties of the Lyapunov functions (monotonously moving to fixed points). 

The basins of attraction of the coexisting CA (strange attractor) and SC are shown in the Fig. 14 for the Poincare crosssection oyf = O.67t(mod27t) in the absence of noise [169]. The value of the maximal Lyapunov exponent for the CA is 0.0449. The presence of the control function effectively doubles the dimension of the phase space (compare (35) and (37)) and changes its geometry. In the extended phase space the attractor is connected to the basin of attraction of the stable limit cycle via an unstable invariant manifold. It is precisely the complexity of the structure of the phase space of the auxiliary Hamiltonian system (37) near the nonhyperbolic attractor that makes it difficult to solve the energy-optimal control problem. 

It is convenient to formulate the problem so that its solution would not change with smooth changes of variables. For example, let us determine the limits for In s)lt and In 0 (t, e)/t at t - 00 (i.e the Lyapunov indices for these functions). For smooth rough two-dimensional systems, if e is sufficiently small we will obtain... 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Let us address these in turn, without being entirely formal. The sensitive dependence on initial conditions can be taken to mean that if a pair of initial points of phase space is given which are separated by any finite amount, no matter how small, then the gap between these solutions grows rapidly (typically exponentially fast) in time. A problem with this concept is that we often think of molecular systems as having an evolution that is bounded by some sort of domain restriction or a property of the energy function the exponential growth for a finite perturbation can therefore only be valid until the separation approaches the limits of the accessible region of phase space. In order to be able to make sense of the calculation of an exponential rate in the asymptotic t oo) sense, we need to consider infinitesimal perturbations of the initial conditions, and this can be made precise by consideration of the Lyapunov characteristic exponents mentioned at the end of this chapter. 

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