Lyapunov equation

In this equation, ax2 and are the standard deviations of the stationary displacement and velocity response of the top floor, respectively, obtained by the Lyapunov equation [249] ... 

If now the so-called Lyapunov-equation Is utilized, the required variahces and 0 2 can be calculated as follows [4],... 

The chaotic nature of individual MD trajectories has been well appreciated. A small change in initial conditions (e.g., a fraction of an Angstrom difference in Cartesian coordinates) can lead to exponentially-diverging trajectories in a relatively short time. The larger the initial difference and/or the timestep, the more rapid this Lyapunov instability. Fig. 1 reports observed behavior for the dynamics of a butane molecule. The governing Newtonian model is the following set of two first-order differential equations ... 

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

Except for simple cases, it is generally a nontrivial task to compute the Lyapunov exponents of a flow. In trying to estimate A(x(0)) in equation 4.59, for example, the exponentially increasing norm, V t), may lead to computer overflow problems. 

Fig. 4.13 Lyapunov exponent versus a for 2.9 < a < 4 for the logistic equation see text.

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... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

On the other hand, it is well known that there is a relationship between Lyapunov exponents and the divergence of the vector field deduced from the differential equations describing a dynamical system. This relation provides a test on the numerical values obtained from the simulation algorithm. This relationship is, according to the definition of Lyapunov exponents ... 

The remaining task lies in the determination of the control matrix X and observer matrix Z such that the sufficient condition for robust performance, Eq. (22.28), holds. A Lyapunov-based approach is employed to obtain these two matrices. After some lengthy and complicated manipulations of Eq. (22.29) and the control structure shown in Fig. 22.3, the following two Riccati equations are derived, whose positive-definite solutions correspond to the control and observer matrices, X and Z. 

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

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... 

Index L is given to GL to distinguish it from the Lyapunov function for closed systems. Strictly speaking, it is not the Lyapunov function, since it cannot be differentiated on the hyperplanes prescribed by the equations 2, = 2 . Therefore, instead of estimating its derivative by virtue of eqn. (152), let us determine its decrease for a finite period of time x. Actually, we will find an ergodicity coefficient [42] for the matrix exp xK... 

Studies of linear systems and systems without "intermediate interactions show that a positive steady state is unique and stable not only in the "thermodynamic case (closed systems). Horn and Jackson [50] suggested one more class of chemical kinetic equations possessing "quasi-ther-modynamic properties, implying that a positive steady state is unique and stable in a reaction polyhedron and there exist a global (throughout a given polyhedron) Lyapunov function. This class contains equations for closed systems, linear mechanisms, and intersects with a class of equations for "no intermediate interactions reactions, but does not exhaust it. Let us describe the Horn and Jackson approach. 

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. 

There exists a second reason why Latora and Baranger have been forced to depart from the adoption of a density equation, thereby rather adopting the supposedly equivalent time evolution of a bunch of trajectories. This is due to the fact that the Lyapunov coefficients are local and might change with moving from one point of the phase space to another. It is important to stress this second reason because it is closely related to the directions which need to be followed to reveal by means of experiments the breakdown of the density perspective, and with it of quantum mechanics, in spite of the fact that so far the predictions of quantum mechanics have been found to fit very satisfactorily the experimental observation. 

The term to the right of the equal sign in Eq. (12.32) is the excess entropy production. Equations (12.31) and (12.32) describe the stability of equilibrium and nonequilibrium stationary states. The term 82S is a Lyapunov functional for a stationary state. 

Equations (12.27) and (12.64) show the stability of the nonequilibrium stationary states in light of the fluctuations Sev The linear regime requires P > 0 and dP/dt < 0, which are Lyapunov conditions, as the matrix (dAJdej) is negative definite at near equilibrium. 

Linear stability analysis does not provide information on how a system will evolve when a state becomes unstable. It does not distinguish between metastable and stable states when multiple local states are possible for given boundary conditions. Boundary conditions affect the value of the Lyapunov functional, and cause changes between stable and metastable states, hence altering the relative stability. An unstable state corresponds to the saddle points of the functional and defines a barrier between the attractors. Approximate solutions of nonlinear evolution equations may help us to understand how the system will behave in time and space. 

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