Lyapunov-type

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. 

To perform the robust optimum design, the OF mean and standard deviation are numerically evaluated with a new procedure based on a Lyapunov type equation. Robustness is formulated as a multiobjective optimization problem, in which both the mean and the standard deviation of the deterministic OF are minimized. The results show a significant improvement in performance control and OF real values dispersion limitation if compared with standard conventional solutions. Some interesting conclusions can be reached with reference to the results obtained for the adopted examples. With reference to TMD efficiency in vibration reduction, the real structural performance obtained by using conventional optimization has a reduced efficiency compared to those obtained when system uncertainty parameters is properly considered. With reference to the obtained robust solutions, it can be noted that they can control and limit final OF dispersion by limiting its standard deviation. Moreover, this goal is achieved by finding optimal solutions in terms of DV that induce an increase in OF mean value. 

We stress that the chaos identified here is not merely a formal result - even deep in the quantum regime, the Lyapunov exponent can be obtained from measurements on a real system. Quantum predictions of this type can be tested in the near future, e.g., in cavity QED and nanomechanics experiments (H. Mabuch et.al., 2002 2004). Experimentally, one would use the known measurement record to integrate the SME this provides the time evolution of the mean value of the position. From this fiducial trajectory, given the knowledge of the system Hamiltonian, the Lyapunov exponent can be obtained by following the procedure described above. It is important to keep in mind that these results form only a starting point for the further study of nonlinear quantum dynamics and its theoretical and experimental ramifications. 

From the results presented in this chapter, more advanced studies from the bifurcation theory can be planed. For example, inside the lobe, the behavior of the reactor is self-oscillating, i.e. an Andronov-Poincare-Hopf bifurcation can be researched from the calculation of the first Lyapunov value, in order to know if a weak focus may appear, or the conditions which give a Bogdanov-Takens bifurcation etc. Finally, it is interesting to remark that the previously analyzed phenomena should be known by the control engineer in order to either avoid them or use them, depending on the process type. 

This system in its linear version (i.e., when e = 0) is a dynamical filter. Suppose that the oscillators interact with each other with the interaction parameter a = 0.9. The frequency 00 of the external driving field varies in the range 0 < < 4.2. The other parameters of the system are A 200, coq 1, c 0.1, and = 0.05. The autonomized spectrum of Lyapunov exponents A-4, >,5 versus the frequency to is presented in Fig. 23. In the range 0 < < 0.2 the system does not exhibit chaotic oscillation. Here, the maximal Lyapunov exponent Xi = 0 and the spectrum is of the type 0, —, —, —, (limit cycles). 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

Conditions (66) and (67) ensure the existence of Lyapunov s convex function for eqns. (17) GGjdNi = fit. With a known type of the potentials /i, for which condition (1) is fulfilled, one can obtain Lyapunov s thermodynamic functions for various (including non-isothermal) conditions. Thus, for an ideal gas and the law of mass action [16]... 

An important conclusion follows from the time monotonic manner (2.31) of changes in values P and d S/dt. In case the system exists near thermody namic equilibrium, the system s spontaneous evolution cannot generate any periodical auto oscillating processes. In fact, periodical processes are described along the closed evolution trajectories, which would make some thermo dynamic parameters (concentration, temperature, etc.) and, as a result, values Ji and Xj return periodically to the same values. This is inconsistent with the one directional time monotonic changes in the P value and with the con stancy of the latter in the stationary point. In terms of Lyapunov s theory of stability, the stationary state under discussion corresponds to a particular point of stable node type (see Section 3.5.2). 

With due regards of the properties of expression (3.6), one can find the Lyapunov functionals for various types of kinetic schemes. This can be done, for example, by consecutive integrating kinetic equations of type (3.10) over rushes of each of the intermediates and combining the results into one expression. Such a procedure is always available for the intermediate linear schemes of the transformations. 

Figure 3.4 Stability types of particular points in the system of linear differential equations dy/dt = ay + bz dz/dt = cy + dz in coordinates (A, y) (according to A. M. Lyapunov).

In general, the presence of a catalyst in the system with the catalytic intermediate linear transformations and, as a consequence, the necessarily existent balances of type (4.1) in respect to aU forms of active centers does not affect the stability of the stationary state in the system. This is due to specific features of these systems where the said mass balance does not hinder writing the Lyapunov functions. 

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. 

The nature of the intramolecular motion may also be identified by studying the way the separation of two trajectories evolves with time [353]. If the motion is regular (quasi-periodic) the separation is linear with time, but exponential if the motion is irregular (chaotic). If the separation is exponential, the rate of the separation — called the Lyapunov characteristic exponent — provides qualitative information concerning the IVR rate for the chaotic trajectories. This type of analysis has been reported, for example, for NO2 [271] and the Cl CHsBr complex [354]. 

Other types of Lyapunov s functions may also be constructed. 

Considerations of this type are characteristic of the Lyapunov method of examination of the stability of a stationary point the function W defined by equation (A18) is an example of the so-called Lyapunov function. Note also that to draw the conclusion about an asymptotic stability of the stationary point (0, 0), in addition to inequality (A 19) deriving from properties of the Lyapunov function and properties of the investigated system, inequality (A20) was also required. 

The conditions (A33) may be shown to include cases (al), (a2), (a3), (d) from Section 5.1. The existence of the Lyapunove function implies the stability of stationary points of this type. 

The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

The application of FSLE was demonstrated in the analysis of experimental (Boffetta et al., 2000) and geophysical flows (Lacorata et ah, 2001). Instead of calculating the average over many particle pairs in Eq. (2.113), one can measure the separation rates as a function of the initial position of the trajectories. This type of analysis has been used for visualizing the location of transport barriers and coherent structures in geophysical flows (Boffetta et ah, 2001 d Ovidio et ah, 2004, 2009). Note that the distribution of finite time Lyapunov (Lapeyre, 2002) exponents also produces similar structures in some range of parameters, namely small initial separation and r (for FSLE) and large time (for FTLE). 

We note that the above results are not limited to the case of linear decay, but also apply to any kind of decay-type or stable reaction dynamics in a flow with chaotic advection (Chertkov, 1999 Hernandez-Garcfa et ah, 2002). In such systems where the reaction dynamics is nonlinear, the decay rate b should be replaced by the absolute value of the negative Lyapunov exponent of the Lagrangian chemical dynamics given by the second equation in (6.25), that represents the average decay rate of small perturbations in the chemical concentration along the trajectory of a fluid element. 

Recall that the roll pattern becomes stable for T > Ti = 4/3. Hence, in the interval Ti < T < Ts the Lyapunov function has 4 local minima, three of them correspond to three types of roll patterns, and one of them corresponds to hexagons. The basins of attractions between them are separated by stable manifolds of some additional saddle-point stationary solutions, corresponding to squares (e.g. R = R2 0, Rs = 0) and skewed hexagons" (e.g. R = R2 7 R3 7 0). Finding the latter solutions is suggested to the readers as an exercise. 

This definition is related to a phase trajectory and as a rest point is a particular type of phase trajectory, this definition also applies to rest points. A rest point is Lyapunov stable if for any > 0 a value of 5 > 0 exists such that after a deviation from this point within 5, the system remains close to it, within the value of Sc, for a long period of time. A rest point is asymptotically stable if it is Lyapunov stable and values of 5 > 0 exist such that after a deviation from this point within 5, the system approaches the rest point at cx). 

The following question arises Since the actual solution z(x, i, ju) of (5.40) can approach only stable stationary solutions s t— °°, which types of solutions are stable We are interested in stability of the stationary solution z(jc, fi) in the sense of Lyapunov, that is, whether for any s >0 there exists (e) such that if... 

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

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