Lyapunov stable

An equilibrium is called asymptotically stable if it is Lyapunov stable and if 35 such that... 

These rotations differ from one another in their stability. One can verify that the stationary solutions K(t) corresponding to the major and the minor axes of inertia (in the case where all the moments of inertia are pairwise different) are Lyapunov-stable and the solution K(t)y which corresponds to the mean axis, is unstable. 

Here, Cq and Co are sets of initial concentrations (t=0) in, respectively, the unperturbed and the perturbed case. The solution c(t, k, Co) is called asymptotically stable if it is Lyapunov stable and a value of 5 exists such that the inequality of Eq. (7.7) results in... 

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). 

If some roots are purely imaginary and the others have a negative real part, the rest point of Eq. (7.1) is Lyapunov stable but not asymptotically stable. 

Now let us define die global stability of rest points. The rest point Co is called a global asymptotically stable rest point within the phase space D if it is Lyapunov stable and for any condition do QD the solution c(t,k,do) approaches Cq at (x>. An analysis of the problem of... 

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). 

First, let us recall some definitions. An equilibrium state O is said to be Lyapunov stable if for any e > 0, there exists 6 > 0 such that any trajectory which starts from a ( -neighborhood of O never leaves its -neighborhood. Otherwise, the equilibrium is said to be unstable. 

Theorem 9.1. If the equilibrium state is Lyapunov stable in the center manifold then the equilibrium state of the original system (9.1.1) is Lyapunov stable as well Moreover if the equilibrium state is asymptotically stable in the center manifold, then the equilibrium state of the original system is also asymptotically stable. 

Investigations performed by Poincare and Lyapunov have shown that only slightly oblate spheroids are stable and for them... 

Figure 3 depicts the spectmm of Lyapunov exponents in a hard-sphere system. The area below the positive Lyapunov exponent gives the value of the Kolmogorov-Sinai entropy per unit time. The positive Lyapunov exponents show that the typical trajectories are dynamically unstable. There are as many phase-space directions in which a perturbation can amplify as there are positive Lyapunov exponents. All these unstable directions are mapped onto corresponding stable directions by the time-reversal symmetry. However, the unstable phase-space directions are physically distinct from the stable ones. Therefore, systems with positive Lyapunov exponents are especially propitious for the spontaneous breaking of the time-reversal symmetry, as shown below. 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

Until further notice we use stable for what is technically called asymptotically stable in the sense of Lyapunov see, e.g., J. La Salle and S. Lefshetz, Stability by Liapunov s Direct Method (Academic Press, New York 1961). 

In the Lyapunov classification they are called stable but not asymptotically stable . In the theory of fluctuations it is more natural to classify this case as unstable, pursuant to the footnote in X.3. 

If the real parts of all eigenvalues e, Ree, <0 are negative, according to the Lyapunov theorem [14, 15] the stationary point is asymptotically stable... 

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. 

Let us give accurate definitions. Let c(t, k, c0) be a solution for eqn. (73) which satisfies the initial condition c(i0) = c0- This solution is called stable (according to Lyapunov) if for any infinitesimal e > 0 there exists values of 5 > 0 such that the inequality... 

So far we have defined the local stability ("there exists such <3 as. . . ). Now let us define the global stability for rest points. The rest point c0 is called globally asymptotically stable (as a whole) within the phase space D if it is stable according to Lyapunov, and for any initial conditions d0e D the solution c(t, k, cLa) tends to approach c 0 at t - oo. 

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 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. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global 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. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

Before we can demonstrate the connection between process control and Eq. (A.20), we need to introduce the concept of Lyapunov functions (Schultz and Melsa. 1967). Lyapunov functions wnre originally designed to study the stability of dynamic systems. A Lyapunov function is a positive scalar that depends upon the system s state. In addition, a Lyapunov function has a negative time derivative indicative of the system s drive toward its stable operating point where the Lyapunov function becomes zero. Mathematically we can describe these conditions as... 

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). 

If the positively defined Lyapunov function exists for a particular kinetic scheme, this scheme has the stationary state that is stable in respect to the concentrations of the intermediates, whether they are close to or far from thermodynamic equilibrium. 

In terms of thermodynamics, the energy dissipation P (or the positively defined Lyapunov function O) has its local minimums in the stable stationary points, and spontaneous jumping between stable stationary states in the system is only allowed when the identical "input" parameters are inherent in two states this may be, for example, common affinity ArE which is given from outside and provides the process. Therefore, we can consider these transitions as related to overcoming some dynamic "potential" barrier (see following). 

X (k < 0) and reside in regions I and II (y > 0, A > 0), which are sepa rated by parabola y /4 = A. In region I, the inequality y - 4A > 0 is additionally fulfilled and, consequently, = 0. For this reason, a minor deviation of the system from the initial point is inevitably followed by turning the system back (evolution) to the same point along curves plotted schematically in Figure 3.4. According to A. M. Lyapunov s definition, region I relates to stable nodes. 

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