Asymptotically stable

One can paraphrase these definitions by saying that a solution (or motion) is stable if all solutions (or motions) which were initially close to it, continue to remain in its neighborhood a solution (or motion) is asymptotically stable if all neighboring solutions (motions) approach it asymptotically. 

If the characteristic exponents of (6-42) have negative real parts, the identically zero solution is asymptotically stable. 

If all roots of the characteristic equation have negative real parts, the point of equilibrium, = 0 is asymptotically stable whatever are the terms Xt. 

This stage of the process refers to the regular behavior. When the solution of a diiference scheme for problem (1) also possesses the properties similar to (2) and (3), the scheme is said to be asymptotically stable. We now deal with the scheme with weights... 

The symmetric scheme with conditionally asymptotically stable for... 

This provides support for the view that the solution is completely distorted. From such reasoning it seems clear that asymptotic stability of a given scheme is intimately connected with its accuracy. When asymptotic stability is disturbed, accuracy losses may occur for large values of time. On the other hand, the forward difference scheme with cr = 1 is asymptotically stable for any r and its accuracy becomes worse with increasing tj, because its order in t is equal to 1. In practical implementations the further retention of a prescribed accuracy is possible to the same value for which the explicit scheme is applicable. Hence, it is not expedient to use the forward difference scheme for solving problem (1) on the large time intervals. 

Thus, the three-layer scheme (13) of accuracy 0 h + r ) possesses the proper asymptotics as oo under the unique restriction tS < 1/2, which is not burdensome. Comparison of the final results with the two-layer scheme (4) reveals some formal advantage of the three-layer scheme over the symmetric two-layer scheme with cr =, which is conditionally asymptotically stable if we imposed the extra constraint t < = 1/V6 A,... 

Proposition 1. Whatever C2 is (i-G., C is a generalized pseudo-inverse (no unique solution) or C2 = (unique solution)), the observer (19) is asymptotically stable if the hypotheses H3 hold. 

Delgado et aL recently demonstrated that time-scale separation is an effective way to localize metabolic control to only a few enzymes. They considered model pathways in which the eigenvalues of the Jacobian of the system are widely separated (i.e., systems with time-scale separation). Their treatment assumes the system possesses a unique, asymptotically stable steady-state and that the reaction steps of the system under analysis are... 

But on an even less cosmological scale there is a deep difficulty involved with such an approach and that is to make a distinction between what is truly asymptotic (stable particles) and only approximately asymptotic (unstable particles). 

The symmetric scheme with <7 = which is absolutely stable in the usual sense yi < (jy0, is conditionally asymptotically stable for r < r0. Being concerned with the explicit scheme for <7 = 0, we observe that the condition of asymptotic stability r < 2 (6 + A)-1 practically coincides with the usual stability condition r < 2 A-1 for small values of 8/A. 

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... 

Fig. 2. Global bifurcation for equations (21)—(23). (a) Hamiltonian reference system, (b) Stable separ-atrix loop arising from the effect of the dissipative" perturbation I2. (c) Asymptotically stable...

The solution c(t, k, c0) is called asymptotically stable if it is stable according to Lyapunov and there exists values of 3 > 0 such that the inequality (74) results in... 

Since rest points are particular cases of the phase trajectories < ( , k, < ) = c0, the above definitions of stability according to Lyapunov are also valid for them. A rest point is stable according to Lyapunov if, for any e > 0 there exists values of 3 > 0 such that after a deviation from this point within 3, the system remains close to it (within the value) for a long period of time. A rest point is asymptotically stable if it is stable according to Lyapunov and there exists values of S > 0 such that after the deviation from this point within 3 the system tends to approach it at t - oo. 

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. 

The stability of the rest point for eqn. (73) depends on the roots of the characteristic equation. The rest point is asymptotically stable if the real parts of all the roots in eqn. (80) are negative. It is unstable if the real part of at least one of the roots is positive. In the case where some roots in eqn. (80) are purely imaginary and the rest of them have a negative real part, the stability cannot be established by using only linear approximations. In this case the rest point of eqn. (77) is stable but not asymptotic. 

Not a single steady-state point in kinetic equations cannot be asymptotically stable in Z) if it does not coincide with a point of G minimum. Indeed, let us denote this steady-state point as Na and assume that it is not the point of G minimum. Then in any vicinity of Na there exist points N for which G(N) < G(N0) (otherwise N0 would be a point of local minimum). But a solution of the kinetic equations whose initial values are such values of N, since G(N) < G(N0), at t - oo cannot tend to N0 G(N) can only diminish with time. Consequently, NQ is not an asymptotically stable rest point in D. In its vicinity in D there exists such N points that, coming from these points, solutions for kinetic equations do not tend to Na at t - oo. 

We have proved that any positive PDE N is asymptotically stable in the polyhedron D (it is even a "node ). In this point constructed above the G dissipation function, a minimum of free energy is achieved and the point of minimum is unique. Whence we obtain that TV is a point of minimum G and a unique positive PDE in D. 

For linear sets of differential equations having an cu-invariant limited polyhedron, an eigenvalue for the matrix of the right-hand side can be either zero or have a negative real part, i.e. after eliminating linear laws of conservation, a steady-state point of these systems becomes asymptotically stable. 

A question arises in what cases is a unique and asymptotically stable steady state realized ... 

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