Asymptotically stable solution

From a mathematical viewpoint when an ODE solution is altered slightly for instance, changing the initial conditions, a set of new curves may or may not show a different behaviour from the original one. From this fact, the heuristic basis for the concepts of stability and instability may be found. Moreover, a third possibility - asymptotically stable solutions - can arise when the altered solutions tend to the original one over a period of time, i.e., time nullifies any changes made to original solution (Martinez-Luaces, 2009e). 

According to this theorem, a solution of Eg. (3) may be computed by tracing a trajectory of the system of differential equations (56) till it approaches a constant solution curve that corresponds to a stationary point of E. Since the asymptotical stability is a local property, a curve emanating from x will approach an asymptotically stable solution xsx of Eg. (55a) with reliability only if x and w belong to the stability domain of that solution function. Thus, theoretically the situation is similar to that of Newton s method. In practice, however, most trajectories flow into a stability domain. So a curve tracing represents a systematic search. This is a considerable advantage for the computational practice. 

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. 

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... 

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. 

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. 

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... 

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. 

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. 

The bursting dynamics ends in a different type of process, referred to as a global (or homoclinic) bifurcation. In the interval of coexisting stable solutions, the stable manifold of (or the inset to) the saddle point defines the boundary of the basins of attraction for the stable node and limit cycle solutions. (The basin of attraction for a stable solution represents the set of initial conditions from which trajectories asymptotically approach the solution. The stable manifold to the saddle point is the set of points from which the trajectories go to the saddle point). When the limit cycle for increasing values of S hits its basin of attraction, it ceases to exist, and... 

Here, the asymptotic solution is a neutrally stable solution in Fig. 4.7-as one would expect. However, one also notices a constantly growing wave-front ahead of the asymptotic solution. For this case also, there are three modes with the leading mode neutrally stable and the other two modes are highly stable, given by oi = (0.3498239,0.0) 02 = (0.2149177,0.1454643) and 03 = (0.1604025,0.2593028). 

Finally we comment briefly on weakly elastic fluids. (See [26] and section 3.2.) We assume that the given Newtonian solution v satisfies ] v i < (ciRe), where Ci is some constant depending only on the domain of the flow this condition ensures that v is asymptotically Liapunov stable. (See e.g., [67].) Then, the viscoelastic solution (v ,Tt) close to (v, 0) is linearly (asymptotically) stable for c > 0 small enough. 

Remark. If Ai-I-A2>l then there is no positive solution of (3.2) and hence no interior equilibrium. In this case E2 is a globally asymptotically stable rest point. If A1-I-A2 < 1, there exists a unique interior rest point and E2 is unstable. 

Proof. Let 7 = (a (/), (/), 0) be the orbitally asymptotically stable periodic orbit of period T given by Theorem 5.4. (We have already noted that if there are several orbits then one must be asymptotically stable, by our assumption of hyperbolicity.) Let the Floquet multipliers of 7, viewed as a solution of (3.1), be 1 and p, where 0periodic orbit, define p( 3) by... 

With our standing hypotheses that all rest points are hyperbolic, Proposition 4.1 implies that if E does not exist then either E2 is asymptotically stable (all eigenvalues of the Jacobian are negative) or Ei exists and is asymptotically stable (all eigenvalues are negative). The results to follow establish that if , i—, 2, is asymptotically stable then it attracts all solutions (is globally asymptotically stable for positive initial conditions). Therefore, when E does not exist, one of the rest points E or E2 attracts all solutions of (3.2). 

It is easy to see that the numbers (yj(l) -1), i = 1,2, are the Floquet exponents corresponding to the identically zero periodic solution of (3.2). Consequently, the solution is asymptotically stable when both exponents are negative and unstable when one of the exponents is positive. Proposition 3.1 says more than this it states that competitors, is washed out of the chemostat if/(I) < 1, but this outcome has nothing to do with competition since it occurs even in the absence of the other competitor. As our main interest is in the effects of competition, we assume hereafter that... 

Consequently, [ is asymptotically stable unstable) if A12 < 0 (A12 > 0). An analogous statement holds for the periodic solution 2-... 

Although stability may in principle be computed, the calculation is extremely complicated. Numerical calculations suggest the asymptotic stability of the limit cycle, but the stability has not been rigorously established. Assuming that the solution is asymptotically stable, a secondary bifurcation can be shown to occur. The argument is quite technical and requires a form of a Poincare map in the appropriate function space it is analogous to the bifurcation theorem used in Chapter 3 for bifurcation from a simple eigenvalue. The principal theorem takes the form of a bifurcation statement. 

Imagine the reactor is initially at this steady state and at t 0 we perturb the temperature and concentration by small amounts. We would like to know whether or not the system returns to the steady state after this initial condition perturbation. If so, we call the steady-state solution (asymptotically) stable. If not, we call the steady state unstable. Obviously we can solve numerically the nonlinear differential equations to answer this question, but then we answer the question on a case-by-case basis. By linearizing the nonlinear differential equations, we can gain further insight without resorting to full numerical solution. Consider the Taylor series expansion of the.nonlinear functions f, fz... 

The solution to the linear model, Equation 6,43, is asymptotically stable if and only if the eigenvalues of J have strictly negative real parts. 

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