Domain of attraction

Fig. 6.9. Domains of attraction for the two stable branches of stationary-state solution systems which have initial extents of conversion lying within the shaded region evolve to the lower branch those in the unshaded region approach the highest extent of reaction. In the region of multiple stationary states, the middle branch acts as a separatrix.

It may also seem sensible, if there are multiple solutions, to ask which of the states is the most stable In fact, however, this is not a valid question, partly because we have only been asking about very small disturbances. Each of the two stable states has a domain of attraction . If we start with a particular initial concentration of A the system will move to one or other. Some initial conditions go to the low extent of reaction state (generally those for which 1 — a is low initially), the remainder go to the upper stationary state. The shading in Fig. 6.9 shows which initial states go to which final stationary state. It is clear from the figure that the middle branch of (unstable) solutions plays the role of a boundary between the two stable states, and so is sometimes known as a separatrix (in one-dimensional systems only, though). 

This linear equation for 80 can easily be solved when 0(t) is known. It is called the linearized or variational equation associated with (3.1). When it turns out that the solutions of (3.5) tend to zero as x->oo it follows that this particular solution 0(t) of (3.1) is stable for small perturbations, or locally stable . Clearly (3.5) cannot tell anything about global stability, i.e., the effect of large perturbations. One can only conclude from the local stability that 0(t) has a certain domain of attraction every solution starting inside this domain will tend to 0(t) for large t. In this chapter, however, we postulate (3.4), which guarantees global stability. 

Fig. 34. The macroscopic rate equation for a bistable system and the domains of attraction.

In fact, it is clear from fig. 34 that there is a larger domain of attraction Da such that every solution (t) with (0) in Da tends to (pa. Two macrostates starting at two neighboring points in Da — A a will first move away from each other, but subsequently approach one another again, until they both end up in (pa. This is clear from fig. 28 and also from the variational equation (X.3.5). Accordingly the fluctuations about such a (t) will first grow 0, but subsequently decrease again. Hence they can still be described by the -expansion and there is still a relation between macrostates and suitably chosen mesostates. 

Higher order terms can be obtained by writing the inner and outer solutions as expansions in powers of e and solving the sets of equations obtained by comparing coefficients. This enzymatic example is treated extensively in [73] and a connection with the theory of materials with memory is made in [82]. The essence of the singular perturbation analysis, as this method is called, is that there are two (or more in some extensions) time (or spatial) scales involved. If the initial point lies in the domain of attraction of steady states of the fast variables and these are unique and stable, the state of the system will rapidly pass to the stable manifold of the slow variables and, one might... 

Stability on the initial state of the system. For the case of a linear, unstable system with bounded inputs and without external disturbances, Zheng and Morari (1995) have developed an algorithm that can determine the domain of attraction for the initial state of the system. 

As the statement of the theorem is a bit technical, we offer Figure 5.1 as a geometrical description of the result. The somewhat lengthy proof is postponed until the end of this section. Essentially, Theorem 5.1 says that if A[2 is positive (meaning that 1 is unstable) then there exists a fixed point , of P, corresponding to a periodic solution of (3.2), which has strong stability properties. The domain of attraction of , denoted by... 

Proof of Theorem C.3. If there were an attracting periodic orbit, then one could find a point x in its domain of attraction such that x periodic orbit. As / is a limit point of the positive orbit through x, there exists T > 0 such that Xrest point by Theorem C.2(b), contradicting our assumption that it converges to a nontrivial periodic orbit. ... 

Herex + denotes the set (xo,xi,..., xn ). It is assumed that the potential, C(x ), has a well whose minimum is at some point and which is surrounded by a domain of attraction, separated from the outside space by a potential barrier. 

Figure 14.3 is a contour plot that shows two such minima, representing the stable reactant and product configurations, and the transition surface (hne in the figure) that separate their domains of attraction. The minimum of V ) on this surface is the saddle point This dividing TV-dimensional surface is defined in the (TV -+ 1 (-dimensional space by the relation... 

Upon increasing R further, we find it difficult to obtain equilibrated 3-headed spins. Starting with initial data corresponding to a 3-headed spin we find that during the transient, two spots actually collide, leading to the annihilation of one spot and the mode collapses to a steady state 2-headed spin. Thus, for increasing R, the 3-headed spins are either unstable or have a small domain of attraction. 

For values of g that differ from a particular fixed point g (x), but lie within its domain of attraction (values that tend to g at t- oo), the excess quantity 3/(t g) can be written as a product. 

We conclude that as long as the mean waiting time and the variance of the jumps are finite, parabolic scaling leads to the Brownian motion in the limit e 0. The macroscopic equation for the density of particles is a scale-invariant diffusion equation. Infinite variance of jumps in the domain of attraction of a stable law leads to Ldvy processes, Levy flights. In the limit e 0, the particle position X (t) becomes self-similar with exponent 1/a. Recall that the random process X(t) is self-similar, if there exists a scaling exponent H such that X t) and e X(t/e) have the same distributions for any scaling parameter e. In this case we write... 

The predictor/corrector algorithm in Diva includes a stepsize control in order to minimize the number of predictor and corrector steps. Finally, the continuation package contains methods for the computation of the dominating eigenvalues of DAEs. This allows a stability analysis of the steady state solutions and a detection of local bifurcations for large sparse systems. As the continuation method is embedded into a dynamic simulator, the user has the opportunity to switch interactively from continuation to time integration. This allows additional investigations of transient behaviour or domains of attraction with the same simulation tool[2]. 

Condition 4 implies that the initial value rest point IIo = 0 for Eq. (7.16). This rest point is asymptotically stable by virtue 3 . 

In the first process, the Brownian motion velocity and presence of electrolytes influences the increase in viscosity of the immobilizing media, the coagulation velocity, the domain of attraction forces, and the concentration of colloidal solution. Consequently, from the Smoluchowski equation (Pomogailo and Kestelman 2005), the rate constant of particle coagulation, k, is inversely proportional to the viscosity of the media, r ... 

To make the picture clearer, we imagine a circular tube through which the orbit C threads (Fig. 3.1). We want to define X) for each A inside the tube. We use here a language appropriate to a three-dimensional state space, but actually we are working with an (>2)-dimensional system. Let G denote this w-dimen-sional tubular region containing all neighborhoods of C. The domain of attraction of C is assumed to contain G inside it. The tube may be thin to the extent that the perturbation is weak. 

Therefore, Oe J " is a domain of attraction and we can say that all initial states entering in set O will remain within the set at all future instances, thus O is an invariant set. 

Hence, the classical Newton process converges very rapidly if the initial guess belongs to the domain of attraction of a stationary point or if an iterate falls into this domain. Exactly that property... 

A numerical procedure should always be tested by using different initial guesses. In particular the robustness, i.e. the influence of small perturbations of the guess to the outcome, should be examined. Since in particular the descent methods behave like quasi-Newton methods in the vicinity of a minimizer, differences between them will become evident only if the initial guesses are chosen outside of the domain of attraction. (Recall, descent methods have just been created for that case ). Therefore, a descent method should also be tested with initial guesses far away from a minimizer. 

A limit cycle C( ) is defined as a closed trajectory, i.e. a periodic solution to the equations of motion, with the property that there exists a domain 9)c around C( ) so that all trajectories starting within % approach C(t) as oo.% can be denoted as the domain of attraction and C(t) as an attractor . A special case of an attractor is an infinitesimally small limit cycle , i.e. a stable focus. 

Conclusion From the Poincare-Bendixon theorem it now follows that within Sc a limit cycle, i.e. a periodic solution of the equations of motion C (t), must exist and that all other solutions starting from any point within Sc approach this limit cycle. Sc is a domain of attraction for C (t). 

---