Globally stable

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Exercise. Show that (3.4) guarantees that all solutions of (3.1) are globally stable. Construct an example where all solutions are globally stable and yet (3.4) does not hold. 

The solution curves in the (, i )-plane are sketched in fig. 29. Evidently all solutions tend to the stationary point, so that they are globally stable. 

The second stationary solution is 0 = 0 and it is unstable. Hence 0s is not globally stable although all solutions 0(t) with initial value 0(0) >0 are attracted by it. The condition (X.3.4) is not obeyed. 

It was demonstrated in X.5 that the O-expansion applies to multivariate master equations, provided that the macroscopic equations possess a single stationary solution, and that it is globally stable. The difference with the one-variable case was that no general method exists to solve the macroscopic equations. With respect to unstable situations, however, there is the added difference that the variety of possible instabilities is much larger than for a single variable. 

For y/3 1, one steady state exists and the regime is globally stable for all values of the Lewis number Lw. For 1 < y/3 < (y/3) and for sufficiently low values of Lewis number the system is again globally stable. Evidently for these conditions only one steady state occurs. For Lw > Lw, undamped oscillations exist. For supercritical values of y/3, y/3 > (y/3), and < a single steady state is stable or unstable according to the value of Lewis number. In the domain l>min < < max>... 

We can also present simpler estimates for vm when the inequality (142) is fulfilled and Z)0 contains a unique and globally stable steady-state point. Let us apply the Hirsch theorem [29, p. 185]... 

Thus if the flow velocity in a completely flowing (homogeneous) system is higher than a certain value, the balance polyhedron contains a unique steady-state point that is globally stable, i.e. every solution for the kinetic equations (139) lying in Da tends to it at t - oo. Note that a critical value for the flow velocity at which this effect is obtained can depend on the choice of balance polyhedron (gas pressure). 

As seen, a fast variable here must consider only 8Z. At any 0BZ and cA 0, the equation for 0Z has a unique and asymptotically globally stable steady-state solution... 

The local and global stable unstable manifold theorems (see, e.g.. Ref. 24, pp. 136-140) tell us the following are the (un)stable manifolds) ... 

As in the model of Section 2, the problem can be studied on its omega limit set with three rest points Eq,Ei,E2. A local stability analysis and, for some special cases, the asymptotic behavior of solutions were given in [E]. However, the populations cannot invade each other simultaneously El and E2 cannot be simultaneously unstable), so the persistence theory does not hold [E]. Moreover, for Michaelis-Menten dynamics, when one of the boundary rest points is locally stable and the other unstable, the locally stable one is globally stable [HWE]. In particular, the oscillation observed in the case of system (3.2) does not occur with (3.4). Indeed, the delayed system seems to behave much like the simple chemostat. 

Note that the definition of stable equilibrium is based on small disturbances certain large disturbances may fail to decay. In Example 2.2.1, all small disturbances to X = -1 will decay, but a large disturbance that sends x to the right of X = 1 will not decay—in fact, the phase point will be repelled out to -l-oo. To emphasize this aspect of stability, we sometimes say that x = -1 is locally stable, but not globally stable. 

The graph of flQ) is a straight line with a negative slope (Figure 2.2.4). The corresponding vector field has a fixed point where /((2) = 0, which occurs at 2 = CVg. The flow is to the right where /(G) >0 and to the left where /(G) <0. Thus the flow is always toward Q —it is a stable fixed point. In fact, it is globally stable, in the sense that it is approached from Q all initial conditions. 

A sinusoidally forced 7 C-circuit can be written in dimensionless form as X + X = A sin cot, where to > 0. Using a Poincare map, show that this system has a unique, globally stable limit cycle. 

Since P has s lope less than 1, it intersects the diagonal at a unique point. Furthermore, the cobweb shows that the deviation of x from the fixed point is reduced by a constant factor with each iteration. Hence the fixed point is unique and globally stable. 

Actually, for r < 1, we can show that every trajectory approaches the origin as z —> oo the origin is globally stable. Hence there can be no limit cycles or chaos for r[Pg.315]

Much of this picture is familiar. The origin is globally stable for r < 1. At r = 1 the origin loses stability by a supercritical pitchfork bifurcation, and a symmetric pair... 

Consider the map x ., =sinx . Show that the stability of the fixed point x = 0 is not determined by the linearization. Then use a cobweb to show that x = 0 is stable—in fact, globally stable. 

Prove that x = 0 is a globally stable fixed point for the map = -sinx . (Hint Draw the line = —x on your cobweb diagram, in addi-... 

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