Locally stable

In principle, energy landscapes are characterized by their local minima, which correspond to locally stable confonnations, and by the transition regions (barriers) that connect the minima. In small systems, which have only a few minima, it is possible to use a direct approach to identify all the local minima and thus to describe the entire potential energy surface. Such is the case for small reactive systems [9] and for the alanine dipeptide, which has only two significant degrees of freedom [50,51]. The direct approach becomes impractical, however, for larger systems with many degrees of freedom that are characterized by a multitude of local minima. 

Proposition 1. For e sufficiently small the periodic orbit 7(t) of (7) survives as aperiodic orbit 7 (t) = rj(t)+0(e), of (10) having the same stability type as 7(t), and depending on e in a C2 manner. Moreover, the local stable and unstable manifolds Wlsoc( ye(t)) and Wfoc e(t)) of 7 (t) remain also e-close to the local stable and unstable manifolds WfoMt)) and Wfioc( (t)) of ft), respectively. 

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

Figure 16. The stability of a steady state, as determined by a rate balance plot [287]. Left panel The rate of synthesis and consumption vcon of a substrate S. Right panel The (net) flux difference vnet vco . The steady state value S° is locally stable After transient perturbation...

Crucial for the later analysis, the decision whether a state is locally stable or not is entirely determined by the slope of the zero-crossing at the steady state (the partial derivative). If the net flux of Eq. (65) has a positive slope, any infinitesimal perturbation will be amplified. 

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Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

The local stability in the neighborhood of the second set of turning points is simply deduced because no new HB point is found the intermediate branch is locally unstable, whereas the partially ignited branch and fully ignited branch are locally stable. The temperature range for self-sustained oscillations is larger at this higher pressure. 

As the pressure increases further, a second HB point (HB2) appears at the extinction point E and shifts toward the other HB HBi) point. An example is shown for 4 atm in Fig. 26.1c. Ignition Ii is no longer oscillatory, because the stationary partially ignited branch becomes locally stable in the vicinity of /i. Time-dependent simulations indicate that the two HB points are supercritical, i.e., self-sustained oscillations die and emerge at these points with zero amplitude. In this case, the first extinction Ei defines again the actual extinction of the system. 

Stability relates to the behaviour of a system when it is subjected to a small perturbation away from a given stationary state (or if fluctuations occur naturally). If the perturbation decays to zero, the system has some in-built tendency to return back to the same state. In this case it is described as locally stable. (The qualification local means that very large perturbations may have different consequences.) We will introduce the relatively simple mathematical techniques required to determine this local stability of a given state in Chapter 3. It will also be useful before then to reduce the reaction rate equations (2.1)—(2.3) to their simplest possible form by introducing dimensionless variables and quantities. 

If the value of the reactant decay rate e is not very small, higher-order correction terms will become significant more quickly. Exact (i.e. precisely computed) concentration histories will not be well appproximated by the pseudo-stationary forms (3.72) and (3.73) even when the state is locally stable. During any possible period of oscillatory behaviour, the number of oscillations will naturally decrease as e increases, as expressed by eqn (3.79). In addition to this, however, the time for the first excursion to develop, which... 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

For given values of the rate constants kx — and K, the behaviour of this reduced system is determined by the constant value of c. If the concentration of C is maintained at a sufficiently low value, the unique stationary-state solution (ass, bss) is locally stable. As c is increased, however, there may be a Hopf bifurcation, at c = c say. For c > c, then, the stationary state will be unstable and surrounded by a stable limit cycle whose amplitude grows in some non-linear fashion with c. This bifurcation behaviour is illustrated in Fig. 13.18. 

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. 

Exercise. Show that (VI.9.7) has one unstable solution. All other solutions are locally stable but not globally. 

Fig. 27. Two locally stable stationary solutions separated by an unstable one.

Now consider the evolution in a situation as in fig. 34. There are three stationary macrostates 0fl, 0j>, c, of which 0a and 0C are locally stable and (j)b is unstable. Of course, even the pure macroscopist would not regard (j)b as a realizable state, on the ground that a system in (j>b would be caused to move into either 0a or 0C by the smallest perturbation. Systems having a macroscopic characteristic as in fig. 34 are called bistable . There are numerous examples the ones that occur most often in the literature are the laser (section 9 below), the tunnel diode 0, and the Schlogl reaction (X.3.6). The macroscopic rate equation for this reaction is... 

Now examine the bistable case on the mesoscopic level. First consider the locally stable solution a. As a x, 0W>a) < 0 there will be a domain Aa around 4>a where (X.3.4) holds. Any macrostate 0(f) that starts at a 0(0) inside Aa... 

Dependence of the energy of the grain on is shown on Fig. 3. For the Hamiltonian with no spin relaxation that we have considered so far all the values of correspond to the stable physical states. If one turns on a finite spin relaxation, then only the minima of E( ) will describe stable or locally stable states. In this case, from Fig. 1 we see that, for h < Aq/2 only the... 

In what way is an unsteady kinetic model investigated to elucidate whether the rest point is locally stable In this case a combination of approaches, which can be called a "rite , is used ... 

The simplest situation arises for bistable systems in which the basic pattern is a front, i.e. an interfacial region that connects the two locally stable stationary states and propagates in space.10 Thereby one of the two states expands on the expense of the other one. Two experimental examples of potential fronts propagating along the... 

From now on, it will be assumed that the system is locally stable, i.e. that all the eigenvalues have a negative real part, viz. 

A free radical chain reaction mechanism gives rise to locally unstable equations if branching processes are not compensated for by termination processes. A straight chain radical mechanism gives rise to equations both locally stable and potentially stiff. The stiffness is due to the large differences of reactivity between molecular and radical species, but also between the free radicals themselves. 

After evaluating this solution, we obtain oj, and then, from one of (11.11), we calculate the critical value of r. Figure 11.7 shows the frequency u> (upper panel) and the critical value of r (lower panel) as functions of d> (1). We can say that the real parts of ( will be positive, and thus (11.8) will be unstable, if and only if the actual delay r° is greater than r. For example, for 4> (1) = 2 and the set 7 and A values, r 44.049 and any r° > r triggers periodic oscillations following some perturbation in the system. Otherwise, the system is locally stable. 

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