Stable nodes

Residue curves can only originate from, terminate at, or be deflected by the pure components and azeotropes in a mixture. Pure components and azeotropes that residue curves move away from are called unstable nodes (UN), those where residue curves terminate are called stable nodes (SN), and those that deflect residue curves are called saddles (S). 

As an example, consider the residue curve map for the nonazeotropic mixture shown in Eigure 2. It has no distillation boundary so the mixture can be separated into pure components by either the dkect or indkect sequence (Eig. 4). In the dkect sequence the unstable node (light component, L) is taken overhead in the first column and the bottom stream is essentially a binary mixture of the intermediate, I, and heavy, H, components. In the binary I—H mixture, I has the lowest boiling temperature (an unstable node) so it is recovered as the distillate in the second column and the stable node, H, is the corresponding bottoms stream. The indkect sequence removes the stable node (heavy component) from the bottom of the first column and the overhead stream is an essentially binary L—I mixture. Then in the second column the unstable node, L, is taken overhead and I is recovered in the bottoms. 

If Xi and A,2are real numbers and both have negative values, the values of the exponential terms and hence the magnitudes of the perturbations away from the steady-state conditions, c, and T, will reduce to zero, with increasing time. The system response will therefore decay back to its original steady-state value, which is therefore a stable steady-state solution or stable node. 

Stable free radical polymerization (SFRP), 20 442, 443 Stable node(s)... 

Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength and saturation 6 of the ATPase reaction. Shown are the transitions to instability via a saddle node (SN) and a Hopf (HO) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive > 0. Within region (ii), the metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.

Real, both — ve Stable node (monotonic approach)... 

FiC. 3.4. Representations of the different singular points in the concentration phase plane (a) stable node, sn (b) stable focus, sf (c) unstable focus uf (d) unstable node, un (e) saddle. point, sp. 

When the dimensionless reactant concentration is slightly greater than the lower root of eqn (3.65), i.e. /i > /if, the stationary state must be an unstable focus, which becomes stable as /i passes through /if. Below this the state is first a stable focus then, as /i approaches zero, a stable node. 

These have only one true stationary-state solution, ass = pss = 0 (as t - go), corresponding to complete conversion of the initial reactant to the final product C. This stationary or chemical equilibrium state is a stable node as required by thermodynamics, but of course that tells us nothing about how the system evolves in time. If e is sufficiently small, we may hope that the concentrations of the intermediates will follow pseudo-stationary-state histories which we can identify with the results of the previous sections. In particular we may obtain a guide to the kinetics simply by replacing p by p0e et wherever it occurs in the stationary-state and Hopf formulae. Thus at any time t the dimensionless concentrations a and p would be related to the initial precursor concentration by... 

The full curves in the /c-/r parameter plane, expressed by (4.46) and (4.47), are shown in Fig. 4.3. There are two closed loops emerging from the origin. The outer loop, given by the upper root in (4.46) and the lower root with 0 < 6SS < 1, also touches the k axis at (fi = 0, k = 1). Outside this locus the stationary state is a stable node inside this loop it is a focus (we will discuss stability within this region in the next subsection). The maximum in this curve occurs for k = (3 + y/S) exp[ — (3 — /5)] 1.787. For larger values of the dimensionless reaction rate constant, e.g. for high ambient temperatures, no damped oscillatory states will be found. 

The zero conversion state is always a stable node. 

This is thus a stable node or a stable focus, depending on the size of the wave velocity c. For c > 2, the eigenvalues are real for lower wave velocities with... 

Fig. 2.1. A stable node. Four solutions of the Lotka model, equations (2.1.22)—(2.1.23) with the distinctive parameter Kp/401 = 2 are presented. The starting point of each trajectory is...

The way in which the solution rij(f) approaches its stationary value n = rii(oo) for a system with two degrees of freedom can be easily illustrated in a phase space (tia,wb) after eliminating time t - Fig. 2.1. This type of singular point is called a stable node. 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

A feature that, to our knowledge when we discovered it, had not been seen before in forced oscillators (Marek and his co-workers have also observed it (M. Marek, personal communication)) is the folding that occurs in the left side of the 3/2 and 2/1 resonance horns. Within these folds there are two sets of stable nodes and two sets of saddles, so that bistability between the two sets exists. There are also cases of bistability between subharmonic responses of period 3 and a torus in the top of the period 3 resonance horns. In addition to the implication of bistability, the fold in the side of the 3/2 resonance horn may be of mathematical significance. Aronson et al. (1986) put forth the mathematical conjecture that if the period 3 resonance horn is a simple disc-... 

Fig. 2. Types of stationary points on the plane, (a), (c), (e) Stable nodes (b), (d), (f) unstable nodes (g) saddle point (h) stable focus (i) unstable focus, (k) whirl.

The character of trajectories is illustrated in Fig. 2(c)where v is a straight line specified by the equation = C12/C22. In both cases the rest point is also called a stable node. 

The principle of detailed equilibrium accounts for the specific features of closed systems. For kinetic equations derived in terms of the law of mass/ surface action, it can be proved that (1) in such systems a positive equilibrium point is unique and stable [22-25] and (2) a non-steady-state behaviour of the closed system near this positive point of equilibrium is very simple. In this case even damped oscillations cannot take place, i.e. the positive point is a stable node [11, 26-28]. 

A dynamic bifurcation occurs when the dynamic behavior of the solution to a system undergoes a qualitative change. For example, a subcritical Hopf bifurcation occurs when a dynamic system changes from a stable node to a limit cycle. Again, AUTO can be used to determine parameter changes that cause this bifurcation to occur. 

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