Point unstable node

The analysis of linearized sytem thus allows, when conditions (1)—(3) are met, us to find the shape of phase trajectories in the vicinity of stationary (singular) points. A further, more thorough examination must answer the question what happens to trajectories escaping from the neighbourhood of an unstable stationary point (unstable node, saddle, unstable focus). In a case of non-linear systems such trajectories do not have to escape to infinity. The behaviour of trajectories nearby an unstable stationary point will be examined in further subchapters using the catastrophe theory methods. 

Fig. 13.7 Zero- and infinite-isoclinic lines and the singular point of the velocity of a small drop. The external electric field is absent (S2 = 0) G- /Vi = /V2 = 0 /Vi/V2 < 0 c-0 < Ni/V2 < 1 cf - N1N2 > 1.5 1,3- infinite-isoclinic lines 2 - zero-isoclinic lines 4 - separatix of singular point A-saddle 5 - separatix of singular point B-saddle C -singular point - unstable node.

The whole concentration space can be filled with one or more residue curve bundles. Each residue curve bundle has its own initial point (unstable node) and its own final point (stable node). Various bundles differ from each other by initial or final points. 

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. 

If we consider the well-stirred system, the stationary state has two Hopf bifurcation points at /r 2, where tr(U) = 0. In between these there are two values of the dimensionless reactant concentration /r 1>2 where the state changes from unstable focus to unstable node. In between these parameter values we can have (tr(U))2 — 4det(U) > 0, so there are real roots to eqn (10.76). 

If the dimensionless rate constant satisfies inequality (10.78), the well-stirred system again has two Hopf bifurcation points fi and n. However, within the range of reactant concentrations between these, the uniform state also changes character from unstable focus to unstable node at n and n 2, as shown in Fig. 10.10. 

Note here, without proof, one of the synergetic theorems about limit cycles [14, 15] a stable limit cycle contains at least one singular point or the unstable node of focus-type exists. 

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. 

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.

Phase trajectories extend far from the singular point. It is an unstable node... 

Let Bt be an unstable node or a focus. Then will be a trace of the matrix for the linear approximation at this point. 

As S continues to increase we reach a point marked by an open circle. Here, the equilibrium point undergoes a saddle-node bifurcation. Somewhere before this bifurcation, the unstable focus point has turned into an unstable node with two positive real eigenvalues. In the saddle-node bifurcation, one of these eigenvalues... 

Fig. 4.7(c). For Dar> 0.166 (Fig. 4.7(d)), only three singular points remain in the system pure MeOH which is a stable node at any Damkohler number pure isobutene which is a saddle point at any Damkohler number and the above-mentioned unstable node which is located outside the triangle. 

Figure 4.27 shows residue curve maps for the reactive reboiler at three different Damkohler numbers. In the nonreactive case (Da = 0 Fig. 4.27(a)), the map topology is structured by one unstable node (pure B), one saddle point (pure C), and one stable node (pure A). Since pure A is the only stable node of nonreactive distillation, this is the feasible bottom product to be expected in a continuous distillation process. 

In analogous manner, residue curve maps of the reactive membrane separation process can be predicted. First, a diagonal [/e]-matrix is considered with xcc = 5 and xbb = 1 - that is, the undesired byproduct C permeates preferentially through the membrane, while A and B are assumed to have the same mass transfer coefficients. Figure 4.28(a) illustrates the effect of the membrane at nonreactive conditions. The trajectories move from pure C to pure A, while in nonreactive distillation (Fig. 4.27(a)) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point This is due to the fact that the membrane changes the effective volatilities (i.e., the products xn a/a) of the reaction system such that xcc a. ca > xbbO-ba-... 

At Da = 0.4 (Fig. 4.29(b)), the two saddle points from the pure vertices move into the composition triangle. The stable node from the 1,4-BD vertex moves to the kinetic azeotrope at x = (0.0328, 0.6935). Pure water and pure THF now become stable nodes. The unstable node between water and THF remains unmoved, and forms two separatrices with the two saddle points. Thereby, the whole composition space is divided into three subspaces which have each a stable node, namely pure water, pure THF and the kinetic azeotrope. 

The effect of a Knudsen-membrane on process behavior is illustrated in Fig. 4.30(a), which is valid at nonreactive conditions. Compared to Fig. 4.29(a), the unstable node on the THF-water edge is moved closer to the water vertex by application of the Knudsen-membrane, while the two saddle points and stable node are not affected. 

Direction of residue curves, pointing from lower to higher temperatures. An unstable node is a point from which trajectories emerge the lowest boiler. A stable node is a point to which trajectories end up the highest boiler. A saddle is an intermediate transition point (intermediate boiler) to which trajectories go and leave. 

Figure A.2 (left) shows the construction of a distillation for an ideal ternary system in which A and C are the light (stable node) and the heavy (unstable node) boilers, while B is an intermediate boiler (saddle). The initial point xiA produces the vapor y, that by condensation gives a liquid with the same composition such that the next point is xi 2 = y,, etc. Accordingly, the distillation line describes the evolution of composition on the stages of a distillation column at equilibrium and total reflux from the bottom to the top. The slope of a distillation line is a measure of the relative volatility of components. The analysis in RCM or DCM leads to the same results.

Stable node O Unstable node Saddle point... 

Fig. 5.8. Phase plane portraits of different possible steady-state singularities (i) stable node, trajectories approach singular point without overshoot (ii) stable focus showing damped oscillatory approach (iii) unstable focus showing divergent oscillatory departure (iv) unstable node showing direct departure (v) saddle point x showing insets and outsets and...

The other three fixed points can be shown to be an unstable node and two saddles. A computer-generated phase portrait is shown in Figure 6.6.7. 

Find the index of a stable node, an unstable node, and a saddle point. 

Solution As shown previously, the system has four fixed points (0,0) = unstable node (0,2) and (3,0) = stable nodes and (1,1) = saddle point. The index at each of these points is shown in Figure 6.8.9. Now suppose that the system had a Q closed trajectory. Where could it lie ... 

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