Stationary points 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. 

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). 

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 form of the solutions to the simplified model were analysed by examining the existence and types of the pseudo-stationary points of the equations for d0/dr = d 3/dr = 0 and values of e in the range 0—1 (r = Figure 29 shows the oscillation of a multiple-cool-flame solution about the locus of such a pseudo-stationary point, Sj. The initial oscillation is damped while Si is a stable focus. The changing of Si into a unstable focus surrounded by a stable limit cycle leads to an amplification of the oscillation which approaches the amplitude of the limit cycle. When Si reverts to a stable focus, and then a stable node, the solution approaches the locus of the pseudo-stationary point. In this way an insight may be gained into the oscillatory behaviour of multiple cool flames. 

Fig. 61. Phase trajectories in the vicinity of the stationary point of an unstable node type.

If eigenvalues are real and different, then a given dynamical system is locally (in the vicinity of a stationary point) equivalent to a certain structurally stable gradient system (this is an unstable node when 2, > 0, X2 > 0 a saddle when A, > 0, X2 < 0 or Xt < 0, X2 > 0 a stable node when Xx < 0, X2 > 0). In the remaining cases a dynamical system is not locally equivalent to a gradient system. 

In terms of the residue curve equation, this means algebraically solving for points where x = y. Generally speaking, these stationary points may be classified into three main types a stable node, an unstable node, and a saddle point, depicted in... 

As with the distillation ROMs, the profiles lying outside the MET may not be physically achievable, but the relevance of this global map is veiy important, and will be highlighted in subsequent sections. Note that it is also possible to identify stable, unstable, and saddle nodes, and each of these stationary points nature and location provide insight into the behavior of the curves (refer to Section 2.5.2). 

Points of pure components and azeotropes are stationary or singular points of residue curve bundles. At these points, the value dXi /dt in Eq. (1.11) becomes equal to zero. A stationary point at which all residue curves come to an end is called a stable node (the temperature increases in the direction of this point). A specific point at which all residue curves start is called an unstable node (the temperature... 

Figure 1.4. Types of stationary points of three-component mixtures (a) one-component stable node, (b) one-component unstable node, (c) one-component saddle, (d) two-component stable node, (e) two-component unstable node, (f) two-component saddle, (g) three-component stable node, (h) three-component unstable node, and (i) three-component saddle. Arrows, direction of residium curves.

The totality of all bonds characterizes the mixture s structure. The bond serves as the elementary nonlocal characteristic of the residue curve bundle structure. Bonds form bond chains. The bond chains of maximum length connect the unstable node A and the stable node A+ of the distillation region Reg". Let s call a polyhedron formed by aU stationary points of one maximum-length bond chain and containing aU components of the mixture a distillation subregion Reg. ... 

The distillation region Reg°° is a polyhedron formed by all stationary points of the totality of aU maximum-length bond chains connecting the same unstable node of the composition space with the same stable node (it will be designated ). The examples of distillation regions Reg° are 12 4, 12 = 2 (at Fig. 1.7a),... 

Each line of a structural matrix corresponds to the /th stationary point and each colunm to the th one. Diagonal elements a, = 1 (it is accepted conditionally that each specific point is bonded to itself). The components are labeled 1,2,3 binary azeotropes are designated by two-digit numbers, 12,13,23 and the ternary azeotrope by a three-digit number, 123. Zero column corresponds to an unstable node N and zero line to the stable iV+ one (except for the diagonal elements). Structural matrices provide an opportunity to easily single out all maximum-length... 

If all the trajectories coming from the stationary point, in this case, such stationary point is called the unstable node (vertex 1). The stationary point to which the trajectories get in is called the stable node A+ (vertex 3). At last, the stationary point that all trajectories bend around is called a saddle point S (vertex 2). 

As far as the stable node of boundary element Ai, A2... A (Regu) is stationary point A< (Ai = Np) and unstable node of boundary element A< +i,A< +2 - Am (Regfl) is stationary point Ak+i (Ak+i = and as far as there is bond A Ak+i Ak and Ak+i are adjacent stationary points of one bonds chain), separation into considered subsets of stationary points meets the rule of connectedness (i.e., it is feasible). In exactly the same way, it is possible to show that splits with one distributed pseudocomponent are feasible. It is noteworthy that the boundary elements Ai, A2... A and A +i, A +2... A are curvilinear, and three... 

At nonsharp separation, the stationary points of section working regions, except the stable node N+, are located outside the concentration simplex (the direction of trajectory from the product is accepted). At sharp separation, other stationary points - trajectory tear-off points x from the boundary elements of concentration simplex - are added to the stable node. These are the saddle points S and, besides that, if the product point coincides with the vertex corresponding to the lightest or to the heaviest component, then this point becomes an unstable node N. ... 

The stationary points of this bundle are located both in the boundary elements of simplex and inside it, at reversible distillation trajectories. The number of such stationary points of the bundle is equal to the difference between the number of the components of the mixture being separated n and the number of the components of section product k plus one. Stationary points of the bundle of top or bottom section are one unstable node A (it exists inside the simplex only in the product point, if product is a pure component or an azeotrope) one stable node A+ (it is located at the boundary element, containing one component more than the product if A < n — 1) the rest of the stationary points of the bundle are saddle points S. The first (in the course of the trajectory) saddle point (5 ) is located at the product boundary element (if product is pure component or azeotrope, then the saddle point coincides with the unstable node N and with product point). The second saddle point (S ) is located at the boundary element, containing product components and one additional component, closest to product... 

Water and 1-butanol form a heterogeneous azeotrope and an immiscibility gap over a limited region of ternary compositions exists. The stability of the stationary points of the system and the distillation line map modeled by UNIQUAC are shown on Figure 3a. One distillation boundary, miming from methanol (unstable node) to the binary heteroazeotrope (saddle) divides the composition space in two regions. The system belongs to Serafimov s topological class 1.0-2 (Hilmen, 2002). 

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

When considering systems with many stationary states, it is important to investigate the stability of the latter. Stability of a stationary state is directly connected to the thermal stability of the reactor. It may happen that a small perturbation of the system takes it out of the unstable state. The process will convert into the other one, now stable. In this case calculations (Fig. 3.30) show that two out of three possible stationary states are stable for them the Jacobian matrix eigenvalues are real and of the same sign (stable node). The third stationary state has real, but negative, Jacobian eigenvalues (saddle point). A comparison of these results with the plots shown in Fig. 3.31 allows one to conclude a stationary state is stable if a slope of the heat elimination curve is smaller than a slope of the heat liberation. 

In Section 5.1, we have provided some evidence for the existence of saddle-node bifurcations, as a (local) mechanism accounting for the creation of pairs of solutions of our reaction-diffusion system (3) with Dirichlet boundary conditions. As the normal form theory [65, 66] shows, at least one of these two solutions is unstable. From a physical point of view, one can guess that, in the limit D oo, any stationary solution should be stable. However, as D is decreased, or equivalently, as the size of the system is increased, the system becomes approximately translationally invariant, and the stationary solutions are very likely to be unstable. As shown in this section, this transition involves not only (stationary) saddle-node bifurcations, but (oscillatory) Hopf bifurcations as well [62,104]. 

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