Stationary points stable node

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. 

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.

An important conclusion follows from the time monotonic manner (2.31) of changes in values P and d S/dt. In case the system exists near thermody namic equilibrium, the system s spontaneous evolution cannot generate any periodical auto oscillating processes. In fact, periodical processes are described along the closed evolution trajectories, which would make some thermo dynamic parameters (concentration, temperature, etc.) and, as a result, values Ji and Xj return periodically to the same values. This is inconsistent with the one directional time monotonic changes in the P value and with the con stancy of the latter in the stationary point. In terms of Lyapunov s theory of stability, the stationary state under discussion corresponds to a particular point of stable node type (see Section 3.5.2). 

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. 58. Phase trajectories for the system (5.2) in the vicinity of the stationary point of a stable node type.

The notions of a limit point and a limit set will be first exemplified by the linear systems considered in Section 5.1. In the case of a stable node (al), (a2), (a3) and a stable focus (d) the limit set (attractor) consists of one point, the stationary point, which is approached by all trajectories. 

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. 

When 0stationary state (1, 1) is a stable node. When a = otj equation (6.71) has two equal real roots, kr = k2 < 0. The value kt is determined from the requirement of vanishing of the discriminant A of the quadratic equation (6.71). This is a sensitive state when a increasing exceeds the value ax, a change in the phase portrait of the system (6.67) takes place the stationary state is, for < a < a0 = (y — 1) a stable focus (the real part of kx >2 is negative). In accordance with what we established in Chapter 5, the catastrophe does not alter a phase portrait nearby a stationary point but is of a global character. 

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... 

It is evident from the superimposed data points in Figure 4.12 that they are veiy closely aligned with the predictions of the NRTL model. The experimental results indicate that the profiles in fact move past the stable node predicted by the UNIQUAC model (highlighted by the curved arrow), and do not pinch at this point as one would expect if using this model. The predictions of the location of stationary points made by the UNIQUAC model are therefore false and those made by the NRTL are much better suited to this particular system. 

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... 

At Ty es " 312.5, a saddle point and a stable node appear in the phase-plane, corresponding to the two other stationary-states. Shortly after this, the limit cycle grows sufficiently large to touch the separatrix of the saddle point and form a homoclinic orbit. The period of motion around this cycle becomes infinite and beyond this point oscillations are not found. There is thus a sudden quenching of large amplitude oscillations. For all residence times greater than = 313 the system sits at the lowest, stable state. There is also hysteresis at this limit. 

This phenomenon, which is well-known in the living world, corresponds to the substantial temporary amplification of an initial perturbation when this exceeds a critical threshold value (e.g. propagation of signals along nerves). The transient behaviour that subsequently returns the system to its original stationary state is almost independent of the applied perturbation. For an initial departure below the threshold, we observe, on the other hand, an ordinary relaxation, with a monotonous decrease with respect to time (28). This process occurs when two fixed points, namely a stable node and a saddle point, are sufficiently close. 

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. 

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. 

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

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