Stable node equilibrium state

In Chap. 12 we will study the global bifurcations of the disappearance of saddle-node equilibrium states and periodic orbits. First, we present a multidimensional analogue of a theorem by Andronov and Leontovich on the birth of a stable limit cycle from the separatrix loop of a saddle-node on the plane. Compared with the original proof in [130], our proof is drastically simplified due to the use of the invariant foliation technique. We also consider the case when a homoclinic loop to the saddle-node equilibrium enters the edge of the node region (non-transverse case). 

Fig. 12.2.2. A nontransverse tangency of the unstable and strong-stable manifolds of a saddle-node fixed point may be obtained by a small time-periodic perturbation of the system with an on-edge homoclinic loop to a saddle-node equilibrium state, as shown in Fig. 12.1.4.

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 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]. 

All mentioned above surfaces of equilibrium states contain special points of stable nodes type. 

We should note that all equilibrium states listed above are stable nodes. It is obvious from Figure 15 in which substantial parts of roots of characteristic equation calculated on the... 

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). 

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

Fig. 8.1.3. A nontransverse homociinic loop F to a saddle node. The separatrix enters the equilibrium state along its strongly stable manifold.

The phase portraits for systems of dimension two and higher are illustrated in Figs. 11.2.4-11.2.7, respectively. Here, when l2 < 0, there are two rough equilibrium states a stable node and a saddle that approach each other as i2M... 

This is a one-dimensional system which may have stable and unstable equilibrium states corresponding to stable and saddle equilibrium states of the entire system (12.4.6) or (12.4.7). The evolution along Meq is either limited to one of the stable points, or it reaches a small neighborhood of the critical values of X. Recall, that we consider x as a governing parameter for the fast system and critical values of x are those ones which correspond to bifurcations of the fast system. In particular, at some x two equilibrium states (stable and saddle) of the fast system may coalesce into a saddle-node. This corresponds to a maximum (or a minimum) of x on Meq, so the value of x cannot further... 

The Cherry flow is a flow on a two-dimensional torus with two equilibrium states a saddle and an unstable node both unstable separatrices are stable one stable separatrix is a-limit to a node and the other lies in the closure of the unstable separatrices and it is P -stable [see Fig. 13.7.4(a)]. The closure of the unstable separatrices is a quasiminimal set which contains the saddle O and a continuum of unclosed P-stable trajectories. The rotation number for such flows is defined in the same way as for flows on a torus without equilibrium states. Since there is no periodic orbits in a Cherry flow. 

Fig. 13.7.20. The bifurcation diagram for the case where both equilibrium states of the heteroclinic cycle are saddles (see Fig. 13.7.12) provided Ai,2 > 0, i/i > 1, 1/2 < 1 and U1U2 > 1- The system has one limit cycle in regions 1-3, two limit cycles in region 4, none in regions 5-7. A pair of limit cycles are born from a saddle-node on the curve SN the unstable one becomes a homoclinic loop on the curve L2, whereas the stable limit cycle terminates on Li.

The second example exhibits a stable equilibrium state which merges with a saddle to spawn a saddle-node. Denote by F, the only imstable trajectory leaving the saddle-node as f -f 00, and its limit set by n(F). If Cl T) is an... 

Both cases have much in common in the sense that the imstable set of both bifurcating equilibrium states is one-dimensional. If the unstable set of the critical equilibrimn state is of a higher-dimension, then the subsequent picture may be completely different. Figure 14.3.1 depicts such a situation. When the imstable cycle shrinks into the equilibrium state we have a dilemma the representative point may jump either to the stable node 0 or to the stable node 02- Therefore this dangerous boundary must be classified as dynamically indefinite. 

When the equilibrium state is topologically saddle, condition (C.2.8) distinguishes between the cases of a simple saddle and a saddle-focus. However, when the equilibrium is stable or completely unstable, the presence of complex characteristic roots does not necessarily imply that it is a focus. Indeed, if the nearest to the imaginary axis (i.e. the leading) characteristic root is real, the stable (or completely imstable) equilibrium state is a node independently of what other characteristic roots are. 

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