Unstable node fixed point

As already mentioned, problems of this nature had appeared as early as in the twenties in connection with the phenomenon of transition from synchronization to an amplitude modulation regime. A rigorous study of this bifurcation was initiated in [3], under the assumption that the dynamical system with the saddle-node is either non-autonomous and periodically depending on time, or autonomous but possessing a global cross-section (at least in that part of the phase space which is under consideration). Thus, the problem was reduced to the study of a one-parameter family of C -diffeomorphisms (r > 2) on the cross-section, which has a saddle-node fixed point O at = 0 such that all orbits of the unstable set of the saddle-node come back to it as the number of iterations tends to -hoo (see Fig. 12.2.1(a) and (b)). 

Fig 12.2.1. The unstable manifold of the saddle-node fixed point may be a smooth curve (a) or a non-smooth curve (b). In the latter case the tangent vector oscillates without a limit when a point on reaches O from the side of node region. 

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.

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

Fixed points boiling points of pure components and azeotropes. They can be nodes (stable and unstable) and saddles. 

A distillation boundary connects two fixed points node, stable or unstable, to a saddle. The distillation boundaries divide the separation space into separation regions. The shape of the distillation boundary plays an important role in the assessment of separations. 

The bifurcation at r is a saddle-node bifurcation, in which stable and unstable fixed points are bom out the clear blue sky as r is increased (see Section 3.1). 

If we continue to increase p, the stable and unstable fixed points eventually coalesce in a saddle-node bifurcation at // = 1. For p > 1 both fixed points have disappeared and now phase-locking is lost the phase difference 0 increases indefinitely, corresponding to phase drift (Figure 4.5.1c). (Of course, once 0 reaches 2jt the oscillators are in phase again.) Notice that the phases don t separate at a uniform rate, in qualitative agreement with the experiments of Hanson (1978) 0 increases most slowly when it passes under the minimum of the sine wave in Figure 4.5.1 c, at 0 = r/2, and most rapidly when it passes under the maximum at 0 = -kI2. ... 

A < 0. The parabola - 4A = 0 is the borderline between nodes and spirals star nodes and degenerate nodes live on this parabola. The stability of the nodes and spirals is determined by t. When t < 0, both eigenvalues have negative real parts, so the fixed point is stable. Unstable spirals and nodes have t > 0. Neutrally stable centers live on the borderline t = 0, where the eigenvalues are purely imaginary. 

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. 

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 ... 

Show that each of the following fixed points has an index equal to -bl. a) stable spiral b) unstable spiral c) center d) star e) degenerate node... 

Solution By the argument above, it suffices to find conditions under which the fixed point is a repeller, i.e., an unstable node or spiral. In general, the Jacobian is... 

Figure 26. Skeleton bifurcation diagram in the t/-p parameter plane for the model equation (16). Shown are Hopf and saddle-node bifurcations (SUN = saddle-unstable-node bifurcation) as well as the border of the focus-node transition (dashed line) mixed-mode wave forms exist close to the dark region (which marks the region where a fixed point is a ShQ nikov saddle focus). The phase portraits sketch the Unear stability of the fixed point(s). (Reprinted with permission from M. T. M. Koper and P. Gaspard, J. Chem. Phys. 96, 7797, 1992. Copyright 1992, American Institute of Physics.)...

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. 

Equation (6.21) has the same fixed points as (6.20) except that their stability is reversed. Thus, a fixed point which is a stable node in equation (6.20) becomes an unstable node for equation (6.21). 

As an example, consider the residue curve map for a ternary system with a minimumboiling binary azeotrope of heavy (H) and light (L) species, as shown in Figure 7.23. There are four fixed points one unstable node at the binary azeotrope (A), one stable node at the vertex for the heavy species (H), and two saddles at the vertices of the light (L) and intermediate (I) species. 

Unstable node is a fixed point for which the gradient of / evaluated at x" is zero and the Hessian matrix at x" is positive definite (Leon, 1998), namely. 

Thus, the saddle-node bifurcation leads to the appearance of a pair of fixed points, stable and unstable, whereas the period-doubling leads to the appearance of an unstable orbit of period two. An unstable fixed point vanishes at y = 0 (corresponding to a single-circuit homoclinic loop) when 0 = 0 and 0 = 7T. An unstable orbit of period two approaches y = 0 (a double homoclinic loop) when... 

Consider the other hypothetical example. Let a two-dimensional diffeo-morphism at e = 0 have a phase portrait as shown in Fig. 14.3.2. Here, O2 and O3 are stable fixed points, and 0 is a saddle. The unstable set Wq of the saddle-node O intersects transversely the stable manifold Wq ... 

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