Saddle-node fixed point

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

Fig. 11.3.3. Bifurcations of a saddle-node fixed point for the case I2 > 0. Observe that iterations of the points near the ghost of the saddle-node (in (C)) become more dense (the expansion rate in horizontal direction is hardly larger than 1).

Fig. 11.3.4. Bifurcations of a saddle-node fixed point for the case h < 0.

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)). 

Recall that a saddle-node fixed point or periodic orbit has one multiplier equal to +1 and the rest of the multipliers lies inside the unit circle. The diffeomorphism (the Poincare map) near the fixed point may be represented... 

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. 

The smooth case corresponds, in particular, to a small time-periodic perturbations of an autonomous system possessing a homoclinic loop to a saddle-node equilibrium (see the previous section). Indeed, for a constant time shift map along the orbits of the autonomous system the equilibrium point becomes a saddle-node fixed point and the homoclinic loop becomes a smooth closed invariant curve, but the transversality of to F is, obviously, preserved under small smooth perturbations. 

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.

Fig. 12.4.2. The option of chaotic behavior resulted from the disappearance of a saddle-node fixed point of the corresponding Poincare map, assuming the contraction condition is not satisfied but the big lobe condition holds each leaf of the foliation must intersect at least two of the connected components of n S.

As we mentioned. Lemma 12.3 may be reformulated as a possibility to embed a sufficiently smooth one-dimensional map near a simple saddle-node fixed point into a smooth one-dimensional flow for ji > 0. An analogous result was proved in [74] in connection with the problem on the appearance of... 

Let us derive the equation of the curve of saddle-node fixed points. Since one of the eigenvalues of such points equals 1, plugging A = 1 into (C.6.2) yields... 

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

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... 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

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 saddle-node bifurcation is the basic mechanism by which fixed points are created and destroyed. As a parameter is varied, two fixed points move toward each other, collide, and mutually annihilate. 

The prize for most inventive terminology must go to Abraham and Shaw (1988), who write of a blue sky bifurcation. This term comes from viewing a saddle-node bifurcation in the other direction a pair of fixed points appears out of the clear blue sky as a parameter is varied. For example, the vector field... 

Now imagine we start decreasing the parameter r. The line r-x slides down and the fixed points approach each other. At some critical value r = r, the line becomes tangent to the curve and the fixed points coalesce in a saddle-node bifurcation (Figure 3.1.6b). For r below this critical value, the line lies below the curve and there are no fixed points (Figure 3.1.6c). 

It s easy to understand why saddle-node bifurcations typically have this algebraic form. We just ask ourselves how can two fixed points of x = /(x) collide and disappear as a parameter r is varied Graphically, fixed points occur where the graph of /(x) intersects the x-axis. For a saddle-node bifurcation to be possible, we need two nearby roots of /(x) this means /(x) must look locally bowl-shaped or parabolic (Figure 3.1.7). 

Please note the important difference between the saddle-node and transcritical bifurcations in the transcritical case, the two fixed points don t disappear after the bifurcation—instead they just switch their stability. 

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). 

To summarize the results so far, we plot the bifurcation curves h = +hfr) in the (r,h) plane (Figure 3.6.2). Note that the two bifurcation curves meet tangentially at (r, /i) = (0,0) such a point is called a cusp point. VJe also label the regions that correspond to different numbers of fixed points. Saddle-node bifurcations occur all along the boundary of the regions, except at the cusp point, where we have a codimension-2 bifurcation. (This fancy terminology essentially means that we have had to tune two parameters, h and r, to achieve this type of bifurcation. Un-... 

An outbreak can also be triggered by a saddle-node bifurcation. If the parameters r and k drift in such a way that the fixed point a disappears, then the population will jump suddenly to the outbreak level c. The situation is made worse by the hysteresis effect—even if the parameters are restored to their values before the outbreak, the population will not drop back to the refuge level. 

For each of the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a saddle-node bifurcation occurs at a critical value of r, to be determined. Finally, sketch the bifurcation diagram of fixed points x versus r. 

The next exercises are designed to test your ability to distinguish among the various types of bifurcations—it s easy to confuse them In each case, find the values of r at which bifurcations occur, and classify those as saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifurcation diagram of fixed points X vs. r. 

The square-root scaling law found above is a very general feature of systems that are close to a saddle-node bifurcation. Just after the fixed points collide, there is a saddle-node remnant or ghost that leads to slow passage through a bottleneck. 

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. ... 

Figure 5.2.8 shows that saddle points, nodes, and spirals are the major types of fixed points they occur in large open regions of the (A, t) plane. Centers, stars, degenerate nodes, and non-isolated fixed points are borderline cases that occur along curves in the (A,t) plane. Of these borderline cases, centers are by far the most important. They occur very commonly in frictionless mechanical systems where energy is conserved. 

---