Saddle fixed point

Figure 37. Development of a potential front from the interplay of local dynamics and spatial coupling. The solid line is the original profile. The dot-dashed line indicates the homogenizing effect of the migration currents on the initial spatial profile. The arrows indicate the reestablishment of the spatial profiles (dotted line) by the local dynamics. The dashed line indicates the saddle fixed point. Both effects result in the motion of the interface to the left.

This result is due to Palis, who had fotmd that two-dimensional diffeomor-phisms with a heteroclinic orbit at whose points an unstable manifold of one saddle fixed point has a quadratic tangency with a stable manifold of another saddle fixed point can be topologically conjugated locally only if the values of some continuous invariants coincide. These continuous invariants are called moduli. Some other non-rough examples where moduli of topological conju-gacy arise are presented in Sec. 8.3. 

Fig. 8.2.2. A nontransverse homoclinic orbit to a saddle fixed point.

Fig. 8.2.3. A structurally unstable heteroclinic cycle including two saddle fixed points.

Fig. 8.3.1. A nontransverse heteroclinic trajectory between two saddle fixed points.

However, a similar classification of two-dimensional diffeomorphisms, or of three-dimensional fiows, is not that trivial. Let us illustrate this with an example. Consider a diffeomorphism T which has two saddle fixed points 0 and O2 with the characteristic roots )Ai) < 1 and i > 1 at (z = 1,2). Suppose that Wq and have a quadratic tangency along a heteroclinic orbit as shown in Fig. 8.3.1. The quadratic tangency condition implies that all similar diffeomorphisms form a surface of codimension-one in the space of all diffeomorphisms with a C -norm. 

Fig. 10.2.6. Geometrically, there is no difference between a critical node hp < 0 (a) and a rough stable node. However, a quantitative comparison can be made with respect to the rate of convergence of nearby trajectories to the origin. A similar observation also applies to a rough saddle fixed point and a critical saddle with /2p+i >0 (b).

Fig. 10.6.2. The fixed point is a center at e = 0. The saddle fixed points form a heteroclinic cycle.

The saddle equilibrium states are the saddle fixed points of the shift map, and respectively, their separatrices are the invariant manifolds. Returning to the original (non-rescaled) variables we find that the fixed points must lie apart from the origin at some distance of order e. If the third iteration (10.6.2) of the map (10.6.1) were the shift map of the reduced system (10.6.5), then the above theorem would follow from our arguments because the fixed points Oi, O2,03 of the third iterations correspond to the cycle of period three of the original map. 

We have shown the existence of infinitely many saddle fixed points of the map T which correspond to saddle periodic orbits (with two-dimensional unstable and m-dimensional stable manifolds) of the system. Those with even fc s have a negative unstable multiplier, and their unstable manifolds are non-orientable. The periodic orbits corresponding to odd fc s have a positive unstable multiplier, and hence, orientable unstable manifolds. 

Remark 1. Following exactly the same procedure as in Appendbc B of Part I for the trajectories near a saddle fixed point of a diffeomorphism, one may show also that... 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

As a first step toward a TST treatment of the stochastically driven dynamics, it is crucial to assume, just as in the autonomous case, that the deterministic dynamics has a fixed point that marks the location of an energetic barrier between reactants and products. In the case of Eq. (13), the fixed point is given by a saddle point q0 of the potential U(q). The reaction rate is determined by the... 

In particular, the TS trajectory remains bounded for all times, which satisfies the general definition. The constants c and c in Eq. (39) depend on the specific choice of the TS trajectory. Because the saddle point of the autonomous system becomes a fixed point for large positive and negative times, one might envision an ideal choice to be one that allows the TS trajectory to come to rest at the saddle point both in the distant future and in the remote past. However, this is impossible in general because the driving force will transfer energy into or out of the bath modes in such a way that... 

With the identification of the TS trajectory, we have taken the crucial step that enables us to carry over the constructions of the geometric TST into time-dependent settings. We now have at our disposal an invariant object that is analogous to the fixed point in an autonomous system in that it never leaves the barrier region. However, although this dynamical boundedness is characteristic of the saddle point and the NHIMs, what makes them important for TST are the invariant manifolds that are attached to them. It remains to be shown that the TS trajectory can take over their role in this respect. In doing so, we follow the two main steps of time-independent TST first describe the dynamics in the linear approximation, then verify that important features remain qualitatively intact in the full nonlinear system. 

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]. 

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. 

FIGURE 9 Stroboscopic phase portraits for the points on figure 8 labelled (a)-(e). (a) Below the third root of unity point (labelled F in figure 8) the phase portrait is structurally a period three phase locked torus (b) above point F, the period I fixed point in the centre is now stable and the phase locked torus has disappeared (c)-(e) before, during, and after a period 3 homoclinic bifurcation to the right of point F oil cut, = 3.97 for each, and A/Ao = 5.90, 5.93 and 5.95 for (c)-(e) respectively. The period 3 phase locked torus is transformed to a free torus as the stable manifold of each saddle crosses the unstable manifold of an adjacent saddle. 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

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. 

Wiggins et al. [22] pointed out that one can always locally transform a Hamiltonian to the form of Eq. (1.38) if there exists a certain type of saddle point. Examination of the associated Hamilton s equations of motion shows that q = Pn = 0 is a fixed point that defines an invariant manifold of dimension 2n — 2. This manifold intersects with the energy surface, creating a (2m — 3)-dimensional invariant manifold. The latter invariant manifold of dimension 2m — 3 is an excellent example of an NHIM. More interesting, in this case the stable and unstable manifolds of the NHIM, denoted by W and W ,... 

---