Unstable fixed point

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

Fig. 4.4 A schematic representation of the pitchfork bifurcation a stable fixed point bifurcates into a period-2 limit cycle plus an unstable fixed point.

Neglecting the two previous, and now unstable, fixed point solutions r = 0 and... 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

Two alternatives present themselves in formulating algorithms for the tracking of segments of stable and unstable manifolds. The first involves observing the initial value problem for an appropriately chosen familv of initial conditions, henceforth referred to as simulation of invariant manifolds. A second generation of algorithms for the computation of invariant manifolds involves numerical fixed point techniques. 

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. 

Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below... 

Figure 3. Classical phase portraits (upper panel), residual quantum wavefunctions (middle panel), and ionization probability versus time (in units of the period T) (bottom panel). The parameters are (A) F = 5.0, iv = 0.52 (B) F = 20, iv = 1.04 and (C) F = 10 and u> = 2.0. Note that the peak structure of the final wavefunction reflects both stable and unstable classical fixed points. For case C, the peaks are beginning to coalesce reflecting the approach of the single-well effective potentiai (see text). 

Figure Jh Homoclinic tangle associated with the fixed point at (—a, 0) for case A. Near the fixed point, the solid line gives the unstable direction while the dashed line is the stable direction. The size of Planck s constant h is shown to illustrate that several states can be supported by the single structure. An estimate of the number of states is given by the number of h boxes needed to cover the structure.

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

As remarked above, we may define the unstable manifold W of the fixed point set A E for the total space of a holomorphic line bundle over E. More generally, let X be a surface containing a curve E. By identifying a tubular neighborhood of E with the total space of the normal bundle of E in X, we can define the unstable manifold W. In fact, this can be defined intrinsically as follows. Let tt X "] S "X be the Hilbert-Chow morphism. We define... 

Furthermore, the oscillation starting with even larger amplitudes gets the kinetic energies until it reaches a limit where the oscillation induces strong shock waves and dissipates its kinetic energies (see Fig. 1). Thus we conclude that the model which has the stable limit cycle near the transition has another unstable fixed point with a larger amplitude. The transition therefore is induced by the... 

One fixed point, stable or unstable, in the stroboscopic map corresponds to a periodic solution of period t. If there are two distinct points, each mapping into the other, we have a periodic solution of twice the period. Figure 30, drawn with a continuous fill-in, illustrates the way in which different kinds of periodic solution can coexist. The invariant circle (actually ovoid with a pointed... 

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 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. 

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