Structurally stable fixed point

Theorem C.2. Structurally stable equilibrium states of the averaged system correspond to structurally stable periodic orbits of the original system ifx is a structurally stable equilibrium state in (C.3.9), then the Poincare map (C.3.8) for the system (C.3.7) has a structurally stable fixed point close to x for all sufficiently small p. ... 

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. 

Basins for damped double-well oscillator) Suppose we add a small amount of damping to the double-well oscillator of Example 6.5.2. The new system is X = y, y = -by + x-x, where 0 < b 1. Sketch the basin of attraction for the stable fixed point (x, y ) =(1,0). Make the picture large enough so that the global structure of the basin is clearly indicated. 

The last panel in Fig. 3.4 shows a situation qualitatively different from the previous ones. Here, the FN nullclines intersect at three points. Simple linear stability analysis reveals that the central one is unstable, whereas the lateral ones are both stable. Since there are no more attracting structures in this simple dynamical system, the final state of the FN dynamics will be one of these stable fixed points, depending on the initial condition. This situation is called bistability. 

The two-dimensional (c,T) phase space for each of these systems can be partitioned into basins of attraction of the distinct attractors. Naturally, the basin structure is more complicated than that of the one-dimensional model studied in Sec. 3, but it is still quite simple. In the one-dimensional model the two stable fixed points were separated by an unstable fixed point, which formed a boundary separating the basins of attraction of the fixed points. In the GK model for the specified parameter setting the situation differs in that one of the stable fixed points has undergone a Hopf bifurcation to yield a stable limit cycle. The basin boundary in the two-dimensional space now consists of the unstable fixed point along with its stable manifolds these are shown in Fig. 1 and labelled Bi and B2. 

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. 

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.

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. 

A beautiful classical theory of unimolecular isomerization called the reactive island theory (RIT) has been developed by DeLeon and Marston [23] and by DeLeon and co-workers [24,25]. In RIT the classical phase-space structures are analyzed in great detail. Indeed, the key observation in RIT is that different cylindrical manifolds in phase space can act as mediators of unimolecular conformational isomerization. Figure 23 illustrates homoclinic tangling of motion near an unstable periodic orbit in a system of two DOFs with a fixed point T, and it applies to a wide class of isomerization reaction with two stable isomer... 

Hyperbolic fixed points also illustrate the important general notion of structural stability. A phase portrait is structurally stable if its topology cannot be changed by an arbitrarily small perturbation to the vector field. For instance, the phase portrait of a saddle point is structurally stable, but that of a center is not an arbitrarily small amount of damping converts the center to a spiral. 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

In fact, resonant fixed points are not restricted to only saddles and stable (completely unstable) points. An example of the other structure is given by the map... 

C.3. 21. Prove that if the origin is a structurally stable equilibrium state of the system (C.3.3), then the corresponding fixed point of the map (C.3.2) is structurally stable as well. Furthermore, show that the topological types of the equilibrium state of (C.3.3) and the fixed point of (C.3.2) are the same. ... 

Therefore, for all small fi the roots a will be close to those of (C.3.10). Thus, the fixed point will be structurally stable. Moreover, it has the same topological type as the equilibrium state of the averaged system. ... 

This experiment established the nuclear model of the atom. A key point derived from this is that the electrons circling the nucleus are in fixed stable orbits, just like the planets around the sun. Furthermore, each orbital or shell contains a fixed number of electrons additional electrons are added to the next stable orbital above that which is full. This stable orbital model is a departure from classical electromagnetic theory (which predicts unstable orbitals, in which the electrons spiral into the nucleus and are destroyed), and can only be explained by quantum theory. The fixed numbers for each orbital were determined to be two in the first level, eight in the second level, eight in the third level (but extendible to 18) and so on. Using this simple model, chemists derived the systematic structure of the Periodic Table (see Appendix 5), and began to... 

In principle, more or less stable BRC structures can be obtained in heterogeneous biomimics, especially when adsorption and catalytic sites are combined, i.e. active sites perform both functions fixation and transformation of the substrate. To put it another way, the above enumerated restrictions typical of homogeneous catalysis are absent in heterogeneous mimic-substrate complexes, where acidic-basic sites are fixed in required points of the active site. 

Stable phases in the rare earth oxide systems are tabulated and discussed. New data on the structure of sesquioxides quenched from the melt are reported. The structural interrelations between the A, B, and C type sesquioxides and the fiuorite dioxides are pointed out. The sequences of several intermediate oxides in the CeO, PrO., and TbO, systems are observed to be related to the fluorite structure and the C form sesquioxide with respect to the metal atom positions. A hypothetical homologous series of the general formula Mn02n i, related to the fluorite structure and the A form sesquioxide with a more or less fixed oxygen lattice, is suggested. 

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