Degenerate fixed point

Figure 23 shows sections of with J fixed, C2 = 0 and various values of L/N, where V = 27 is the total polyad action. There is an obvious conical point, X, at K, L) = (7,0). In addition, the concentric contours degenerate to points Y and Z at K, L) = (—7, V). It is important for what follows to explore the character of the point X. As a preliminary, the Hamiltonian may be reduced from the above seven-parameter form, to one involving two essential parameters, both of which vary with the total action. The appropriate definitions [14, 28, 29], modified to conform with the present (7, K) notation, are... 

Thus, the value of the equilibrium population of the logistic mapping at the first bifurcation is x = x = 2/3. Indeed, /a (2/3) = 2/3 is a period-1 fixed point or a degenerate period-2 cycle. 

Figure 6.8 Intersections of the degenerated diffraction cone (20 = 90°) and receiving slit plane. Diffraction cones are produced by the incident ray from fixed point Ai = 0, 0) to points on the sample A2 having only an axial compo nent. The black rectangle represents the receiving slit. (Reprinted from Ref. 54. Permission of the International Union of Crystallography.)...

When there s only one eigendirection, the fixed point is a degenerate node. A... 

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. 

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. 

The borderline cases (centers, degenerate nodes, stars, or non-isolated fixed points) are much more delicate. They can be altered by small nonlinear terms, as weTl see in Example 6.3.2 and in Exercise 6.3.11. 

In Exercise 6.8.1, you are asked to show that spirals, centers, degenerate nodes and stars all have / = +1. Thus, a saddle point is truly a different animal from all the other familiar types of isolated fixed points. 

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... 

This degenerate case typically arises when a nonconservative system suddenly becomes conservative at the bifurcation point. Then the fixed point becomes a nonlinear center, rather than the weak spiral required by a Hopf bifurcation. See Exercise 8.2.11 for another example. 

Such a critical fixed point is called a complex degenerate) saddle. Its stable manifold is y = 0, and the unstable manifold is given by x = 0, as shown in Fig. 10.2.6(b). Here, in the critical case, the trajectories behave qualitatively identical to those nearby the rough unstable cycle shown in Fig. 10.2.7(b). 

Theorem 11.4 shows essentially that outside the narrow sector bounded by 1 and 2, the bifurcation behavior does not differ from that of equilibrium states (see Sec. 11.5) fixed points correspond to equilibrium states, and the invariant curves correspond to periodic orbits. However, the transition from the region D2 to the region Dq occurs here in a more complicated way. In the case of equilibrium states the regions D2 and Do are separated by a line on which a stable and an imstable periodic orbits coalesce thereby forming a semi-stable cycle. In the case of invariant closed curves, the existence of a line corresponding to a semi-stable invariant closed curve is possible only in very degenerate cases (for example, when the value of R does not depend on as... 

The Jacobian of the Henon map is constant and equal to h. Therefore, when 6 > 0, the Henon map preserves orientation in the plane, whereas orientation is reversed when 6 < 0. Note also that if 6 < 1, the map contracts areas, so the product of the multipliers of any of its fixed or periodic points is less than 1 in absolute value. Hence, in this case the map cannot have completely unstable periodic orbit (only stable and saddle ones). On the contrary, when b > 1, no stable orbits can exist. When 6 = 1, the map becomes conservative. At b = 0, the Henon map degenerates into the above logistic map, and therefore one should expect some similar bifurcations of the fixed points when b is suflSciently small. 

The curve AH is given by (C.6.9)-(C.6.10). Since the Jacobian of the map (C.6.1) is no longer constant, one should expect that the corresponding bifurcation of the birth of the invariant curve will be non-degenerate at the fixed point. To make sure, let us compute the first Lyapimov value L. ... 

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