Degenerate saddle-node

Fig. 12.1.7. The boundary of the stability region of the periodic orbit born at the bifurcation of the on>edge homoclinic loop to a degenerate saddle-node corresponds to the homoclinic loop of the border saddle equilibrium state Oi.

Remark. This statement remains valid (with obvious modifications) also in the case of the on-edge homoclinic loop to a degenerate saddle-node. In this case, /i is a vector of parameters (of dimension equal to the number of zero Lyapunov values plus one), and an additional bifurcation parameter e is introduced as before. A stable periodic orbit exists when the saddle-node disappears (the region /j> Dq m our notations), or when e > hhomi/j) fjL Do. Here, the surface e = hhomifJ ) corresponds to the homoclinic loop of the border saddle equilibrium Oi, as illustrated in Fig. 12.1.7. 

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. 

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. 

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