VECTOR FIELDS
Hyperbolic sector
Separatrix
Elliptic sector
Elliptic sector
Hyperbolic sector
Parabolic sector
Figure 9.5 A typical isolated critical point.
sector. Those with only parabolic sectors are called nodes. Since a parabolic sector can be stable or unstable, nodes can be stable or unstable. The other types of sectors are unstable so the only stable critical points are the stable focus, node, and center. A critical point with only elliptic sectors is called a rose. An example is the dipole (Figure 9.2). A critical point with only hyperbolic sectors is called a cross point. Saddle points are cross points with four sectors. Still more complicated types are possible with an infinite number of sectors, and there may be nonisolated critical points. We shall always assume that critical points are isolated and have a finite number of sectors.
Exercises
4. Verify that the system of differential equations corresponding to the vector field V(x, y) = (x + y, -x + y) has solution x = Ke^t sin(t), y = Ke^t cos(t). What type of critical point does V have at the origin? Is it stable or unstable?
5. Analyze the examples of critical points in Exercise 2 into sectors of different types.
6. The number of elliptic sectors plus the number of hyperbolic sectors is always even. Verify this for the examples of critical points illustrated above, and then prove it.