Stability

  At least three notions of stability apply to a fixed point: local stability , global stability, and linear stability . Here we will discuss local stability and linear stability. Linear stability often, but not always, implies local stability. The additional ingredient needed is hyperbolicity. This turns out to be quite general: hyperbolicity plus a linearization procedure is usually sufficient to analyze the stability of an attracting set, whether it be a fixed point, periodic orbit, or strange attractor.

The notion of the local stability of an orbit is straightforward.  A fixed point is locally stable if solutions based near remain close to for all future times. Further, if the solution actually approaches the fixed point, i.e., as , then the orbit is called asymptotically stable .

  
Figure 4.8: (a) A stable fixed point. (b) An asymptotically stable fixed point.

Figure 4.8(a) shows a center that is stable, but not asymptotically stable. Centers commonly occur in conservative systems. Figure 4.8(b) shows a sink, an asymptotically stable fixed point that commonly occurs in a dissipative system. A fixed point is unstable if it is not stable. A saddle and source are examples of unstable fixed points (see Fig. 4).