A Fractal Example in Ordinary Differential Equations

Derinition. Let f be a continuous flow in the plane, i.e., f is a continuous mapping of DR x E onto E (where DR is the real line and E is the plane) satisfying f(O, x) = x and f(t, f(s, x)) = f(s + t, x) for all x in E and all s, t in DR. The orbit of a point x, <(x), is the set f(R, x). If a sequence of points xi converges to a point y, we say also that the sequence &(xj) converges to 6(y). This concept of convergence is not unique, since if we choose different sample points zj from the orbits &(xj), they might converge to a point w with A(w) #* &(y). Even for continuous flows in the plane with no fixed points, i.e., with orbits which are closed sets (curves running from oo to oo), this convergence need not be unique. For flows of this type, there may be open sets in E that are invariant under the flow (i.e., they consist of whole orbits) in which this convergence is unique. In that case, the orbits making up these open sets are called regular. Any other orbit is called a separatrix.