Instant Chaos Is Chaos in Slow Motion

Instant chaos is the onset of chaotic behaviour as a local bifurcation directly from a trivial steady state. We describe a systematic method for constructing examples of instant chaos, by scaling spatial variables and time. In this way we generalize properties of examples previously studied by other authors. We show that whenever a chaotic attractor of limited amplitude is obtained using a scaling property then it appears in slow motion}for any set S transverse to the vector field, the return time to S tends to infinity as we approach the bifurcation point. When instant chaos appears for a family of vector fields with a nontrivial scaling property, if it is not in slow motion then the amplitude of the chaotic attractor becomes arbitrarily large around the bifurcation point. We use this method to obtain the Lorenz attractor in a bifurcation directly from an asymptotically stable equilibrium.