Bifurcations
by Ralph Herman Abraham

This is a minimal picture gallery of dynamical bifurcations: only three drawings. All are from Dynamics, the Geometry of Behavior bu Ralph H. Abraham and Christopher D. Shaw, 4th edn. Santa Cruz, CA: Aerial Press, 2000. Also known as "DGB".

The Hopf

The simplest example of a subtle bifurcation.

Discovered by Henri Poincare ca 1880.

A point attractor of a flow in two dimensions subtly changes into a periodic attractor.

See panel 17.1.7 on p. 495 of DGB.

The Fold

The simplest example of a catastrophic bifurcation.

Discovered by Henri Poincare ca 1880.

A point attractor of a flow in one dimension appears out of the blue.

See panel 18.1.8 on p. 517 of DGB.

The Blue Loop

The simplest example of an explosive bifurcation.

A point attractor of a flow in two dimensions expolodes into a periodic attractor.

See panel 21.1.5 on p. 595 of DGB.