About bifurcations

Dynamical systems are much employed in the mathematical modeling and computer simulation of the simpler systems found in nature. Such models usually have some control parameters, variable coefficients for example, which are tuned by the modeler so as to obtain the best fit between some given experimental data and the simulated data output by the model. A model with controls is called a dynamical scheme, or alternately, a parameterized family of dynamical systems. We consider here a scheme of iterations.

When the control parameters of the scheme are fixed, we then have a single dynamical system, one member of the parametrized family. And if the controls are then moved, the portrait will be changed. When the controls are moved smoothly and gradually, the portrait may be seen to also change smoothly and gradually. Sometimes, however, the portrait undergoes a radical change even when the controls are moved very gently. Such an event is called a bifurcation.

Bifurcations are certainly the most important features of a scheme, and locating them is a difficult job for the experimentalist. One might begin a study of bifurcations by looking at some exemplary cases, and most elementary texts do just this. The simple examples fall into three categories:

• subtle bifurcations, in which the change is not immediately striking,
• catastrophic bifurcations, in which a basin suddenly appears or disappears, and
• explosive bifurcations, in which an attractor suddenly expands or contracts.