Definition:Bifurcation

Definition

A bifurcation is a sudden change in the nature of an attractor or repellor as the defining mapping or flow undergoes a change with respect to changes in the defining equations.

Hopf Bifurcation

A Hopf bifurcation is a bifurcation in which a family of flows $\map {x_\lambda} t$, indexed by a real bifurcation parameter $\lambda$, has an attractor consisting of:

a fixed point replaced by a circle

a repelling fixed point for a small change in the index.

Flip Bifurcation

A flip bifurcation is a bifurcation in which a family of mappings $T_\lambda$, indexed by a real bifurcation parameter $\lambda$, has an repelling fixed point replaced by a pair of periodic points of period $2$, forming an attractor.

Also see

• Results about bifurcations can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bifurcation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bifurcation