Definition:Bifurcation/Flip

Definition

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.

Period Doubling

The change in the periodic orbits under a flip bifurcation is known as a period doubling.

Also known as

A flip bifurcation is also known as a period doubling bifurcation.

Examples

Arbitrary Example

Consider the mapping:

$\forall x \in \R: \map {T_\lambda} x = x^2 - \paren {1 + \lambda}$

$T_\lambda$ has a flip bifurcation at $\lambda = 0$.

Also see

• Definition:Feigenbaum Constants

• Results about flip 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