Period Doubling Bifurcation
Definition
Easy Definition
Period-doubling bifurcation is a bifurcation where the period of a periodic orbit doubles or halves depending on the change of a parameter in a dynamical system.
Hard Definition
$$ \dot{x} \mapsto f \left( x , r \right) \qquad , x \in \mathbb{R}^{n} , r \in \mathbb{R}^{1} $$ Let’s assume the dynamical system $f$ is $x$ and $\alpha$ smooth. If $\bar{x}$ is a hyperbolic fixed point of this system, let one of the eigenvalues of the Jacobian matrix $D f \left( \bar{x} \right)$ be $\lambda_{k}$. A bifurcation associated with the appearance or disappearance of $\lambda_{k} = -1$ is called a period-doubling bifurcation1.
Diagram
Explanation
Period-doubling bifurcation is also called flip bifurcation and is one of the first phenomena encountered through bifurcation diagrams in examples like the logistic family.
Especially notable is when the period doubles to form an infinite sequence called a period-doubling cascade1, which is directly connected to the concept of chaos. The occurrence period of parameters approaching chaos converges to a certain constant, known as Feigenbaum’s universality.