Period-doubling bifurcation

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In mathematics, a Period doubling bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. The hallmark of this is a Floquet multiplier of -1.

[edit] Example

Bifurcation diagram for the modified Phillips curve.

Consider the following logistical map for a modified Phillips curve:

$\pi_{t} = f(u_{t}) + a \pi_{t}^e$
$\pi_{t+1} = \pi_{t}^e + c (\pi_{t} - \pi_{t}^e)$
$f (u) = β 1 + β 2 e - u$
$b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0$

where $π$ is the actual inflation, $π e$ is the expected inflation, u is the level of unemployment, and $m - π$ is the money supply growth rate. Keeping $\beta_{1} = -2.5, \ \beta_{2} = 20, \ c = 0.75$ and varying $b$ , the system undergoes period doubling bifurcations, and after a point becomes chaotic, as illustrated in the bifurcation diagram on the right.

[edit] Period-halving bifurcation

Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos.

A Period halving bifurcation in a dynamical system is a bifurcation in which the system switches to a new behavior with half the period of the original system. A series of period-halving bifurcations leads the system from chaos to order.