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.

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[edit] Example

Bifurcation diagram for the modified Phillips curve.
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) = \beta_{1} + \beta_{2} e^{-u} \,
 b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0

where π is the actual inflation, πe 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.
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.

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