Period-doubling bifurcation

In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system. Period doubling bifurcations can also occur in continuous dynamical systems, namely when a new limit cycle emerges from an existing limit cycle, and the period of the new limit cycle is twice that of the old one.

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Examples

Consider the following logistical map for a modified Phillips curve:

\pi_{t} = f(u_{t}) %2B a \pi_{t}^e
\pi_{t%2B1} = \pi_{t}^e %2B c (\pi_{t} - \pi_{t}^e)
f(u) = \beta_{1} %2B \beta_{2} e^{-u} \,
b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0

where \pi is the actual inflation, \pi^e is the expected inflation, u is the level of unemployment, and m - \pi 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.

Period-halving bifurcation

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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