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
Consider the following logistical map for a modified Phillips curve:
<math> \pi_{t} = f(u_{t}) + a \pi_{t}^e </math>
<math> \pi_{t+1} = \pi_{t}^e + c (\pi_{t} - \pi_{t}^e) </math>
<math> f(u) = \beta_{1} + \beta_{2} e^{-u} </math>
<math> b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0 </math>
where <math> \pi </math> is the actual inflation, <math> \pi^e </math> is the expected inflation, u is the level of unemployment, and <math> m - \pi </math> is the money supply growth rate. Keeping <math> \beta_{1} = -2.5, \ \beta_{2} = 20, \ c = 0.75 </math> and varying <math>b</math>, 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
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.
[edit] External links
- The Flip (Period Doubling) Bifurcation in Discrete Time, Dynamic Processes by Elmer G. Wiens
