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| In [[mathematics]], a '''period doubling bifurcation''' in a discrete [[dynamical system]] is a [[Bifurcation theory|bifurcation]] in which the system switches to a new behavior with twice the [[period (physics)|period]] of the original system. That is, there exists two points such that applying the dynamics to each of the points yields the other point. Period doubling bifurcations can also occur in continuous
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| dynamical systems, namely when a new [[limit cycle]] emerges from an existing limit cycle,
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| and the period of the new limit cycle is twice that of the old one. | |
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| ==Examples==
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| * [[Logistic map]]
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| * Logistical map for a modified [[Phillips curve]]
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| [[Image:Bifurc pi empl.jpg|thumb|right|300px|Bifurcation diagram for the modified Phillips curve.]]
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| Consider the following logistical map for a modified [[Phillips curve]]:
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| <math> \pi_{t} = f(u_{t}) + a \pi_{t}^e </math>
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| <math> \pi_{t+1} = \pi_{t}^e + c (\pi_{t} - \pi_{t}^e) </math>
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| <math> f(u) = \beta_{1} + \beta_{2} e^{-u} \,</math>
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| <math> b > 0, 0 \leq c \leq 1, \frac {df} {du} < 0 </math>
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| where :
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| * <math>\pi</math> is the actual [[inflation]]
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| * <math> \pi^e </math> is the expected inflation,
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| * u is the level of unemployment,
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| * <math> m - \pi </math> is the [[money supply]] growth rate.
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| 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 [[Chaos theory|chaotic]], as illustrated in the bifurcation diagram on the right.
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| * [[Complex quadratic polynomial|Complex quadratic map]]
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| [[File:Bifurcation1-2.png|right|thumb|300px|Bifurcation from period 1 to 2 for [[complex quadratic polynomial|complex quadratic map]]]]
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| ==Period-halving bifurcation==
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| [[Image:Chaosorderchaos.png|400px|right|thumb|Period-halving bifurcations (L) leading to order, followed by period doubling bifurcations (R) leading to chaos.]]
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| A '''period halving bifurcation''' in a dynamical system is a [[Bifurcation theory|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 theory|chaos]] to order.
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| ==See also==
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| *[[Feigenbaum constants]]
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| ==References==
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| * {{cite book | title=Elements of Applied Bifurcation Theory | volume=112 | series=Applied Mathematical Sciences | first=Yuri A. | last=Kuznetsov | edition=3rd | publisher=[[Springer-Verlag]] | year=2004 | isbn=0-387-21906-4 | zbl=1082.37002 }}
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| ==External links==
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| *[http://www.egwald.com/nonlineardynamics/onedimensionaldynamics_1.php#flipbifurcationconditions The Flip (Period Doubling) Bifurcation] in Discrete Time, Dynamic Processes
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| [[Category:Nonlinear systems]]
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| [[Category:Bifurcation theory]]
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