en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

2014-03-28T20:17:23Z

WP:CHECKWIKI error fixes + other fixes using AWB (10065)

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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~~		Hello, my name is Andrew and my spouse doesn't like it at all. What me and my family love psychic ([http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ hknews.classicmall.com.hk]) is bungee leaping but I've been taking on new things lately. For years he's been residing in Alaska and he doesn't plan on altering it. He is an information officer.
	~~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.~~

	~~==Examples==~~
	* [~~[Logistic map]]~~
	* Logistical map for a modified [[Phillips curve]]
	~~[[Image:Bifurc pi empl.jpg\|thumb\|right\|300px\|Bifurcation diagram for the modified Phillips curve.]]~~
	~~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,
	* <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 [[Chaos theory\|chaotic]], as illustrated in the bifurcation diagram on the right.~~

	* [[Complex quadratic polynomial\|Complex quadratic map]]
	~~[[File:Bifurcation1~~-~~2.png\|right\|thumb\|300px\|Bifurcation from period 1 to 2 for [[complex quadratic polynomial\|complex quadratic map]]]]~~

	~~==Period~~-~~halving bifurcation==~~
	~~[[Image:Chaosorderchaos.png\|400px\|right\|thumb\|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 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.~~

	~~==See also==~~
	*[[Feigenbaum constants]]

	~~==References==~~
	* {{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 }}~~

	~~==External links==~~
	*[http://~~www~~.~~egwald~~.com~~/nonlineardynamics/onedimensionaldynamics_1~~.~~php#flipbifurcationconditions The Flip (Period Doubling~~) ~~Bifurcation]~~ in ~~Discrete Time, Dynamic Processes~~
	~~[[Category:Nonlinear systems]]~~
	~~[[Category:Bifurcation theory]]~~

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes, added orphan tag

2009-12-16T22:27:35Z

remove Erik9bot category,outdated, tag and general fixes, added orphan tag

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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
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.

==Examples==
* [[Logistic map]]
* Logistical map for a modified [[Phillips curve]]
[[Image:Bifurc pi empl.jpg|thumb|right|300px|Bifurcation diagram for the modified Phillips curve.]]
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,
* <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 [[Chaos theory|chaotic]], as illustrated in the bifurcation diagram on the right.

* [[Complex quadratic polynomial|Complex quadratic map]]
[[File:Bifurcation1-2.png|right|thumb|300px|Bifurcation from period 1 to 2 for [[complex quadratic polynomial|complex quadratic map]]]]

==Period-halving bifurcation==
[[Image:Chaosorderchaos.png|400px|right|thumb|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 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.

==See also==
*[[Feigenbaum constants]]

==References==
* {{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 }}

==External links==
*[http://www.egwald.com/nonlineardynamics/onedimensionaldynamics_1.php#flipbifurcationconditions The Flip (Period Doubling) Bifurcation] in Discrete Time, Dynamic Processes
[[Category:Nonlinear systems]]
[[Category:Bifurcation theory]]

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en>Yobot: WP:CHECKWIKI error fixes + other fixes using AWB (10065)

en>SmackBot: remove Erik9bot category,outdated, tag and general fixes, added orphan tag