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| | I'm Bernie and I live with my husband and our 2 children in Rasta, in the south part. My hobbies are Homebrewing, Art collecting and Coin collecting.<br><br>Also visit my homepage ... Fifa 15 Coin Generator; [http://www.jr-graph.fr/pixelpost/index.php?showimage=1 www.jr-graph.fr], |
| In [[mathematics]], particularly in [[dynamical systems]], a '''bifurcation diagram''' shows the possible long-term values (equilibria/fixed points or periodic orbits) of a system as a function of a [[Bifurcation theory|bifurcation parameter]] in the system. It is usual to represent stable solutions with a solid line and unstable solutions with a dotted line.
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| ==Bifurcations in 1D discrete dynamical systems==
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| ===Logistic map===
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| {{See also|Dynamical systems|List of chaotic maps}}
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| [[Image:LogisticMap BifurcationDiagram.png|300px|thumb|right|Bifurcation diagram of the [[logistic map]]]]
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| [[Image:diagram bifurkacji anim small.gif|300px|thumb|right|Animation showing the formation of bifurcation diagram]]
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| [[Image:Circle map bifurcation.jpeg|thumb|right|Bifurcation diagram of the [[circle map]]. Black regions correspond to [[Arnold tongues]].]]
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| An example is the bifurcation diagram of the [[logistic map]]:
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| :<math> x_{n+1}=rx_n(1-x_n). \,</math>
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| The bifurcation parameter ''r'' is shown on the horizontal axis of the plot and the vertical axis shows the possible long-term population values of the logistic function.
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| The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a [[period-doubling bifurcation]].
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| The ratio of the lengths of successive intervals between values of ''r'' for which bifurcation occurs [[convergent series|converges]] to the [[first Feigenbaum constant]].
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| ===Real quadratic map===
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| {{See also|Complex quadratic polynomial}}
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| The map is <math>x_{n+1}=x_n^2-c</math>.
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| ==Symmetry breaking in bifurcation sets==
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| [[Image:Asymbif.gif|300px|thumb|right|Symmetry breaking in [[pitchfork bifurcation]] as the parameter epsilon is varied. epsilon = 0 is the case of symmetric pitchfork bifurcation.]]
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| In a dynamical system such as
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| :<math> \ddot {x} + f(x;\mu) + \epsilon g(x) = 0</math>,
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| which is [[structurally stable]] when <math> \mu \neq 0 </math>, if a bifurcation diagram is plotted, treating <math> \mu </math> as the bifurcation parameter, but for different values of <math> \epsilon </math>, the case <math> \epsilon = 0</math> is the symmetric pitchfork bifurcation. When <math> \epsilon \neq 0 </math>, we say we have a pitchfork with ''broken symmetry.'' This is illustrated in the animation on the right.
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| == See also ==
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| * [[Bifurcation theory]]
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| * [[Feigenbaum constants]]
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| * [[Phase portrait]]
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| * [[Complex_quadratic_polynomial#Critical_curves|Skeleton of bifurcation diagram]]
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| ==References==
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| *Paul Glendinning, "Stability, Instability and Chaos", Cambridge University Press, 1994.
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| *Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.
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| ==External links==
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| * [http://yuval.bar-or.org/index.php?item=4 Logistic Map Simulation]. A Java applet simulating the Logistic Map by Yuval Baror.
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| * [http://www.egwald.com/nonlineardynamics/logisticsmapchaos.php The Logistic Map and Chaos]
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| [[Category:Chaos theory]]
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| [[Category:Bifurcation theory]]
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| [[de:Bifurkationsdiagramm]]
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I'm Bernie and I live with my husband and our 2 children in Rasta, in the south part. My hobbies are Homebrewing, Art collecting and Coin collecting.
Also visit my homepage ... Fifa 15 Coin Generator; www.jr-graph.fr,