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| {{Other uses|Pitchfork (disambiguation)}}
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| In [[bifurcation theory]], a field within [[mathematics]], a '''pitchfork bifurcation''' is a particular type of local bifurcation. Pitchfork bifurcations, like [[Hopf bifurcation]]s have two types - supercritical or subcritical.
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| In continuous dynamical systems described by [[Ordinary differential equation|ODEs]]—i.e. flows—pitchfork bifurcations occur generically in systems with [[symmetry in mathematics|symmetry]].
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| ==Supercritical case==
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| [[Image:Pitchfork bifurcation supercritical.svg|180px|right|thumb|Supercritical case: solid lines represent stable points, while dotted line
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| represents unstable one.]]
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| The [[normal form (bifurcation theory)|normal form]] of the supercritical pitchfork bifurcation is
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| :<math> \frac{dx}{dt}=rx-x^3. </math>
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| For negative values of <math>r</math>, there is one stable equilibrium at <math>x = 0</math>. For <math>r>0</math> there is an unstable equilibrium at <math>x = 0</math>, and two stable equilibria at <math>x = \pm\sqrt{r}</math>.
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| ==Subcritical case==
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| [[Image:Pitchfork bifurcation subcritical.svg|180px|right|thumb|Subcritical case: solid line represents stable point, while dotted lines
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| represent unstable ones.]]
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| The [[normal form (bifurcation theory)|normal form]] for the subcritical case is
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| :<math> \frac{dx}{dt}=rx+x^3. </math> | |
| In this case, for <math>r<0</math> the equilibrium at <math>x=0</math> is stable, and there are two unstable equilbria at <math>x = \pm \sqrt{-r}</math>. For <math>r>0</math> the equilibrium at <math>x=0</math> is unstable.
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| ==Formal definition== | |
| An ODE
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| :<math> \dot{x}=f(x,r)\,</math> | |
| described by a one parameter function <math>f(x, r)</math> with <math> r \in \Bbb{R}</math> satisfying:
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| :<math> -f(x, r) = f(-x, r)\,\,</math><!-- the \,\, is to force no-hinting in the antialiasing mode of pngtex - please do not remove it! --> (f is an [[odd function]]),
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| :<math>
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| \begin{array}{lll}
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| \displaystyle\frac{\part f}{\part x}(0, r_{o}) = 0 , &
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| \displaystyle\frac{\part^2 f}{\part x^2}(0, r_{o}) = 0, &
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| \displaystyle\frac{\part^3 f}{\part x^3}(0, r_{o}) \neq 0,
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| \\[12pt]
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| \displaystyle\frac{\part f}{\part r}(0, r_{o}) = 0, &
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| \displaystyle\frac{\part^2 f}{\part r \part x}(0, r_{o}) \neq 0.
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| \end{array}
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| </math>
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| has a '''pitchfork bifurcation''' at <math>(x, r) = (0, r_{o})</math>. The form of the pitchfork is given
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| by the sign of the third derivative:
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| :<math> \frac{\part^3 f}{\part x^3}(0, r_{o})
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| \left\{
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| \begin{matrix}
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| < 0, & \mathrm{supercritical} \\
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| > 0, & \mathrm{subcritical}
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| \end{matrix}
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| \right.\,\,
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| </math><!-- the \,\, is to force no-hinting in the antialiasing mode of pngtex - please do not remove it! -->
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| ==References==
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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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| *S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos", Springer-Verlag, 1990.
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| == See also ==
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| * [[Bifurcation theory]]
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| * [[Bifurcation diagram]]
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| [[Category:Bifurcation theory]]
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