|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Other uses|Pitchfork (disambiguation)}}
| | The title of the writer is Numbers. My family life in Minnesota and my family enjoys it. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. Since she was 18 she's been operating as a receptionist but her marketing never comes.<br><br>My blog [http://3bbc.com/index.php?do=/profile-548128/info/ 3bbc.com] |
| 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.
| |
| | |
| In continuous dynamical systems described by [[Ordinary differential equation|ODEs]]—i.e. flows—pitchfork bifurcations occur generically in systems with [[symmetry in mathematics|symmetry]].
| |
| | |
| ==Supercritical case==
| |
| [[Image:Pitchfork bifurcation supercritical.svg|180px|right|thumb|Supercritical case: solid lines represent stable points, while dotted line
| |
| represents unstable one.]]
| |
| The [[normal form (bifurcation theory)|normal form]] of the supercritical pitchfork bifurcation is | |
| :<math> \frac{dx}{dt}=rx-x^3. </math>
| |
| 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>.
| |
| | |
| ==Subcritical case==
| |
| [[Image:Pitchfork bifurcation subcritical.svg|180px|right|thumb|Subcritical case: solid line represents stable point, while dotted lines
| |
| represent unstable ones.]]
| |
| The [[normal form (bifurcation theory)|normal form]] for the subcritical case is
| |
| :<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.
| |
| | |
| ==Formal definition==
| |
| An ODE
| |
| :<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:
| |
| :<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]]),
| |
| | |
| :<math> | |
| \begin{array}{lll}
| |
| \displaystyle\frac{\part f}{\part x}(0, r_{o}) = 0 , &
| |
| \displaystyle\frac{\part^2 f}{\part x^2}(0, r_{o}) = 0, &
| |
| \displaystyle\frac{\part^3 f}{\part x^3}(0, r_{o}) \neq 0,
| |
| \\[12pt]
| |
| \displaystyle\frac{\part f}{\part r}(0, r_{o}) = 0, &
| |
| \displaystyle\frac{\part^2 f}{\part r \part x}(0, r_{o}) \neq 0.
| |
| \end{array}
| |
| </math>
| |
| | |
| has a '''pitchfork bifurcation''' at <math>(x, r) = (0, r_{o})</math>. The form of the pitchfork is given
| |
| by the sign of the third derivative:
| |
| | |
| :<math> \frac{\part^3 f}{\part x^3}(0, r_{o})
| |
| \left\{
| |
| \begin{matrix}
| |
| < 0, & \mathrm{supercritical} \\
| |
| > 0, & \mathrm{subcritical}
| |
| \end{matrix}
| |
| \right.\,\,
| |
| </math><!-- the \,\, is to force no-hinting in the antialiasing mode of pngtex - please do not remove it! -->
| |
| | |
| ==References== | |
| *Steven Strogatz, "Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering", Perseus Books, 2000.
| |
| *S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos", Springer-Verlag, 1990.
| |
| | |
| == See also ==
| |
| | |
| * [[Bifurcation theory]]
| |
| * [[Bifurcation diagram]]
| |
| | |
| [[Category:Bifurcation theory]]
| |
The title of the writer is Numbers. My family life in Minnesota and my family enjoys it. One of the extremely best things in the world for me is to do aerobics and now I'm attempting to earn money with it. Since she was 18 she's been operating as a receptionist but her marketing never comes.
My blog 3bbc.com