Pitchfork bifurcation
From Wikipedia, the free encyclopedia
In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcations have two types - supercritical or subcritical.
In flows, that is, continuous dynamical systems described by ODEs, pitchfork bifurcations occur generically in systems with symmetry.
Contents |
[edit] Supercritical case
The normal form of the supercritical pitchfork bifurcation is
For negative values of r, there is one stable equilibrium at x = 0. For r > 0 there is an unstable equilibrium at x = 0, and two stable equilibria at 
.
[edit] Subcritical case
The normal form for the subcritical case is
In this case, for r < 0 the equilibrium at x = 0 is stable, and there are two unstable equilbria at 
. For r > 0 the equilibrium at x = 0 is unstable.
[edit] Formal definition
An ODE
described by a one parameter function f(x,r) with 
satisfying:
(f is an odd function),
![\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/5/c/b/5cb285a832f1af13d268993fc742ad4d.png)
has a pitchfork bifurcation at (x,r) = (0,ro). The form of the pitchfork is given by the sign of the third derivative:
[edit] 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.
