Pitchfork bifurcation

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.

upercritical case



180px|right|thumb|Supercritical case: solid lines represents stable points, while dotted linesrepresents unstable one.The normal form of the supercritical pitchfork bifurcation is: frac{dx}{dt}=rx-x^3. 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 x = pmsqrt{r}.

ubcritical case



180px|right|thumb|Subcritical case: solid lines represents stable points, while dotted linesrepresents unstable one.The normal form for the subcritical case is: frac{dx}{dt}=rx+x^3. In this case, for r<0 the equilibrium at x=0 is stable, and there are two unstable equilbria at x = pmsqrt{-r}. For r>0 the equilibrium at x=0 is unstable.

Formal definition

An ODE: dot{x}=f(x,r), described by a one parameter function f(x, r) with r in Bbb{R} satisfying:: -f(x, r) = f(-x, r),, (f is an odd function),

:egin{array}{lll}displaystylefrac{part f}{part x}(0, r_{o}) = 0 , &displaystylefrac{part^2 f}{part x^2}(0, r_{o}) = 0, &displaystylefrac{part^3 f}{part x^3}(0, r_{o}) eq 0,\ [12pt] displaystylefrac{part f}{part r}(0, r_{o}) = 0, &displaystylefrac{part^2 f}{part r part x}(0, r_{o}) eq 0.end{array}

has a pitchfork bifurcation at (x, r) = (0, r_{o}). The form of the pitchfork is givenby the sign of the third derivative:

: frac{part^3 f}{part x^3}(0, r_{o})left{ egin{matrix} < 0, & mathrm{supercritical} \ > 0, & mathrm{subcritical} end{matrix} ight.,,

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


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