Pitchfork bifurcation

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For other uses, see Pitchfork (disambiguation).

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 continuous dynamical systems described by ODEs—i.e. flows—pitchfork bifurcations occur generically in systems with symmetry.

Supercritical case

Supercritical case: solid lines represent stable points, while dotted line represents 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=\pm {\sqrt {r}}.

Subcritical case

Subcritical case: solid line represents stable point, while dotted lines represent unstable ones.

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=\pm {\sqrt {-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 {\mathbb {R}} satisfying:

-f(x,r)=f(-x,r)\,\,  (f is an odd function),
{\begin{array}{lll}\displaystyle {\frac {\partial f}{\partial x}}(0,r_{{o}})=0,&\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}(0,r_{{o}})=0,&\displaystyle {\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{{o}})\neq 0,\\[12pt]\displaystyle {\frac {\partial f}{\partial r}}(0,r_{{o}})=0,&\displaystyle {\frac {\partial ^{2}f}{\partial r\partial x}}(0,r_{{o}})\neq 0.\end{array}}

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

{\frac {\partial ^{3}f}{\partial x^{3}}}(0,r_{{o}})\left\{{\begin{matrix}<0,&{\mathrm {supercritical}}\\>0,&{\mathrm {subcritical}}\end{matrix}}\right.\,\,

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

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