Tay Bridge disaster

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my site; wellness [continue reading this..] 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

• Bifurcation theory
• Bifurcation diagram