Pitchfork bifurcation

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


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Bifurcation theory — is the mathematical study of changes in the qualitative or topological structure of a given family. Examples of such families are the integral curves of a family of vector fields or, the solutions of a family of differential equations. Most… …   Wikipedia

  • Bifurcation diagram — In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long term values (equilibria/fixed points or periodic orbits) of a system as a function of a bifurcation parameter in the system. It is usual to represent …   Wikipedia

  • Saddle-node bifurcation — In the mathematical area of bifurcation theory a saddle node bifurcation or tangential bifurcation is a local bifurcation in which two fixed points (or equilibria) of a dynamical system collide and annihilate each other. The term saddle node… …   Wikipedia

  • Catastrophe theory — This article refers to the study of dynamical systems. For other meanings, see catastrophe. In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more… …   Wikipedia

  • Supercritical — may refer to:* Critical mass, the smallest amount of fissile material needed for a sustained nuclear chain reaction * Critical temperature, Tc, the temperature above which distinct liquid and gas phases do not exist * Supercritical fluid, a… …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • Bicycle and motorcycle dynamics — A computer generated, simplified model of bike and rider demonstrating an uncontrolled right turn. An …   Wikipedia

  • Katastrophentheorie (Mathematik) — Die mathematische Katastrophentheorie beschäftigt sich mit unstetigen, sprunghaften Veränderungen von bestimmten dynamischen Systemen. Diese können, auch wenn sie unter bestimmten Voraussetzungen einen stabilen Zustand anstreben, bei Änderungen… …   Deutsch Wikipedia

  • Manipur University — Established June 5, 1980 Type Public Vice Chancellor Prof H Nandakumar [1] …   Wikipedia

  • Théorie des bifurcations — La théorie des bifurcations, en mathématiques et en physique est l étude des systèmes dynamiques. Une bifurcation intervient lorsqu un petit changement d un paramètre physique produit un changement majeur dans l organisation du système. Sommaire… …   Wikipédia en Français

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”