Pitchfork bifurcation
Encyclopedia
In bifurcation theory
, a field within mathematics
, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcation
s have two types - supercritical or subcritical.
In flows, that is, continuous dynamical systems described by ODE
s, pitchfork bifurcations occur generically in systems with symmetry
.
The normal form of the supercritical pitchfork bifurcation is
For negative values of
, there is one stable equilibrium at 
. For 
there is an unstable equilibrium at 
, and two stable equilibria at 
.
The normal form for the subcritical case is
In this case, for
the equilibrium at 
is stable, and there are two unstable equilbria at 
. For 
the equilibrium at 
is unstable.
described by a one parameter function
with 
satisfying:
(f is an odd function),
has a pitchfork bifurcation at
. The form of the pitchfork is given
by the sign of the third derivative:
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
, a field within mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a pitchfork bifurcation is a particular type of local bifurcation. Pitchfork bifurcations, like Hopf bifurcation
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...
s have two types - supercritical or subcritical.
In flows, that is, continuous dynamical systems described by ODE
ODE
ODE may refer to:* Ohio Department of Education, the state education agency of Ohio* Omicron Delta Epsilon, an international honor society in the field of economics* Open Dynamics Engine, a real-time physics engine...
s, pitchfork bifurcations occur generically in systems with symmetry
Symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. It is actually the same as invariance: the property that something does not change under a set of transformations....
.
Supercritical case
For negative values of
, there is one stable equilibrium at . For there is an unstable equilibrium at , and two stable equilibria at .Subcritical case
In this case, for
the equilibrium at is stable, and there are two unstable equilbria at . For the equilibrium at is unstable.Formal definition
An ODEdescribed by a one parameter function
with satisfying: (f is an odd function),has a pitchfork bifurcation at
. The form of the pitchfork is givenby the sign of the third derivative: