From Wikipedia, the free encyclopedia
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 .
Supercritical case
The normal form of the supercritical pitchfork bifurcation is
d
x
d
t
=
r
x
−
x
3
.
{\displaystyle {\frac {dx}{dt}}=rx-x^{3}.}
For negative values of
r
{\displaystyle r}
x
=
0
{\displaystyle x=0}
r
>
0
{\displaystyle r>0}
x
=
0
{\displaystyle x=0}
x
=
±
r
{\displaystyle x=\pm {\sqrt {r}}}
Subcritical case
The normal form for the subcritical case is
d
x
d
t
=
r
x
+
x
3
.
{\displaystyle {\frac {dx}{dt}}=rx+x^{3}.}
In this case, for
r
<
0
{\displaystyle r<0}
x
=
0
{\displaystyle x=0}
x
=
±
−
r
{\displaystyle x=\pm {\sqrt {-r}}}
r
>
0
{\displaystyle r>0}
x
=
0
{\displaystyle x=0}
An ODE
x
˙
=
f
(
x
,
r
)
{\displaystyle {\dot {x}}=f(x,r)\,}
described by a one parameter function
f
(
x
,
r
)
{\displaystyle f(x,r)}
r
∈
R
{\displaystyle r\in {\mathbb {R}}}
−
f
(
x
,
r
)
=
f
(
−
x
,
r
)
{\displaystyle -f(x,r)=f(-x,r)\,\,}
odd function ),
∂
f
∂
x
(
0
,
r
o
)
=
0
,
∂
2
f
∂
x
2
(
0
,
r
o
)
=
0
,
∂
3
f
∂
x
3
(
0
,
r
o
)
≠
0
,
∂
f
∂
r
(
0
,
r
o
)
=
0
,
∂
2
f
∂
r
∂
x
(
0
,
r
o
)
≠
0.
{\displaystyle {\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
)
{\displaystyle (x,r)=(0,r_{o})}
∂
3
f
∂
x
3
(
0
,
r
o
)
{
<
0
,
s
u
p
e
r
c
r
i
t
i
c
a
l
>
0
,
s
u
b
c
r
i
t
i
c
a
l
{\displaystyle {\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
![{\displaystyle {\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}}}](https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f731517921c602ea11ababfdad008d8c839e1aa7)
