In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior.
Normal forms are often used for determining local bifurcations in a system. All systems exhibiting a certain type of bifurcation are locally (around the equilibrium) topologically equivalent to the normal form of the bifurcation. For example, the normal form of a saddle-node bifurcation is
-
d
x
d
t
=
μ
+
x
2
{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}}
where
μ
{\displaystyle \mu }
-
d
x
d
t
=
r
ln
x
+
x
−
1
{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1}
near
x
=
1
{\displaystyle x=1}
-
d
u
d
t
=
R
u
−
u
2
+
O
(
u
3
)
{\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Ru-u^{2}+O(u^{3})}
with the transformation
u
=
r
2
(
x
−
1
)
,
R
=
r
+
1
{\displaystyle u={\frac {r}{2}}(x-1),R=r+1}
See also canonical form for use of the terms canonical form, normal form, or standard form more generally in mathematics.
References
- Strogatz, Steven. "Nonlinear Dynamics and Chaos". Westview Press, 2001. p. 52.
Further reading
- Guckenheimer, John; Holmes, Philip (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Section 3.3, ISBN 0-387-90819-6
- Kuznetsov, Yuri A. (1998), Elements of Applied Bifurcation Theory (Second ed.), Springer, Section 2.4, ISBN 0-387-98382-1
- Murdock, James (2006). "Normal forms". Scholarpedia. 1 (10): 1902. Bibcode:2006SchpJ...1.1902M. doi:10.4249/scholarpedia.1902.
- Murdock, James (2003). Normal Forms and Unfoldings for Local Dynamical Systems. Springer. ISBN 978-0-387-21785-7.