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 } is the bifurcation parameter. The transcritical bifurcation
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} can be converted to the normal form
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
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.