Normal form (dynamical systems)

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

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}}$

where ${\displaystyle \mu }$ is the bifurcation parameter. The transcritical bifurcation

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1}$

near ${\displaystyle x=1}$ can be converted to the normal form

${\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=\mu u-u^{2}+O(u^{3})}$

with the transformation ${\displaystyle u=x-1,\mu =r+1}$.[1]

See also canonical form for use of the terms canonical form, normal form, or standard form more generally in mathematics.

References

1. 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. Retrieved 4 December 2016.
• Murdock, James (2003). Normal Forms and Unfoldings for Local Dynamical Systems. Springer. ISBN 978-0-387-21785-7.