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

\frac{d x}{d t} = μ + x^{2}

where $μ$ is the bifurcation parameter. The transcritical bifurcation

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

near $x = 1$ can be converted to the normal form

\frac{d u}{d t} = R u - u^{2} + O (u^{3})

with the transformation $u = \frac{r}{2} (x - 1), R = r + 1$ .

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.

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.

This mathematics-related article is a stub. You can help Wikipedia by adding missing information.

[strogatz-1] Strogatz, Steven. "Nonlinear Dynamics and Chaos". Westview Press, 2001. p. 52.

Normal form (dynamical systems)

References

Further reading