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

dxdt=μ+x2

where μ is the bifurcation parameter. The transcritical bifurcation

dxdt=rln⁡x+x−1

near x=1 can be converted to the normal form

dudt=μu−u2+O(u3)

with the transformation u=x−1,μ=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. doi:10.4249/scholarpedia.1902. Bibcode: 2006SchpJ...1.1902M.
* Murdock, James (2003). Normal Forms and Unfoldings for Local Dynamical Systems. Springer. ISBN 978-0-387-21785-7.

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