Normal form (dynamical systems)

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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

where is the bifurcation parameter. The transcritical bifurcation

near can be converted to the normal form

with the transformation .[1]

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


  1. ^ Strogatz, Steven. "Nonlinear Dynamics and Chaos". Westview Press, 2001. p. 52.

Further reading[edit]

  • 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.