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

[math]\displaystyle{ \frac{\mathrm{d}x}{\mathrm{d}t} = \mu + x^2 }[/math]

where [math]\displaystyle{ \mu }[/math] is the bifurcation parameter. The transcritical bifurcation

[math]\displaystyle{ \frac{\mathrm{d}x}{\mathrm{d}t} = r \ln x + x - 1 }[/math]

near [math]\displaystyle{ x=1 }[/math] can be converted to the normal form

[math]\displaystyle{ \frac{\mathrm{d}u}{\mathrm{d}t} = \mu u - u^2 + O(u^3) }[/math]

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

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

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.