Normal form (dynamical systems) explained

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

(d x)/(d t) = \mu + x ^{2}

where

\mu is the bifurcation parameter. The transcritical bifurcation (d x)/(d t) = r ln x + x - 1

near

x =1 can be converted to the normal form (d u)/(d t) = \mu u - u ^{2} + O(u ^{3)}

with the transformation

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.

Further reading

• • • Murdock. James. Normal forms. Scholarpedia. 10.4249/scholarpedia.1902. 2006. 1. 10. 1902. 2006SchpJ...1.1902M. free.
• Book: Murdock. James. Normal Forms and Unfoldings for Local Dynamical Systems. 2003. Springer. 978-0-387-21785-7.

Notes and References

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