In mathematics, the normal form of a

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...

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 In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated fun ...

to the normal form of the bifurcation. For example, the normal form of a saddle-node bifurcation is :

\frac = \mu + x^2

where

\mu

is the bifurcation parameter. The transcritical bifurcation :

\frac = r \ln x + x - 1

near

x=1

can be converted to the normal form :

\frac = \mu u - u^2 + O(u^3)

with the transformation

u = x -1, \mu = r + 1

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

canonical form In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an ...

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

References

Further reading