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

: \frac{\mathrm{d}x}{\mathrm{d}t} = \mu + x^2

where \mu is the bifurcation parameter. The transcritical bifurcation

: \frac{\mathrm{d}x}{\mathrm{d}t} = r \ln x + x - 1

near x=1 can be converted to the normal form

: \frac{\mathrm{d}u}{\mathrm{d}t} = \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 for use of the terms canonical form, normal form, or standard form more generally in mathematics.

References

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Further reading

  • {{citation | last1 = Guckenheimer | first1 = John | last2 = Holmes | first2 = Philip | title = Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields | year = 1983 | publisher = Springer | isbn = 0-387-90819-6 | at = Section 3.3}}
  • {{citation | last1 = Kuznetsov | first1 = Yuri A. | title = Elements of Applied Bifurcation Theory | year = 1998 | publisher = Springer | edition = Second | isbn = 0-387-98382-1 | at = Section 2.4}}
  • {{cite journal|last1=Murdock|first1=James|title=Normal forms|journal=Scholarpedia|doi=10.4249/scholarpedia.1902|date=2006|volume=1|issue=10|page=1902|bibcode=2006SchpJ...1.1902M|doi-access=free}}
  • {{cite book|last1=Murdock|first1=James|title=Normal Forms and Unfoldings for Local Dynamical Systems|date=2003|publisher=Springer|isbn=978-0-387-21785-7}}

Category:Bifurcation theory

Category:Dynamical systems

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