Bogdanov-Takens bifurcation

Bogdanov-Takens bifurcation

In bifurcation theory, a field within mathematics, a Bogdanov-Takens bifurcation is a well-studied example of a bifurcation with co-dimension two, meaning that two parameters must be varied for the bifurcation to occur. It is named after R. I. Bogdanov and Floris Takens, who independently and simultaneously described this bifurcation.

A system "y"' = "f"("y") undergoes a Bogdanov-Takens bifurcation if it has a fixed point and the linearization of "f" around that point has a double eigenvalue at zero (assuming that some technical nondegeneracy conditions are satisfied).

Three codimension-one bifurcations occur nearby: a saddle-node bifurcation, an Andronov-Hopf bifurcation and a homoclinic bifurcation. All associated bifurcation curves meet at the Bogdanov-Takens bifurcation.

The normal form of the Bogdanov-Takens bifurcation is: egin{align}y_1' &= y_2, \y_2' &= eta_1 + eta_2 y_1 + y_1^2 pm y_1 y_2.end{align}

It has also been found the existence of a codimension-three degenerate Takens-Bogdanov bifurcation, also known as Dumortier-Roussarie-Sotomayor bifurcation.

References

*Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373-388, 1981.
*Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995.
*Takens, F. "Forced Oscillations and Bifurcations." Comm. Math. Inst. Rijksuniv. Utrecht 2, 1-111, 1974.
*Dumortier F., Roussarie R., Sotomayor J. and Zoladek H., Bifurcations of Planar Vector Fields, Lecture Notes in Math. vol. 1480, 1-164, Springer-Verlag (1991).

External links

*


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Bifurcation theory — is the mathematical study of changes in the qualitative or topological structure of a given family. Examples of such families are the integral curves of a family of vector fields or, the solutions of a family of differential equations. Most… …   Wikipedia

  • R. I. Bogdanov — Rifkat Ibragimovich Bogdanov (born 1950) is a Russian mathematician known for contributions to nonlinear dynamical systems, bifurcation theory, and differential geometry. Ethnic Tatar. In his work on bifurcations of limit cycles and versal… …   Wikipedia

  • Floris Takens — (born November 12, 1940) is a Dutch mathematician known for contributions to the theory of chaotic dynamical systems. Together with David Ruelle he predicted that fluid turbulence could develop through a strange attractor, a term they coined, as… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • SYSTÈMES DYNAMIQUES DIFFÉRENTIABLES — Sans doute née avec le mémoire que Poincaré écrivit en 1881 «sur les courbes définies par des équations différentielles», où l’étude quantitative (analytique) locale des équations différentielles dans le champ complexe est remplacée par leur… …   Encyclopédie Universelle

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”