Hopf bifurcation theorem


Consider a planar system of ordinary differential equations, written in such a form as to make explicit the dependence on a parameter μ:

x = f1(x,y,μ)
y = f2(x,y,μ)

Assume that this system has the origin as an equilibrium for all μ. Suppose that the linearization Df at zero has the two purely imaginary eigenvalues λ1(μ) and λ2(μ) when μ=μc. If the real part of the eigenvalues verify

ddμ((λ1,2(μ)))|μ=μc>0

and the origin is asymptotically stablePlanetmathPlanetmath at μ=μc, then

roman]enumerateμcisabifurcationpoint;forsomeμ1¯suchthatμ1<μ<μc,theoriginisastablefocus;forsomeμ2¯suchthatμc<μ<μ2,theoriginisunstable,surroundedbyastablelimitcyclewhosesizeincreaseswithμ.Thisisasimplifiedversionofthetheorem,correspondingtoasupercriticalHopfbifurcation.SometimestheHopftheoremiscalledPoincaré-Andronov-Hopf theoremsinceitwasindependentlydiscoveredbyAndronovin1929andHopfin1943andPoincaréhaddiscussionofsuchresultin1892.[HK]

References

  • HK
Hale,JackH.&Kocak,Hüseyin:DynamicsandBifurcations.Springer-Verlag,NewYork,1991.TitleHopf bifurcation theoremCanonical nameHopfBifurcationTheoremDate of creation2013-03-22 13:18:45Last modified on2013-03-22 13:18:45OwnerDaume (40)Last modified byDaume (40)Numerical id8AuthorDaume (40)Entry typeTheoremClassificationmsc 34C05SynonymPoincaré-Andronov-Hopf