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(ℜ(λ1,2(μ)))|μ=μc>0 |
and the origin is asymptotically stable at μ=μc, then
roman]enumerateμcisabifurcationpoint;forsomeμ1∈ˉℝsuchthatμ1<μ<μc,theoriginisastablefocus;forsomeμ2∈ˉℝsuchthatμc<μ<μ2,theoriginisunstable,surroundedbyastablelimitcyclewhosesizeincreaseswithμ.Thisisasimplifiedversionofthetheorem,correspondingtoasupercriticalHopfbifurcation.SometimestheHopftheoremiscalledUnknown node type: emsinceitwasindependentlydiscoveredbyAndronovin1929andHopfin1943andPoincaréhaddiscussionofsuchresultin1892.Unknown node type: citeUnknown node type: sectionHale,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 |