A codimension two bifurcation with a third order Picard-Fuchs equation. (English) Zbl 0571.34021

The authors show that for small \((\mu_ 1,\mu_ 2)\) lying in a certain cone the two parameter family of planar vectorfields \[ (1)\quad dx/dt=2xy+\mu_ 1x+\mu_ 2x^ 3,\quad dy/dt=1/4-x^ 2-y^ 2 \] has a unique limit cycle which is born from a Hopf bifurcation and dies in a saddle connection. Since (1) is a small perturbation of a Hamiltonian vectorfield \(X_ H\), applying the averaging method reduces the problem to showing that the function \[ \eta (h)=\int_{\gamma h}x^ 3 dy/\int_{\gamma h}x dy \] where \(\gamma_ h\) is a smooth compact component of the h level set of H, is strictly monotonic for \(-1/12<h<0\) or \(0<h<1/12\). This is done by showing that \(\eta\) satisfies a third order Picard-Fuchs equation.


34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Arnol’d, V.I, Lectures on bifurcations in versal families, Russian math. surveys, 27, 54-123, (1972) · Zbl 0264.58005
[2] Arnol’d, V.I; Arnol’d, V.I, Geometrical methods in the theory of ordinary differential equations, (1983), Springer-Verlag New York · Zbl 0569.58018
[3] Arnold, V, Algebraic unsolvability of the problem of Lyapunov stability and the problem of topological classification of singular points of an analytic system of differential equations, Funct. anal. appl., 4, 173-180, (1970) · Zbl 0236.34042
[4] Bogdanov, R.I, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funct. anal. appl., 9, 144-145, (1975) · Zbl 0447.58009
[5] Bogdanov, R.I, Bifurcation of the limit cycle of a family of plane vector fields, Selecta math. soviet, 1, 373-387, (1981) · Zbl 0518.58029
[6] Bogdanov, R.I, Versal deformation of a singularity of a vector field on the plane in the case of zero eigenvalues, Selecta math. soviet, 1, 389-421, (1981) · Zbl 0518.58030
[7] Bogdanov, R.I, Orbital equivalence of singular points of vector fields, Funct. anal. appl., 10, 316-317, (1976) · Zbl 0351.58010
[8] Bogdanov, R.I, (), 51-84
[9] Brieskorn, E; Knörrer, H, Ebene algebraische kurven, (1981), Birkhäuser Basel · Zbl 0508.14018
[10] Carr, J, Applications of centre manifold theory, () · Zbl 0464.58001
[11] {\scJ. Carr, S. N. Chow, and J. Hale}, Abelian integrals and bifurcation theory, J. Differential Equations, in press. · Zbl 0587.34033
[12] Chow, S.N; Hale, J.K, Methods of bifurcation theory, (1982), Springer-Verlag New York
[13] Guckenheimer, J, On a codimension two bifurcation, () · Zbl 0482.58006
[14] Guckenheimer, J; Holmes, P, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, () · Zbl 0515.34001
[15] Horozov, E.I, Versal deformations of equivariant vector fields with \(Z\)_{2} or \(Z\)3 symmetry, (), 163-192, [Russian]
[16] Il’yashenko, Yu.S, The multiplicity of limit cycles arising from perturbations of the form \(w′ = P2Q1\) of a Hamilton equation in the real and complex domain, (), 191-202, No. 2 · Zbl 0494.34018
[17] Il’yashenko, Yu.S, Zeros of special abelian integrals in a real domain, Funct. anal. appl., 11, 309-311, (1977) · Zbl 0413.58011
[18] Keener, J.P, Infinite period bifurcation and global bifurcation branches, SIAM J. appl. math., 41, 127-144, (1981) · Zbl 0523.34046
[19] Rauch, J; Lebowitz, A, Elliptic functions, (1973), Williams & Wilkins Baltimore, Md
[20] Takens, F, Forced oscillations and bifurcations, () · Zbl 1156.37315
[21] Takens, F, Singularities of vector fields, Publ. math. I.H.E.S., 43, 47-100, (1974) · Zbl 0279.58009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.