Bifurcation of Limit Cycles from Quadratic Isochrones

Abstract For a one parameter family of plane quadratic vector fields X (.,e) depending analytically on a small real parameter e, we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous (or linearizable) center. Techniques are developed for treating the bifurcations of all orders, and these are applied to prove the following results. For the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three. For one class of nonlinear isochrones, at most one continuous family of limit cycles can emerge, whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge. Moreover, for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center.

[1]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[2]  J. Walker,et al.  Book Reviews : THEORY OF BIFURCATIONS OF DYNAMIC SYSTEMS ON A PLANE A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier J. Wiley & Sons, New York , New York (1973) , 1976 .

[3]  N. N. Bautin,et al.  On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type , 1954 .

[4]  Solomon Lefschetz,et al.  Differential Equations: Geometric Theory , 1958 .

[5]  Christiane Rousseau,et al.  Codimension 2 Symmetric Homoclinic Bifurcations and Application to 1:2 Resonance , 1990, Canadian Journal of Mathematics.

[6]  R. Roussarie,et al.  A note on finite cyclicity property and Hilbert's 16th. Problem , 1988 .

[7]  Stephen P. Diliberto I. ON SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS , 1950 .

[8]  Jan A. Sanders,et al.  A codimension two bifurcation with a third order Picard-Fuchs equation , 1985 .

[9]  Carmen Chicone,et al.  Bifurcation of critical periods for plane vector fields , 1989 .

[10]  Jürgen Moser,et al.  Lectures on Celestial Mechanics , 1971 .

[11]  Konstantin Sergeevich Sibirsky Introduction to the Algebraic Theory of Invariants of Differential Equations , 1989 .

[12]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[13]  K. S. Sibirskiĭ,et al.  Introduction to the algebraic theory of invariants of differential equations , 1988 .

[14]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[15]  G. S. Petrov Elliptic integrals and their nonoscillation , 1986 .

[16]  On the number of critical points of the period , 1986 .

[17]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[18]  P. Byrd,et al.  Handbook of Elliptic Integrals for Engineers and Physicists , 2014 .

[19]  Willie L. Roberts,et al.  On systems of ordinary differential equations , 1961 .

[20]  William F. Langford,et al.  Degenerate Hopf bifurcation formulas and Hilbert's 16th problem , 1989 .

[21]  N. G. Lloyd,et al.  New Directions in Dynamical Systems: Limit Cycles of Polynomial Systems – Some Recent Developments , 1988 .