Homoclinic period blow-up in reversible and conservative systems
It is shown that in conservative systems of autonomous ordinary differential equations each non-degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium possesses an associated family of periodic orbits. The family is parametrized by the period, and the periodic orbits accumulate on the homoclinic orbit as the period tends to infinity. A similar result holds for symmetric orbits in reversible systems. The statements are proved by constructing a one-dimensional equation for the homoclinic solution, which vanishes identically for the special cases. The results extend earlier work by Devaney and Henrard and provide a positive answer to a conjecture of Strömgren.
conservative systems of autonomous ordinary differential equations
scientific article; zbMATH DE number 63820
scientific article