Periodic and homoclinic orbits in conservative and reversible systems

Abstract

One of the main features of the phase portrait of scalar second order equations of the form

$$\ddot y + g\left( y \right) = 0$$

(1)

is the stable appearance of families {r _τ | τ ≥ τ ₀} of τ-periodic orbits r _τ which are symmetric with respect to the y-axis and which accumulate on a homoclinic orbit as τ → ∞. Now the equation (1) is both conservative and reversible, and it is the main purpose of this note to show that similar period blow-ups to a homoclinic orbit appear stably in general conservative or reversible systems. For certain subcases the result is not new : Devaney [1] gave a proof for the reversible case, using slightly stronger hypotheses than ours, and Strömgren [4] conjectured the result for Hamiltonian systems. A proof of Strömgren’s conjecture for the case of two degrees of freedom was given by Henrard [2]. Here we indicate how a technique introduced recently by X.-B. Lin [3] can be used to give a unified proof for both the conservative and the reversible case. The results stated in this note are based on joint work with Bernold Fiedler, and full details will be given in a forthcoming paper [5].