Guido Gentile, Michele V. Bartuccelli, and Jonathan H. Deane
Bifurcation curves of subharmonic solutions
(365K, pdf)

ABSTRACT.  We revisit a problem considered by Chow and Hale on the existence of 
subharmonic solutions for perturbed systems. In the analytic setting, 
under more general (weaker) conditions, we prove their results on the 
existence of bifurcation curves from the nonexistence to the existence 
of subharmonic solutions. In particular our results apply also when one 
has degeneracy to first order -- i.e. when the subharmonic Melnikov 
function vanishes identically. Moreover we can deal as well with the 
case in which degeneracy persists to arbitrarily high orders, in the 
sense that suitable generalisations to higher orders of the 
subharmonic Melnikov function are also identically zero. In general the 
bifurcation curves are not analytic, and even when they are smooth 
they can form cusps at the origin: we say in this case that the curves 
are degenerate as the corresponding tangent lines coincide. 
The technique we use is completely different from that of Chow and Hale, 
and it is essentially based on rigorous perturbation theory.