**Author:**
Livia Corsi and Guido Gentile

**Title:** *
Melnikov theory to all orders and Puiseux series
for subharmonic solutions
*

**Abstract: **
We study the problem of subharmonic bifurcations for analytic systems
in the plane with perturbations depending periodically on time, in the
case in which we only assume that the subharmonic Melnikov function
has at least one zero. If the order of zero is odd, then there is
always at least one subharmonic solution, whereas if the order
is even in general other conditions have to be assumed to
guarantee the existence of subharmonic solutions. Even when such
solutions exist, in general they are not analytic in the perturbation
parameter. We show that they are analytic in a fractional power of the
perturbation parameter. To obtain a fully constructive algorithm
which allows us not only to prove existence but also to obtain
bounds on the radius of analyticity and to approximate the solutions
within any fixed accuracy, we need further assumptions.
The method we use to construct the solution -- when this is possible --
is based on a combination of the Newton-Puiseux algorithm
and the tree formalism. This leads to a graphical representation
of the solution in terms of diagrams. Finally, if the subharmonic
Melnikov function is identically zero, we show that it
is possible to introduce higher order generalisations,
for which the same kind of analysis can be carried out.

**Keywords:**
Subharmonic solution; bifurcation curve; subharmonic Melnikov function;
degeneracy; Puiseux series; Weierstrass preparation theorem;
perturbation theory; tree formalism; diagrammatic rules.

Livia Corsi

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: lcorsi@mat.uniroma3.it

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it