48 research outputs found
Melnikov theory to all orders and Puiseux series for subharmonic solutions
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.Comment: 30 pages, 6 figure