Melnikov theory to all orders and Puiseux series for subharmonic solutions

TL;DR

This paper develops a method combining Newton-Puiseux and tree formalism to analyze subharmonic bifurcations in analytic systems, revealing solutions are often fractional power series in the perturbation parameter and providing constructive algorithms for their approximation.

Contribution

It introduces a novel approach using Newton-Puiseux and diagrammatic methods to study subharmonic solutions, including cases with zero Melnikov functions, extending existing bifurcation analysis.

Findings

Solutions are fractional power series in the perturbation parameter.

Existence of solutions depends on the order of zero of the Melnikov function.

A constructive algorithm for approximating solutions and bounding their analyticity radius.

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.