Mathematics – Dynamical Systems
Scientific paper
2008-07-02
Mathematics
Dynamical Systems
30 pages, 6 figures
Scientific paper
10.1063/1.3009574
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.
Corsi Livia
Gentile Guido
No associations
LandOfFree
Melnikov theory to all orders and Puiseux series for subharmonic solutions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Melnikov theory to all orders and Puiseux series for subharmonic solutions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Melnikov theory to all orders and Puiseux series for subharmonic solutions will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-422041