Electron. J. Diff. Eqns., Vol. 1999(1999), No. 35, pp. 1-15.
### Higher order branching of periodic orbits from polynomial isochrones

B. Toni

**Abstract:**

We discuss the higher order local bifurcations of limit cycles from polynomial
isochrones (linearizable centers) when the linearizing transformation is
explicitly known and yields a polynomial perturbation one-form. Using a method
based on the relative cohomology decomposition of polynomial one-forms
complemented with a step reduction process, we give an explicit formula for the
overall upper bound of branch points of limit cycles in an arbitrary *n* degree
polynomial perturbation of the linear isochrone, and provide an algorithmic
procedure to compute the upper bound at successive orders.

We derive a complete analysis of the nonlinear cubic Hamiltonian isochrone and
show that at most nine branch points of limit cycles can bifurcate in a cubic
polynomial perturbation. Moreover, perturbations with exactly two, three, four,
six, and nine local families of limit cycles may be constructed.
Submitted August 29, 1999. Published September 20, 1999.

Math Subject Classifications: 34C15, 34C25, 58F14, 58F21, 58F30.

Key Words: Limit cycles, Isochrones, Perturbations, Cohomology Decomposition.

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This article is related to another publication in the EJDE:
*Branching of periodic orbits from Kukles isochrones,* by B. Toni,
Vol. 1998(1998), No. 13, pp. 1-10

Bourama Toni

Facultad de Ciencias

Universidad Aut\'onoma Del Estado de Morelos

Av. Universidad 1001, Col. Chamilpa

Cuernavaca 62210, Morelos, Mexico.

Tel: (52)(73) 29 70 20. Fax: (52) (73) 29 70 40.

e-mail address: toni@servm.fc.uaem.mx

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