Symmetric periodic orbits near heteroclinic loops at infinity for a class of polynomial vector fields

For polynomial vector fields in R3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

Integrability of a linear center perturbed by a fourth degree homogeneous polynomial

In this work we study the integrability of a two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fourth degree. We give sufficient conditions for integrability in polar coordinates. Finally we establish a conjecture about the independence of the two classes of parameters which appear in the system; if this conjecture is true the integrable cases found will be the only possible ones.

Integrability of a linear center perturbed by a fifth degree homogeneous polynomial

In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

Me Alexander I... Suuri Ruhtinas Suomen maasa... Wakutus-Kirja... Annettu Borgåsa sinä 15/27 päiwänä Maaliskuusa 1809

Turku 1809, Frenckell

Hans Kejserliga Majestäts Alexander I:s Tal hållet wid Landtdagens slut, i Borgå, den 6/18 Julii 1809 - [Titelsidan]

Åbo 1809, Frenckell

Hans Kejserliga Majestäts Alexander I:s Tal, hållet wid hyllningen i Borgå den 17/29 Martii 1809 - [Titelsidan]

Åbo 1809, Frenckell

Alexander II

London : Methuen & Co.Ltd 1915