Invariant Parallels, Invariant Meridians and Limit Cycles of Polynomial Vector Fields on Some 2-Dimensional Algebraic Tori in $$\mathbb R ^3$$ R 3

Abstract

We consider the polynomial vector fields of arbitrary degree in R 3 having the 2-dimensional algebraic toruswhere l, m and n positive integers, and r ∈ (1, ∞), invariant by their flow. We study the possible configurations of invariant meridians and parallels that these vector fields can exhibit on T 2 (l, m, n). Furthermore we analyze when these invariant meridians or parallels are limit cycles.

Polynomial vector fields on the Clifford torus

“…The analogous result for the invariant straight lines of polynomial vector fields in R 2 was provided before in [1]. The study of the maximum number of meridians and parallels for a torus in R 3 were studied in [9], and for an algebraic torus in [10]. In surfaces of revolution in R 3 the meridians and parallels invariant by polynomial vector fields have been studied in [5].…”

When Parallels and Meridians are Limit Cycles for Polynomial Vector Fields on Quadrics of Revolution in the Euclidean 3-Space

“…The notion of the extactic polynomial goes back to the work of Lagutinskii (see [2] and references therein) and have been used in different papers, see for instance [1,4,6,9].…”

Quadratic three-dimensional differential systems having invariant planes with total multiplicity nine

“…) . The notion of extactic polynomial already appears in the work of Lagutinskii, see [3] and references therein, and have been used as defined in [19] in different papers, see for instance [2,8,14,15,18]. In this work the definition of extactic polynomial and its properties play a fundamental role in the proof of the main theorem, as will be seen ahead.…”