Uniqueness of limit cycles in polynomial systems with algebraic invariants

Abstract

The uniqueness of limit cycles is proved for quadratic systems with an invariant parabola and for cubic systems with four real line invariants. Also a new, simple proof is given of the uniqueness of limit cycles occurring in unfoldings of certain vector fields with codimension two singularities.

Poincaré Bifurcations of Two Classes of Polynomial Systems

“…In addition, the existence of invariant algebraic curves in polynomial systems may influence the number of limit cycles. For example, the planar quadratic systems with one invariant line or conic curve or cubic curve can have at most one limit cycle [1][2][3][4]. In [3], the authors proved that the cubic systems with four invariant lines have at most one limit cycle.…”

“…For example, the planar quadratic systems with one invariant line or conic curve or cubic curve can have at most one limit cycle [1][2][3][4]. In [3], the authors proved that the cubic systems with four invariant lines have at most one limit cycle. In [5,6], the authors proved that a real polynomial system of degree with irreducible invariant algebraic curves has at most 1 + ( − 1)( − 2)/2 limit cycles if is even and ( − 1)( − 2)/2 limit cycles if is odd.…”

“…When = 0, system (2) is a Hamilton system with a family of ovals ℎ = {( , ) ∈ R 2 | ( , ) = 2 + 2 = ℎ, ℎ > 0} . (3) Define the Abelian integral…”

The Distribution of Limit Cycles in Quadratic Systems with Four Finite Singularities

“…By (2.2) this line corresponds with the line y=u*x in system (2.1) connecting O with a singularity at infinity. System (2.3) is of a type studied in [15]. In [15] it was shown that the known solution v=0 of (2.3) can be exploited to bring the system into a simple Lie nard form by introducing the variable z replacing v according to…”

“…The Bogdanov Takens system (see, for instance [6], and quadratic systems with an algebraic invariant curve of order at most two (see [14,15]), are examples of quadratic systems with at most one limit cycle in the whole phase plane. Conjecture 1.1.…”