A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems

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Abstract

In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the existence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in R2 and of the discontinuous piecewise differential systems in R2 formed by two linear differential systems separated by a straight line. For this class of discontinuous piecewise differential systems the study of their algebraic and explicit non-algebraic limit cycles just is starting. Here we provide the first explicit non-algebraic limit cycle for the discontinuous piecewise linear differential systems. Additionally we recall seven open questions related with these types of limit cycles.

Introduction

We start by recalling the definition of the two classes of differential systems whose algebraic and explicit non-algebraic limit cycles we will study.

Let P(x,y) and Q(x,y) be real polynomials in the variables x and y. Then the differential system ẋ=P(x,y),ẏ=Q(x,y),where as usual the dot denotes the derivative with respect to the independent variable t, is a polynomial differential system. The maximum of the degrees of the polynomials P and Q is the degree of the polynomial differential system (1).

Here we consider the discontinuous piecewise differential systems X±:(ẋ,ẏ)=(f±(x,y),g±(x,y)),defined in the half-planes Σ±={(x,y)R2:±x>0}. On the straight line Σ={x=0} the differential system is bivaluated. The straight line Σ is called the straight line of discontinuity when the two vector fields X± do not coincide on it. We use the Filippov conventions for defining the discontinuous piecewise differential system on Σ, see [33]. If f+(0,y)f(0,y)>0 at the point (0,y)Σ we say that (0,y) is a crossing point. If a periodic orbit of a discontinuous piecewise differential system (2) has exactly two crossing points we say that it is a crossing periodic orbit.

A limit cycle (respectively crossing limit cycle) of system (1) (respectively (2)) is an isolated periodic orbit in the set of all periodic orbits (respectively crossing periodic orbits) of system (1) (respectively (2)).

Here if h(x,y) is a real polynomial irreducible in the ring R[x,y] of all real polynomials in the variables x and y, the zero set {h(x,y)=0} is an algebraic curve. An algebraic limit cycle is a limit cycle contained in an algebraic curve of the plane, otherwise such a limit cycle is called non-algebraic. The degree of an algebraic limit cycle is the degree of the irreducible polynomial which defines the algebraic curve containing the limit cycle. It is well known that the orbits of a polynomial differential system (1) are contained in analytic curves, which usually are not algebraic curves.

In general it is a difficult problem to distinguish if a limit cycle is algebraic or not. The proof that the famous limit cycle exhibited in the van der Pol equation in 1926 (see [77]) was not algebraic arrived in 1995, see Odani [76]. The differential equation of van der Pol can be written as a polynomial differential system (1) of degree 3, but we do not know explicitly its limit cycle. More precisely, we do not know the explicit expression of the analytic curve which contains the non-algebraic limit cycle of van der Pol equation. We remark that all the algebraic limit cycles are explicit because we only know them when we provide the algebraic curve containing the limit cycle.

A crossing limit cycle of the piecewise differential system (2) is algebraic if all its points, with the exception of the ones which are in Σ, are contained in algebraic curves of the half-planes Σ±. The degree (n,n+) of an algebraic crossing limit cycle is formed by the degree of the irreducible polynomials defining the algebraic curves which contain the crossing limit cycle, thus n (respectively n+) is the degree of the irreducible polynomial defining the algebraic curve in Σ (respectively Σ+) which contains the piece of the crossing limit cycle in Σ (respectively Σ+).

Section snippets

Some general results on algebraic limit cycles

A nice result on algebraic limit cycles is the following one.

Theorem 1 Bautin–Christopher–Dolov–Kuzmin Theorem

Assume that f(x,y)=0 is a non-singular algebraic curve of degree m, and cx+dy+e=0 a straight line which does not intersect any bounded component of the algebraic curve f(x,y)=0. Let a and b real numbers such that ac+bd0. Consider the polynomial differential system ẋ=af(cx+dy+e)fy,ẏ=bf+(cx+dy+e)fx,of degree m. Then every bounded component of the algebraic curve f(x,y)=0 is a hyperbolic algebraic limit cycle of the polynomial

Algebraic limit cycles of discontinuous piecewise linear differential systems

This is a very new area of research. At this moment we only know the paper of Buzzi, Gasull and Torregrosa [17] dedicated to this problem. In what follows we summarize the main results of [17].

For the family of piecewise linear differential systems (2) being f±(x,y) and g±(x,y) polynomials of degree 1 defined in Σ± the following statements hold.

  • (i)

    If a piecewise linear differential system (2) has an algebraic crossing limit cycle, then every linear differential system either has no equilibrium

Explicit non-algebraic limit cycles of polynomial differential systems

Recently, since 2006 up to now, many articles have been showing explicit non-algebraic limit cycles in polynomial differential systems, i.e. in those articles the authors provided the explicit expression of the analytic curve containing the limit cycle. In this direction Gasull, Giacomini and Torregrosa [39] provided as far as we know the first explicit non-algebraic limit cycle for a polynomial differential system of degree 5. Clearly, if we multiply the right hand part of that quintic

Explicit non-algebraic limit cycles for the discontinuous piecewise linear differential systems

A discontinuous piecewise linear differential system with two pieces separated by a straight line in the plane R2 after a linear change of variables can be written into the form ẋẏ=a11a12a21a22xy+b1b2in x<0;and ẋẏ=a11+a12+a21+a22+xy+b1+b2+in x>0.That is, without loss of generality we can assume that the discontinuity straight line is x=0.

In 1930s Andronov, Vitt and Khaikin studied these discontinuous piecewise linear differential systems in their seminal book [3], where these

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The first author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Spain, Agencia Estatal de Investigación, Spain grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca, Spain grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by National Natural Science Foundation of China grants 11671254 and 11871334.

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