Algebraic invariant curves for the Liénard equation

Abstract

Abstract. Odani has shown that if deg g ≤ deg f then after deleting some trivial cases the polynomial systemẋ = y,ẏ = −f (x)y − g(x) does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of deg f and deg g. We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system. The resultsThe subject of this work is the polynomial Liénard systeṁwhereare polynomials of degrees m and n respectively (a m b n = 0). Odani in [3] investigated algebraic phase curves of (1). He proved that if n ≤ m andthen the vector field (1) does not have any algebraic invariant curve.He also presented the example of Wilson [6] withand the algebraic limit cycleOdani conjectured that if m < n < 2m + 1 then (1) does not have algebraic limit cycles. Here we show that this conjecture is false, and we solve the problem of algebraic limit cycles and algebraic invariant curves for all pairs (m, n) almost completely.In what follows we consider systems of the form (1) with the restriction (2) and denote by A m,n the space of such systems,

On the hyperelliptic limit cycles of Liénard systems

“…We note that in Zoladek's paper [9], he stated that, for m > 1 and n > 1, and (m, n) = (2, 4), there exist Liénard systems of type (m, n) having hyperelliptic limit cycles. However, as shown in [3], it turns out that if (m, n) = (3, 5) then the Liénard systems (2) can not have hyperelliptic limit cycles, which shows that the arguments in the proof of [9] contain some gaps. In [3], Llibre and Zhang also proved that in one of the following cases there exist Liénard systems (2) of the type (m, n) which admit hyperelliptic limit cycles: Combining all the known results obtained up to now and theorem 1.1, we can in fact give a complete description of hyperelliptic limit cycles for any Liénard systems of type (m, n).…”

“…Does a Liénard system (2) with m > 5 and m + 1 < n < 2m have an algebraic limit cycle?In this paper, by developing the main ideas in[3,8,9] we prove the following result, which gives a positive answer to the above open problem.…”

“…In this section, we shall collect some related known results. For the detailed proof of these results, we refer the reader to [3,8,9]. However, for the convenience of the reader, below we give an outline of some proofs.…”

“…Therefore, in this case, it is not possible to have algebraic limit cycles, let alone the hyperelliptic ones. Chavarriga et al [1] and Zoladek [9] also investigated this problem and proved that any Liénard systems of type (0, n), or (1, n), or (2, 4), or (m, m+1) have no algebraic limit cycles.…”

Phase portraits of the quadratic polynomial Liénard differential systems

“…The polynomial Liénard differential systems and their applications also have been analysed by many authors these recent years. Thus some authors studied their limit cycles (see for instance [14,15,17,21,26,27,39,41]), or their algebraic limit cycles (see [28,31,37]), or their invariant algebraic curves (see [7,8,49]), or their canard limit cycles (see [43]), or the shape of their limit cycles (see [46]), or the period function of their centres (see [47]), or their integrability (see [9,30]), or a kind of a generalized Liénard system (see [18]).…”