On the Number of Hyperelliptic Limit Cycles of Liénard Systems

Abstract

In this paper, we present a systematic study of the maximum number, denoted by H(m, n), of hyperelliptic limit cycles of the Liénard systems

$$\begin{aligned} \dot{x}=y, \quad \dot{y}=-f_m(x)y-g_n(x), \end{aligned}$$

where, respectively, $f_m(x)$ and $g_n(x)$ are real polynomials of degree m and n. The main results of the paper are as follows: We give the upper as well as the lower bounds of H(m, n) in all the cases. It turns out that in most cases these bounds are sharp. Furthermore, the configuration of hyperelliptic limit cycles is also explicitly described.

Appendix A

In this appendix, we shall introduce a useful theorem for determining the number of the distinct real roots of a polynomial. See for instance [9, 10] for more details.

We shall first give some notations. Recall that $D_k$ denoted the determinant of the submatrix of Discr(f), formed by the first 2k rows and the first 2k columns, for $k=1,\ldots ,n$. Then the n-tuple $(D_1,D_2,\ldots ,D_n)$ is called the discriminant sequence of polynomial f(x). The sign of $D_i$, with $1\le i\le n$, is what interests us. Then we give the following definitions.

Definition 1

(Sign list) We call the list

$$\begin{aligned} \left[ \hbox {sign}(D_1), \hbox {sign}(D_2), \ldots , \hbox {sign}(D_n)\right] \end{aligned}$$

the sign list of the discrimination sequence $(D_1,D_2,\ldots ,D_n)$.

Definition 2

(Revised sign list) Given a sign list $[s_1,s_2,\ldots ,s_n]$, we construct a new list $[\varepsilon _1,\varepsilon _2,\ldots ,\varepsilon _n]$ as follows:

• If $[s_i,s_{i+1},\ldots ,s_{i+j}] =[s_i,0,0,\ldots ,0, s_{i+j}]$, i.e.,

  $$\begin{aligned} s_i\ne 0,\quad s_{i+1}=\cdots =s_{i+j-1}=0, \quad s_{i+j}\ne 0, \end{aligned}$$

  (17)

  then we define a new list by keeping the two ends unchanged and replacing all these 0’s respectively by $-s_i,-s_i,s_i,s_i, -s_i,-s_i,\ldots , (-1)^{\left[ \frac{j}{2}\right] }s_i$, i.e.,

  $$\begin{aligned}{}[s_i,0,0,\ldots ,0,s_{i+j}] \rightarrow [s_i, -s_i,-s_i,s_i,s_i, -s_i,-s_i,\ldots , (-1)^{\left[ \frac{j}{2}\right] }s_i, s_{i+j}]. \end{aligned}$$

• If the subsection $[s_i,s_{i+1},\ldots ,s_{i+j}]$ doesn’t meet the relation (17), then there are no changes for it.

The determination of the number of the sign changes of the revised sign list is the prerequisites for the use of the theorem. Once we obtain the number of the distinct roots of f(x), and the number of the sign changes of the revised sign list, we can have the number of the distinct real roots of f(x).

Theorem A.1

Given a polynomial f(x) with real coefficients, if the number of the sign changes of the revised sign list of

$$\begin{aligned} \left[ D_1(f),D_2(f),\ldots ,D_n(f)\right] \end{aligned}$$

is v, then the number of the pairs of distinct conjugate imaginary roots of f(x) equals v. Furthermore, if the number of non-vanishing members of the revised sign list is l, then the number of the distinct real roots of f(x) equals $\, l-2v$.