# On the Uniqueness of Limit Cycles in a Generalized Liénard System

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## Abstract

Kooij and Sun (J Math Anal Appl 208:260–276, 1997) proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Liénard system \(dx/dt=h(y)-F(x),\ dy/dt=-g(x)\). We will give a counterexample to their theorem. Moreover, we shall give some sufficient conditions for the existence, uniqueness and hyperbolicity of limit cycles.

## Keywords

Generalized Liénard systems Limit cycle Uniqueness Hyperbolicity## 1 Introduction

### Theorem 1.1

- (i)
\(h(0)=0\),

*h*(*y*) is strictly increasing, and \(h(\pm \infty )=\pm \infty \); - (ii)
\(xg(x)>0\) for \(x\ne 0\) and \(G(\pm \infty )=\infty \);

- (iii)
there exist constants \(x_1\), \(x_2\) with \(x_1<0<x_2\) such that \(F(x_1)=F(0)=F(x_2)=0\) and \(xF(x)<0\) for \(x \in (x_1, x_2)\backslash \{0\}\);

- (iv)
there exist constants \(M>0\),

*K*, \(K_0\) with \(K>K_0\), such that \(F(x)>K\) for \(x \ge M\) and \(F(x)<K_0\) for \(x \le -M\); - (v)one of the following holds:
- (a)
\(G(x_1)=G(x_2)\), or

- (b)\(G(-x) \ge G(x)\) for \(x>0\). Furthermore, let \(W(x):=\int _0^{h^{-1}(F(x))}h(y)dy\), where \(h^{-1}\) is the inverse function of
*h*. Then- (\(\alpha \))
if \(x_2 \le |x_1|\) then \(max_{0 \le x \le x_2} [{G(x)+W(x)}]\ge G(x_1)\),

- (\(\beta \))
if \(0<|x_1|<x_2 \) then \(max_{x_1 \le x \le 0} [{G(x)+W(x)}]\ge G(x_2)\).

- (a)

### Theorem 1.2

(Kooij and Sun [9, Theorem 2.1]) If conditions \((i)-(v)\) of Theorem 1.1 and (2) hold. Then system (1) has exactly one closed orbit, a hyperbolic stable limit cycle.

We shall give an example such that the conditions of Theorem 1.2 are satisfied, but there are at least two limit cycles. Our investigation shows that the conditions of Theorem 1.2 cannot ensure that all closed orbits of system (1) have to intersect both \(x=x_1\) and \(x=x_2\). In fact, we will give an example to show that under the conditions of Theorem 1.2 there may be at least two limit cycles which intersect \(x=x_2\) but do not intersect \(x=x_1\). Therefore, Theorem 1.2 is incorrect. Moreover, we will give some sufficient conditions for the existence, uniqueness and hyperbolicity of limit cycles of system (1).

The idea of the proof of the uniqueness of limit cycles for the classical Liénard system (i.e. system (1) with \(h(y)\equiv y\)), via a comparison of integral curves, appears already in the paper by Liénard [12], and other references in this direction [1, 5, 10, 11, 13, 14, 15, 16]. By utilizing the traditional comparison method, we obtain that system (1) with \(h(y)\equiv y\) has exactly one nontrivial periodic solution which is orbitally stable, however, we cannot show that the limit cycle of system (1) with \(h(y)\equiv y\) is hyperbolic (i.e. exponentially asymptotically stable).

The Proof of Theorem 3.1 for system (1) with \(h(y)\equiv y\) appears for the first time in [4], but the problem is also treated in [2] and generalized in [6] and [15]. The monotonicity assumption on *F*(*x*) is relaxed in [16]. In this paper, we estimate the divergence of corresponding system integrated along a limit cycle and apply suitable transformations ( see, for example, [3, 17, 19, 20]). By this we can show that the limit cycle of system (1) is unique, hyperbolic and stable.

## 2 A Counterexample to Theorem 1.2

In this section we give a counterexample such that the conditions of Theorem 1.2 are satisfied, but there are at least two limit cycles which intersect \(x=x_2\) but do not intersect \(x=x_1\).

### Example 1

- (1)
*g*(*x*) is continuous on \(\mathbf R\), \(xg(x)>0\) for \(x\ne 0\) and \(G(\pm \infty )=\infty \); - (2)
*F*(*x*) is continuously differential on \(\mathbf R\), \(F(0)=0\).

*H*(

*u*) such that (4) has at least two limit cycles. Then construct the function

*g*(

*x*) such that it satisfies the conditions of Theorem 1.2, and after the transformation \(u=\sqrt{2G(x)}\mathrm{sgn}x\), the function

As indicated in Fig. 1, let \(\widehat{POEDG}\) be part of the graph for \(y=H(u)\). \(D=(1,0)\). On arc \(\widehat{OED}\), we have \(H(u)\le 0\) and \(H''(u)>0\). On line segments \(\overline{PO}\), \(\overline{DG}\), we have \(H(u)\ge 0\). On \(\overline{PO}\), \(H'(u)=c_2\), and on \(\overline{DG}\), \(H'(u)=c_1\), with \(-\frac{6}{5}c_2>c_1>-c_2\), \(-c_2<1\). The function *H*(*u*) is continuously differentiable if \(u_P\le u\le u_G\).

- (1)
there exists \(\delta >0\) such that \( H_1(z)\le H_2(z)\) for \(0<z<\delta \), \((H_1(z)\not \equiv H_2(z))\), \(H_1(z)<a\sqrt{z}\), \(H_2(z)>-a\sqrt{z}\ (a<\sqrt{8})\);

- (2)
there exists \(z_0>\delta \) such that \(\int _0^{z_0}(H_1(z)-H_2(z))dz>0\); when \(z \ge z_0\), \(H_1(z) \ge H_2(z)\), \(H_1(z)>-a\sqrt{z}\), \(H_2(z)=H(-\sqrt{2z})=-c_2\sqrt{2z}<\sqrt{2z}<a\sqrt{z}\ (a<\sqrt{8}\)).

*G*. In order that system (4) has at least two limit cycles, we continue to construct the curve \(\widehat{GMN}\) as part of the graph of \(y=H(u)\) where \(H'(u)>0\). Moreover, \(H''(u)=\lambda <0\) on \(\widehat{GM}\), \(H'(u)=c_3>0\) on \(\overline{MN}\), where \(\frac{3}{2}c_3>c_1>-c_2>c_3\). For such curve \(y=H(u)\), there exists \(z_1>z_0\) such that the above condition (2) is satisfied with the role of \(H_1(z)\) and \(H_2(z)\) interchanged. That is, there exists \(z_1>z_0\) such that \(\int _0^{z_1}(H_2(z)-H_1(z))dz>0\); when \(z \ge z_1\), \(H_2(z) \ge H_1(z)\), \(H_2(z)>-a\sqrt{z}\), \(H_1(z)=H(\sqrt{2z})=H(u_M)+c_3\sqrt{2z}<H(u_M)+\sqrt{2z}<a\sqrt{z}\) (here \(z_1\) is sufficiently large, \(a<\sqrt{8}\)).

Using Theorem 1.3 in [21] again, we can construct an outer boundaries \(l_3\supset l_2\) such that any orbit starting at \(l_2\), \(l_3\) will enter the annular region bounded between \(l_3\) and \(l_2\) as time decreases. Thus there must exist at least one limit cycle \(L_2 \supset L_1\) in the annular region between \(l_2\) and \(l_3\).

The construction for \(y=H(u)\) is nearly complete. We continue to extend its graph to both the left and right such that \(H'(u)=c_3\) if \(u>u_N\), \(H''(u)<0\) if \(u<u_P\); \(u_Q\le -2\), \( H'(u)\ge 1\) for \(u\le u_Q\), \(H(u)\in C^1(-\infty , \infty )\). In this way, Eq. (4) has at least two limit cycles.

*g*(

*x*) as follows:

It is clear that the conditions (*i*), (*ii*), (*iv*) and (2) are satisfied. Since \(G(x)=kx^2/2\) for \(x\le 0\), \(G(x)=x^2/2\) for \(x>0\), \(u_Q=-c_0\le -2\), \(u_D=1\), by the transformation \(u=\sqrt{2G(x)}\mathrm{sgn}x\), we have \(x_1=-\frac{1}{2}\), \(x_2=1\), it follows that \(0<|x_1|<x_2\), \(G(x_1)=\frac{c_0}{2}>\frac{1}{2}=G(x_2)\), and \(G(-x)>G(x)\) for \(x>0\). Thus, the conditions (iii) and (\(\beta \)) in (v) are also satisfied. This concludes the construction of the counterexample.

## 3 Existence, Uniqueness and Hyperbolicity of Limit Cycles

In this section we give some sufficient conditions for the existence, uniqueness and hyperbolicity of limit cycles of system (1).

### Theorem 3.1

*F*(

*x*) and

*h*(

*y*) are continuously differentiable on \(\mathbf R\), and condition \((v^*)\) holds if one of the following conditions

- (i)
\(G(x_1)=G(x_2)\);

- (ii)
if \(G(x_1)<G(x_2)\) then \(max_{x_1 \le x \le 0}[{G(x)+W(x)}]\ge G(x_2)\);

- (iii)
if \(G(x_1)>G(x_2)\) then \(max_{0 \le x \le x_2}[{G(x)+W(x)}]\ge G(x_1)\).

This theorem will be proved by showing that if \(\gamma \) is a closed orbit then its characteristic exponent \(\int _{\gamma }-f(x)dt<0\), where \(f(x)=(d/dx)F(x)\). This shows that \(\gamma \) is hyperbolic and stable (see, for example, [3, 17, 19, 20]).

Because two adjacent limit cycles cannot both be stable, the uniqueness of \(\gamma \) follows. In order to estimate the characteristic exponent we need the following lemma by Zeng [19].

### Lemma 3.1

*y*(

*x*), \(\alpha \le x \le \beta \). Then

### Proof of Theorem 3.1

*E*(

*x*,

*y*) with respect to system (1),

*B*,

*C*and

*D*be the points at which \(\gamma \) intersects the line \(x=x_2\), the negative y-axis and the negative x-axis, respectively as time

*t*increases, where \(y_B<0\), \(y_C<0\) and \(x_D<0\). Let

*P*be a point on the arc \(\widehat{BC}\) of \(\gamma \). Then the coordinates \((x_P,y_P)\) of

*P*satisfy \(0\le x_P\le x_2\), \(y_P<0\). Hence, \(h(y_P)<F(x_P)<0\). \(\square \)

Next we prove that the point \((x_1,0)\) is in the interior of \(\gamma \). Suppose it is not the case, then \(x_1\le x_D<0\).

*A*. Let

*B*and

*C*be the intersections of \(\gamma \) with \(x=x_2\) in the first and fourth quadrant( see Fig. 2), respectively. If we denote the arc of \(\gamma \) between

*A*and

*B*by \(\gamma _1\), then applying Lemma 3.1 with \(\alpha =0\) and \(\beta =x_2\) yields

*B*and

*C*, we obtain by condition (2) and \(f(x)=(d/dx)F(x)\)

## Notes

### Acknowledgements

Open access funding provided by University of Helsinki including Helsinki University Central Hospital. We would like to thank the referees for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper. Zhang Daoxiang is grateful to the National Natural Science Foundation of China (11671013). Ping Yan is grateful to the National Natural Science Foundation of China (41730638).

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