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J. Math. Anal. Appl. 343 (2008) 299–309

www.elsevier.com/locate/jmaa

On the existence and uniqueness of limit cycles for generalized Liénard systems

Dongmei Xiao a,∗,1, Zhifen Zhang b,2

a Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China b Department of Mathematics, Peking University, Beijing 100871, China Received 29 June 2007 Available online 29 January 2008 Submitted by J. Mawhin

Abstract

In this paper, we consider a generalized Liénard system dx = φ(y) − F (x), dt dy = −g(x), (0.1) dt

where F is continuous and differentiable on an open interval (b1, a1) with −∞ b1 < 0 < a1 +∞. Assume that there exist a and b with b1 < b < 0 < a < a1 such that xF (x) < 0 as b < x < a, and xF (x) > 0 as a < x < a1 or b1 < x < b. A new uniqueness theorem of limit cycles for the Liénard system (0.1) is obtained. An example is given to show the application of the theorem. © 2008 Elsevier Inc. All rights reserved.

Keywords: Liénard system; Limit cycles; Uniqueness

1. Introduction

The existence and number of limit cycles for planar systems are related to Hilbert 16th problem and self-sustaining oscillatory problems in mathematical models. It is a challenging problem to ﬁnd out conditions so as to guarantee the uniqueness of limit cycles for planar systems. As one knows, there are some systems for which the uniqueness of limit cycles has been extensively studied in the last century, such as Liénard system with the following form:

dx = y − F (x), dt

* Corresponding author. E-mail addresses: [email protected] (D. Xiao), [email protected] (Z. Zhang).

1 Supported by the National Natural Science Foundations of China (No. 10771136). 2 Supported by the National Natural Science Foundations of China (No. 10571002).

0022-247X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2008.01.059

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dy = −x, (1.1) dt as well as the more general form

dx = φ(y) − F (x), dt dy = −g(x), (1.2) dt where the function g(x) is continuous and F (x) is continuously differentiable on an open interval (b1, a1), −∞ b1 < 0 < a1 +∞. The function φ(y) is continuously differentiable on (−∞, +∞). Moreover, it is always assumed that

(A1) xg(x) > 0 for x = 0. Let G(x) = 0x g(s) ds; (A2) φ(0) = 0, φ (y) > 0 for −∞ < y < +∞. Let Φ(y) = 0y φ(s) ds.

There have been various techniques for establishing the uniqueness of limit cycles for (1.1) and (1.2) (see [1,2, 8–17] and references therein). In these sufﬁcient conditions, the following conditions are usually assumed.

(H1) There exist a and b, b1 < b < 0 < a < a1 such that F (b) = F (0) = F (a) = 0. (H2) xF (x) > 0 if a < x < a1 and b1 < x < b,

xF (x) < 0 if b < x < a.

Sansone and Conti in [10] obtained that if the hypotheses (H1) and (H2) hold, and if the derivative of F (x) with respect to x satisﬁes F (x) > 0 as a < x < a1 and b1 < x < b, then for system (1.1) the number of limit cycles, which enclose two points (a, 0) and (b, 0) inside is unique. Recently, Carletti and Villari in [1] further gave the sufﬁcient conditions for uniqueness of limit cycle of (1.1). For system (1.2), in [11] we gave the conditions to guarantee that a limit cycle encloses two points (a, 0) and (b, 0) inside, which generalized the conclusion of Sansone and Conti in [10]. It is an interesting problem what conditions should add to guarantee the uniqueness of limit cycles for (1.2) if the limit cycle encloses only one point of (a, 0) and (b, 0). Zeng et al. in [12] and [13] studied this problem and used the Filippov transformation z = G(x) to reduce system (1.2) to two equations

dz

= Fi(z) − φ(y),

(Ei )

dy

respectively, where Fi(z) = F (xi(z)) ∈ C1[0, zi), xi(z) is the inverse function of z = G(x) for (−1)i+1x 0, i = 1, 2, z1 = G(a1) and z2 = G(b1). In [13] (cf. [13, Theorem 1]), it has been proved that system (1.2) has at most one limit cycle if besides (H1) and (H2), the following hypotheses hold

(H3) if G(b) > G(a) (G(b) < G(a)), then F1(z)F1(z) (F2(z)F2(z), respectively) is nondecreasing (nonincreasing, respectively) as z > G(a) (z > G(b), respectively);

(H4) if G(b) > G(a) (G(b) < G(a)), then F1(z) F2(u) for any pair (z, u) satisfying G(a) < z < u (G(b) < u < z, respectively) and F1(z) = F2(u) (F2(u) = F1(z), respectively).

From the above sufﬁcient conditions in [1,10,12] and [13], we can see that if limit cycles enclose both points (a, 0) and (b, 0), then the monotonicity of F (x) on a1 > x > a and b > x > b1 is sufﬁcient to guarantee the uniqueness of the limit cycles. However, if limit cycles enclose only one of the two points (a, 0) and (b, 0), then to guarantee the uniqueness of the limit cycles, it needs more restrictions on the functions F (x) and G(x), such as assumptions (H3) and (H4).

In this paper, we discuss further the uniqueness of limit cycles of (1.2) when the limit cycles enclose only one of two points (a, 0) and (b, 0). We obtain a new sufﬁcient criterion for the uniqueness of limit cycles of (1.2), which is different from those sufﬁcient conditions appeared in [2,9–15] and [16]. For simplicity of statement, we make the following assumption on the derivative of F (x).

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Fig. 1. When G(b) > G(a), the vertical isocline and a closed orbit of system (Ei ).

Fig. 2. When G(b) < G(a), the vertical isocline and a closed orbit of system (Ei ).

(H5) When b1 < x < a1, f (x) d=ef F (x) = 0 has only two roots x1 and x2, x1 > 0 and x2 < 0. Moreover, z is a unique positive root of F1(z) − F2(z) = 0, which implies that G(xi) z .

Now we state a new uniqueness theorem as follows.

Theorem 1.1. Suppose that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1), (H2), (H5), and its equivalent Eqs. (Ei) satisfy one of the following conditions.

(i) If G(b) > G(a) (which implies G(a) < z < G(b)), then F2(u) > F1(z) for any pair (z, u) satisfying G(a) < z < z , 0 < u < z and F1(z) = F2(u) (see Fig. 1).

(ii) If G(a) > G(b) (which implies G(b) < z < G(a)), then F2(z) > F1(u) for any pair (z, u) satisfying G(b) < z < z , 0 < u < z and F1(u) = F2(z) (see Fig. 2).

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Then

(R1) system (1.2) has at most one limit cycle, and it is stable if it exists; (R2) system (1.2) has a unique stable limit cycle if G(b1) = G(a1) = +∞ and Φ(±∞) = ±∞.

2. Main results

In this section, we ﬁrst give the proof of Theorem 1.1, then compare it with Theorem 1 in [13] and provide an

alternative statement of Theorem 1.1.

We now prove the conclusion (R1) of Theorem 1.1. Our main task is to prove the uniqueness of limit cycles of

(1.2) if it exists, which can be derived as follows: if any one closed orbit of system (1.2) is stable, then system (1.2)

has at most one limit cycle since adjacent closed orbits can not have the same stability. Hence, we will only discuss

the stability of closed orbits of (1.2). Assume that L is a closed orbit of (1.2), then L = L1 ∪ L2, Li is an orbit of Eqs. (Ei ), i = 1, 2. And L1 (L2)

intersects the vertical isocline φ(y) = F1(z) (φ(y) = F2(z), respectively) at A(zA, yA) (B(zB , yB ), respectively). It is easy to see that yA > yB . Let y¯i(z) and yi(z) represent a part of Li , which is located above and below of the vertical isocline φ(y) = Fi(z), respectively (see Fig. 1 or Fig. 2). To determine the stability of L, we have to calculate the integral of divergent of (1.2) along L,

div(1.2) dt = −F (x) dt.

L

L

Utilizing the notations in [13], we denote

V Fi (z), yi (z), y¯i (z) d=ef

Fi (z)

+

Fi (z) .

Fi(z) − φ(yi(z)) φ(y¯i(z)) − Fi(z)

Hence, we have

−F (x) dt = − F1(z) dy − F2(z) dy

L

L1

L2

zA

zB

= − V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz ,

(2.1)

0

0

where dt > 0 along the integration path and dy > 0 along the curves L1 and L2. To estimate the sign of the integral (2.1), we ﬁrst introduce two lemmas. They can be found in [12] and [13]. For

convenience to read, we give the proof of Lemma 2.2

Lemma 2.1. Let 0 c < d zA (or zB ). If Fi(d) − Fi(z) y¯i(z)) dz 0 (or 0, respectively).

0 (or

0) for c < z < d, then

d c

V

(Fi

(z),

yi

(z),

Lemma 2.2. Consider systems (E1) and (E2), if there exist 0 p1 < u < z1, 0 p2 < z < z2 such that

(1) F1(p1) = F2(p2), F1(z1) = F2(z2); (2) F1(u) < 0 if p1 < u < z1, and F2(z) < 0 if p2 < z < z2; (3) F1(u) < F2(z) if F1(u) = F2(z).

Let y¯i(·) and yi(·) be solutions of (Ei), i = 1, 2, which satisfy the following inequalities: φ y1(u) < F1(u) < φ y¯1(u) , as p1 < u < z1; φ y2(z) < F2(z) < φ y¯2(z) , as p2 < z < z2; y¯1(p1) y¯2(p2) and y1(p1) y2(p2).

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Then there exist two functions W¯ 2(z) and W2(z) in the interval (p2, z2) such that

W¯ 2(z) = y¯1 F1−1 F2(z) , W2(z) = y1 F1−1 F2(z) ,

and

z1

z2

V F1(u), y1(u), y¯1(u) du > V F2(z), y2(z), y¯2(z) dz.

p1

p2

Proof. When p1 < u < z1 and p2 < z < z2, we let F1(u) = F2(z), which deﬁne a transformation from variables u into z. Consider Eq. (E1) with the following form:

dy = 1 . (2.2) du F1(u) − φ(y) Then (2.2) can be transformed to

dy = 1

F2(z) ,

(E3)

dz F2(z) − φ(y) F1(F1−1(F2(z)))

and solutions y¯1(u) and y1(u) of (2.2) are transformed to

W¯ 2(z) = y¯1 F1−1 F2(z) and W2(z) = y1 F1−1 F2(z) , respectively, where W¯ 2(z) and W2(z) are solutions of Eq. (E3).

It is clear that W¯ 2(p2) = y¯1(p1) and W2(p2) = y1(p1). From y¯1(p1)

y¯2(p2) and y1(p1)

y2(p2), we have

W2(p2) y2(p2), W¯ 2(p2) y¯2(p2).

(2.3)

Next we prove the inequality of integral. Since

F2(z)

−1

< 1,

F1(F1 (F2(z)))

1

1

<

F2(z) ,

F2(z) − φ(y¯2(z)) F2(z) − φ(y¯2(z)) F1(F1−1(F2(z)))

1

1

>

F2(z) .

F2(z) − φ(y2(z)) F2(z) − φ(y2(z)) F1(F1−1(F2(z)))

Comparing Eqs. (E2) and (E3), and note the initial conditions (2.3), we obtain that

W¯ 2(z) y¯2(z) and W2(z) y2(z)

as z p2 by differential inequality theory. Note that φ (y) > 0. Hence,

φ W¯ 2(z) φ y¯2(z) , φ W2(z) φ y2(z) ,

and

F2(z)

+

F2(z)

>

F2(z)

+

F2(z)

.

F2(z) − φ(W2(z)) φ(W¯ 2(z)) − F2(z) F2(z) − φ(y2(z)) φ(y¯2(z)) − F2(z)

Therefore, we have

z1

z2

V F1(u), y1(u), y¯1(u) du − V F2(z), y2(z), y¯2(z) dz

p1

p2

z2

z2

= V F2(z), W2(z), W¯ 2(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

p2

p2

The proof is completed. 2

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Now we are in a position to prove the uniqueness of limit cycles if it exists. We ﬁrst prove the case (ii) and divide

two cases of yA to discuss. (I) If yA 0, then F1(zA) − F1(z) 0 as 0 < z < zA and F2(zB ) − F2(z) 0 as 0 < z < zB . From Lemma 2.1,

we have

zA

zB

V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

0

0

Hence, the integral of divergent of (1.2) along L is negative, which leads that the closed orbit L is stable. Therefore,

system (1.2) has at most one limit cycle. (II) If yA < 0, then we can take points D(zD, yD) and C(zC, yC) in φ(y) = F1(z) and φ(y) = F2(z), respectively,

such that F1(zD) = F2(zC ) = F1(zA). Note that F1(zA) − F1(z) 0 as 0 < zD < z < zA, F2(G(b)) − F2(z) 0 as 0 < z < G(b) and F2(zB ) − F2(z) 0 as 0 < zC < z < zB . From Lemma 2.1, we have

zA

V F1(z), y1(z), y¯1(z) dz 0,

zD zB

V F2(z), y2(z), y¯2(z) dz 0,

zC

G(b)

V F2(z), y2(z), y¯2(z) dz 0.

(2.4)

0

On the other hand, F1(0) = F2(G(b)) = 0 and F1(zD) = F2(zC). From hypothesis (H5) and condition (ii) of Theorem 1.1, and utilizing Lemma 2.2 we obtain

zD

zC

V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

0

G(b)

(2.5)

Summarizing (2.4) and (2.5), we obtain that

zA

zB

V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

0

0

Hence, the integral of divergent of (1.2) along L is negative, which implies that the closed orbit L is stable. Thus system (1.2) has at most one limit cycle. We complete the proof of conclusion (R1) in Theorem 1.1 for case (ii).

Let y = −Y . Then system (Ei) is transformed to

dz

= −Fi(z) + φ(−Y ),

(Fi )

dY

and the case (i) in Theorem 1.1 is transformed to the case (ii) for system (Fi). The similar arguments can be applied to the case. We can obtain the uniqueness of limit cycles in Theorem 1.1 for case (i). Therefore, we ﬁnish the proof of conclusion (R1) of Theorem 1.1.

We next prove conclusion (R2) of Theorem 1.1. From conclusion (R1), we only need to prove the existence of limit cycles for system (1.2) if we further assume that G(b1) = G(a1) = +∞ and Φ(±∞) = ±∞. The following lemma implies the existence of limit cycles for system (1.2).

Lemma 2.3. Assume that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1), (H2) and (H5). If G(b1) = G(a1) = +∞ and Φ(±∞) = ±∞, then system (1.2) has at least a limit cycle.

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305

Fig. 3. The annular region of positive invariant set of system (1.2) and the vertical isocline φ(y) − F (x) = 0.

Proof. Let

V (x, y) = G(x) + Φ(y), where G(x) = 0x g(s) ds and Φ(y) = 0y φ(s) ds. Then

dV (x, y) = −g(x)F (x). dt (1.2)

Taking 0 < r1 min{a, −b}, we have

dV (x, y) > 0,

dt (1.2)

as V (x, y) = r1.

Thus, the graph of V (x, y) = r1 becomes an intra-boundary of an annular region and the unique equilibrium (0, 0) of

system (1.2) is unstable.

On the other hand, since Φ(±∞) = ±∞, respectively, and G(b1) = G(a1) = +∞, any trajectory of system (1.2) in the region {(x, y): V (x, y) > r1} spirals as t → +∞.

From (H2) and (H5), there exists 0, 0 < 0 1 such that F (a + 0) = k1 > 0, F (b − 0) = k2 < 0, and F (x) k1 as a + 0 x < a1 and F (x) k2 as b1 < x b − 0. Let

l1: x = a + 0, l2: x = b − 0.

Taking a point P (a + 0, y0) on line l1 with y0 < 0, we consider that as time t increases the trajectory Γ (P , t) of system (1.2) with the initial point P intersects l2 at Q(b − 0, yQ) and R(b − 0, yR), respectively, and intersects l1 again at S(a + 0, yS) and T (a + 0, yT ), respectively (see Fig. 3). We claim that y0 < yT < 0 as y0 → −∞. Let

y

x

V1(x, y) = φ(s) − k2 ds + g(s) ds.

0

0

We consider the increment of the function V1(x, y) along the segment of trajectory P QRST of system (1.2)

dV1 = F (x) − k2, dy (1.2)

dV1 = −g(x)(F (x) − k2) .

dx (1.2)

φ(y) − F (x)

(2.6) (2.7)

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Note that F (x) k1 as a + 0 x < a1 and F (x) k2 as b1 < x b − 0. From (2.6), we have

yS

V1(S) − V1(T ) = F (x) − k2 dy > (k1 − k2)(yS − yT ) > 0,

yT

yQ

yR

V1(Q) − V1(R) = F (x) − k2 dy = k2 − F (x) dy > 0.

yR

yQ

Since the functions g(x) and F (x) are bounded for x ∈ (b − 0, a + 0), for arbitrary small > 0 there exists a sufﬁciently large M > 0, when −y0 > M,

dV1 < dx (1.2)

by (2.7). Hence,

V1(P ) − V1(Q) < (a − b + 2 0), Note that

V1(R) − V1(S) < (a − b + 2 0).

V1(P ) − V1(T ) = V1(P ) − V1(Q) + V1(Q) − V1(R) + V1(R) − V1(S) + V1(S) − V1(T ).

Thus,

1 V1(P ) − V1(T ) > (k1 − k2)(yS − yT ) > 0,

2 which implies that y0 < yT < 0 as −y0 is sufﬁciently large.

Therefore, the closed curve by the segment of trajectory P QRST of system (1.2) and the segment T P on line l1 becomes an outer-boundary of an annular region (see Fig. 3). The annular region is a bounded positive invariant set for system (1.2), which does not include any equilibria of system (1.2). Therefore, the existence of limit cycles follows directly from the Poincaré–Bendixson theorem. We ﬁnished the proof of the existence of limit cycle of system (1.2). 2

Remark 2.1. Let us compare Theorem 1 in [13] with Theorem 1.1 here in case G(b) > G(a). Using the notations in this paper, Theorem 1 in [13] can be stated as follows:

Theorem 1. (See [13].) Suppose that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1)–(H4), then system (1.2) has at most a limit cycle, and it is stable if it exists.

The differences between Theorem 1 in [13] and Theorem 1.1 here are as follows.

(1) When z > G(a), in Theorem 1, hypothesis (H3) required, that is, F1(z)F1(z) is nondecreasing. However, in Theorem 1.1, hypothesis (H5) is required, that is, F1(z) is nondecreasing for z > G(a).

(2) Hypothesis (H4) in Theorem 1 requires that F1(z) F2(u) if F1(z) = F2(u) for G(a) < z < u. That implies that they need to compare the slopes of F1(z) and F2(u) outside the interval (0, G(a)); but condition (i) in Theorem 1.1 here requires that F1(z) F2(u) if F1(z) = F2(u) for 0 < u < z and G(a) < z < z . That implies that we need to compare the slopes of F1(z) and F2(u) inside the interval (0, z ). Hence, roughly speaking, our uniqueness

conditions of limit cycles are different from that in [13], and these conditions on the interval (0, G(b)) in case (i)

and on interval (0, G(a)) in case (ii) have strongly geometric intuition.

In general case, the condition (i) or (ii) in Theorem 1.1 is not easy to be veriﬁed since the inverse function of G(x) is hardly expressed explicitly for some continuous functions g(x). Hence, in order to apply Theorem 1.1 more conveniently, one hopes to represent conditions in this theorem directly by means of the original functions of system (1.2). We provide an alternative statement of conclusion (R1) of Theorem 1.1 as follows.

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Corollary 2.4. Assume that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1), (H2) and (H5).

(i) If ba g(x) dx = 0, then system (1.2) has at most one limit cycle (cf. [11, Theorem 2.2]). (ii) If ba g(x) dx = 0, then there exists only a pair of a0 and b0, b1 < b0 < 0 < a0 < a1 such that ba00 g(x) dx = 0

and F (b0) = F (a0). System (1.2) has at most one limit cycle if one of the following conditions hold. (ii.1) If ba g(x) dx < 0 (i.e. G(b) > G(a)), then fg((xx22)) > fg((xx11)) for any pair of (x1, x2) satisfying a < x1 < a0,

b0 < x2 < 0 and F (x1) = F (x2). (ii.2) If ba g(x) dx > 0 (i.e. G(a) > G(b)), then fg((xx22)) > fg((xx11)) for any pair of (x1, x2) satisfying 0 < x1 < a0,

b0 < x2 < b and F (x1) = F (x2).

3. Example and discussion

The existence and uniqueness of limit cycle is one of the most delicate problem in studying of mathematical models. There are various techniques to establish the uniqueness of limit cycles of ecological systems (cf. [3,6,7,11] and references therein). It is the main technique to transform an ecological system into a generalized Liénard system. For example, a general predator–prey system in [11] can be transferred into the generalized Liénard system (1.2) in suitable region of phase plane. And the geometric shape of prey isocline of predator–prey systems can be kept in that of F (x) = φ(y) of the generalized Liénard system (1.2). Some researchers think that the geometric property of prey isocline plays important role in understanding both the globally asymptotically stable of a unique equilibrium and the uniqueness of limit cycles for predator–prey systems. Hwang in [4] showed that local asymptotic stability of a unique positive equilibrium together with the existence of a concave down prey isocline implied that this equilibrium was globally asymptotically stable for certain predator–prey systems. Kuang in [5] pointed out that Freeman conjectured that the existence of a unique unstable equilibrium together with existence of humps of prey isocline implied the existence and uniqueness of the limit cycle for predator–prey systems. The role of our main results (Theorem 1.1 or Corollary 2.4) is to give information about the geometry of F (x) = φ(y) to ensure existence and uniqueness of limit cycles.

We now give an example to show the application of Corollary 2.4. In [7], Ruan and Xiao studied global dynamics of a predator–prey system with nonmonotonic functional response

x˙ = rx 1 − x − xy , K c + x2

y˙ = y μx − D (3.1) c + x2

in the closed ﬁrst quadrant of R2, where r, c, μ, K and D are positive parameters. They gave the conditions that system (3.1) has a unique limit cycle (cf. [7, Theorems 2.4–2.5]). However, there is a gap in the proof of Theorem 2.5. We will point out it in the following discussion of the uniqueness of limit cycle for system (3.1).

It is clear that system (3.1) has a unique positive equilibrium (x1, y1) if

μ2 > 16 cD2, x2 > K > x3,

(3.2)

3

where

μ − μ2 − 4cD2

x1 =

,

2D

y1 = r 1 − x1 K

c + x12 ,

μ + μ2 − 4cD2

x2 =

,

2D

2μ − μ2 − 4cD2

x3 =

.

2D

The nontrivial periodic orbits of system (3.1) must be in the domain E1 if it exists, here

E1 = (x, y): 0 < x < K, 0 < y < +∞ .

To study the uniqueness of periodic orbits of system (3.1), we transfer system (3.1) in E1 into a Liénard system. Let x − x1 = −X, y − y1 = y1(eY − 1) and x dt = (c + x2) dT . Then system (3.1) can be written as

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dX = φ(Y ) − F (X), dT dY = −g(X), (3.3) dT

where φ(Y ) = y1(eY − 1), F (X) = rKX (X2 + (K − 3x1)X + c + 3x12 − 2Kx1), and g(X) = DX(Xx1−−xX1+x2) , here x1 − K < X < x1 and −∞ < Y < +∞.

It is easy to check the following facts:

(a1) Xg(X) > 0 for X ∈ (x1 − K

0) ∪ (0

x1), G(x1) =

x1 0

g(s) ds

=

+∞

and

G(x1

−

K)

=

D2 ((K − x1)(x1 + 2x2 − K) − 2x1x2 ln xK1 ) > 0.

(a2) φ(0) = 0, φ (Y ) > 0 for −∞ < Y < +∞, and φ(−∞) = −y1 and φ(+∞) = +∞.

0x1−K g(s) ds =

If inequalities (3.2) hold, then we have

K − 3x1 > 0, c + 3x12 − 2Kx1 = x1(x2 + 3x1 − 2K) < 0. Let = (K − 3x1)2 − 4(c + 3x12 − 2Kx1) = K2 + 2x1K − 3x12 − 4c. Hence, there exist a and b, 0 < a < x1 and K − x1 < b < 0 such that

F (b) = F (0) = F (a) = 0,

where √

a = −(K − 3x1) + , 2

√ b = −(K − 3x1) − .

2

f (X) d=ef F (X) = 0 has only two roots in the interval (x1 − K x1), which implies that the prey isocline φ(y) = F (x) of system (3.1) has only two humps in the range

E2 = (x, y): x1 − K < x < x1, −∞ < y < +∞ ,

namely a local maximum and a local minimum. Therefore, F (X) satisﬁes hypotheses (H1), (H2) and (H5). In the proof of Theorem 2.5 in [7], authors only proved the uniqueness of limit cycles of system (3.1) if the limit

cycles enclose two points (a, 0) and (b, 0) inside, and did not prove the uniqueness of limit cycles if the limit cycles enclose only one point of (a, 0) and (b, 0) inside.

To apply Corollary 2.4 for system (3.1), we can modify Theorem 2.5 in [7] to the following conclusion.

Theorem 3.1. Suppose that μ2 > 136 aD2 and x2 > K > x3. Then system (3.1) has at most one limit cycle in the interior of the ﬁrst quadrant if one of the following conditions holds.

(i) ba g(X) dX = 0. (ii) If ba g(X) dX < 0 (i.e. G(b) > G(a)), then fg((XX22)) > fg((XX11)) for any pair (X1, X2) satisfying a < X1 < a0, b0 <

X2 < 0 and F (X1) = F (X2), where a0 and b0 with x1 − K < b0 < 0 < a0 < x1 are solutions of ba00 g(X) dX = 0 and F (b0) = F (a0). (iii) If ba g(X) dX > 0 (i.e. G(a) > G(b)), then fg((XX22)) > fg((XX11)) for any pair (X1, X2) satisfying 0 < X1 < a0, b0 < X2 < b and F (X1) = F (X2), where a0 and b0 with x1 − K < b0 < 0 < a0 < x1 are solutions of ba00 g(X) dX = 0 and F (b0) = F (a0).

Acknowledgments

We would like to thank referees for their helpful comments and suggestions.

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J. Math. Anal. Appl. 343 (2008) 299–309

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On the existence and uniqueness of limit cycles for generalized Liénard systems

Dongmei Xiao a,∗,1, Zhifen Zhang b,2

a Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China b Department of Mathematics, Peking University, Beijing 100871, China Received 29 June 2007 Available online 29 January 2008 Submitted by J. Mawhin

Abstract

In this paper, we consider a generalized Liénard system dx = φ(y) − F (x), dt dy = −g(x), (0.1) dt

where F is continuous and differentiable on an open interval (b1, a1) with −∞ b1 < 0 < a1 +∞. Assume that there exist a and b with b1 < b < 0 < a < a1 such that xF (x) < 0 as b < x < a, and xF (x) > 0 as a < x < a1 or b1 < x < b. A new uniqueness theorem of limit cycles for the Liénard system (0.1) is obtained. An example is given to show the application of the theorem. © 2008 Elsevier Inc. All rights reserved.

Keywords: Liénard system; Limit cycles; Uniqueness

1. Introduction

The existence and number of limit cycles for planar systems are related to Hilbert 16th problem and self-sustaining oscillatory problems in mathematical models. It is a challenging problem to ﬁnd out conditions so as to guarantee the uniqueness of limit cycles for planar systems. As one knows, there are some systems for which the uniqueness of limit cycles has been extensively studied in the last century, such as Liénard system with the following form:

dx = y − F (x), dt

* Corresponding author. E-mail addresses: [email protected] (D. Xiao), [email protected] (Z. Zhang).

1 Supported by the National Natural Science Foundations of China (No. 10771136). 2 Supported by the National Natural Science Foundations of China (No. 10571002).

0022-247X/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2008.01.059

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D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

dy = −x, (1.1) dt as well as the more general form

dx = φ(y) − F (x), dt dy = −g(x), (1.2) dt where the function g(x) is continuous and F (x) is continuously differentiable on an open interval (b1, a1), −∞ b1 < 0 < a1 +∞. The function φ(y) is continuously differentiable on (−∞, +∞). Moreover, it is always assumed that

(A1) xg(x) > 0 for x = 0. Let G(x) = 0x g(s) ds; (A2) φ(0) = 0, φ (y) > 0 for −∞ < y < +∞. Let Φ(y) = 0y φ(s) ds.

There have been various techniques for establishing the uniqueness of limit cycles for (1.1) and (1.2) (see [1,2, 8–17] and references therein). In these sufﬁcient conditions, the following conditions are usually assumed.

(H1) There exist a and b, b1 < b < 0 < a < a1 such that F (b) = F (0) = F (a) = 0. (H2) xF (x) > 0 if a < x < a1 and b1 < x < b,

xF (x) < 0 if b < x < a.

Sansone and Conti in [10] obtained that if the hypotheses (H1) and (H2) hold, and if the derivative of F (x) with respect to x satisﬁes F (x) > 0 as a < x < a1 and b1 < x < b, then for system (1.1) the number of limit cycles, which enclose two points (a, 0) and (b, 0) inside is unique. Recently, Carletti and Villari in [1] further gave the sufﬁcient conditions for uniqueness of limit cycle of (1.1). For system (1.2), in [11] we gave the conditions to guarantee that a limit cycle encloses two points (a, 0) and (b, 0) inside, which generalized the conclusion of Sansone and Conti in [10]. It is an interesting problem what conditions should add to guarantee the uniqueness of limit cycles for (1.2) if the limit cycle encloses only one point of (a, 0) and (b, 0). Zeng et al. in [12] and [13] studied this problem and used the Filippov transformation z = G(x) to reduce system (1.2) to two equations

dz

= Fi(z) − φ(y),

(Ei )

dy

respectively, where Fi(z) = F (xi(z)) ∈ C1[0, zi), xi(z) is the inverse function of z = G(x) for (−1)i+1x 0, i = 1, 2, z1 = G(a1) and z2 = G(b1). In [13] (cf. [13, Theorem 1]), it has been proved that system (1.2) has at most one limit cycle if besides (H1) and (H2), the following hypotheses hold

(H3) if G(b) > G(a) (G(b) < G(a)), then F1(z)F1(z) (F2(z)F2(z), respectively) is nondecreasing (nonincreasing, respectively) as z > G(a) (z > G(b), respectively);

(H4) if G(b) > G(a) (G(b) < G(a)), then F1(z) F2(u) for any pair (z, u) satisfying G(a) < z < u (G(b) < u < z, respectively) and F1(z) = F2(u) (F2(u) = F1(z), respectively).

From the above sufﬁcient conditions in [1,10,12] and [13], we can see that if limit cycles enclose both points (a, 0) and (b, 0), then the monotonicity of F (x) on a1 > x > a and b > x > b1 is sufﬁcient to guarantee the uniqueness of the limit cycles. However, if limit cycles enclose only one of the two points (a, 0) and (b, 0), then to guarantee the uniqueness of the limit cycles, it needs more restrictions on the functions F (x) and G(x), such as assumptions (H3) and (H4).

In this paper, we discuss further the uniqueness of limit cycles of (1.2) when the limit cycles enclose only one of two points (a, 0) and (b, 0). We obtain a new sufﬁcient criterion for the uniqueness of limit cycles of (1.2), which is different from those sufﬁcient conditions appeared in [2,9–15] and [16]. For simplicity of statement, we make the following assumption on the derivative of F (x).

D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

301

Fig. 1. When G(b) > G(a), the vertical isocline and a closed orbit of system (Ei ).

Fig. 2. When G(b) < G(a), the vertical isocline and a closed orbit of system (Ei ).

(H5) When b1 < x < a1, f (x) d=ef F (x) = 0 has only two roots x1 and x2, x1 > 0 and x2 < 0. Moreover, z is a unique positive root of F1(z) − F2(z) = 0, which implies that G(xi) z .

Now we state a new uniqueness theorem as follows.

Theorem 1.1. Suppose that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1), (H2), (H5), and its equivalent Eqs. (Ei) satisfy one of the following conditions.

(i) If G(b) > G(a) (which implies G(a) < z < G(b)), then F2(u) > F1(z) for any pair (z, u) satisfying G(a) < z < z , 0 < u < z and F1(z) = F2(u) (see Fig. 1).

(ii) If G(a) > G(b) (which implies G(b) < z < G(a)), then F2(z) > F1(u) for any pair (z, u) satisfying G(b) < z < z , 0 < u < z and F1(u) = F2(z) (see Fig. 2).

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D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

Then

(R1) system (1.2) has at most one limit cycle, and it is stable if it exists; (R2) system (1.2) has a unique stable limit cycle if G(b1) = G(a1) = +∞ and Φ(±∞) = ±∞.

2. Main results

In this section, we ﬁrst give the proof of Theorem 1.1, then compare it with Theorem 1 in [13] and provide an

alternative statement of Theorem 1.1.

We now prove the conclusion (R1) of Theorem 1.1. Our main task is to prove the uniqueness of limit cycles of

(1.2) if it exists, which can be derived as follows: if any one closed orbit of system (1.2) is stable, then system (1.2)

has at most one limit cycle since adjacent closed orbits can not have the same stability. Hence, we will only discuss

the stability of closed orbits of (1.2). Assume that L is a closed orbit of (1.2), then L = L1 ∪ L2, Li is an orbit of Eqs. (Ei ), i = 1, 2. And L1 (L2)

intersects the vertical isocline φ(y) = F1(z) (φ(y) = F2(z), respectively) at A(zA, yA) (B(zB , yB ), respectively). It is easy to see that yA > yB . Let y¯i(z) and yi(z) represent a part of Li , which is located above and below of the vertical isocline φ(y) = Fi(z), respectively (see Fig. 1 or Fig. 2). To determine the stability of L, we have to calculate the integral of divergent of (1.2) along L,

div(1.2) dt = −F (x) dt.

L

L

Utilizing the notations in [13], we denote

V Fi (z), yi (z), y¯i (z) d=ef

Fi (z)

+

Fi (z) .

Fi(z) − φ(yi(z)) φ(y¯i(z)) − Fi(z)

Hence, we have

−F (x) dt = − F1(z) dy − F2(z) dy

L

L1

L2

zA

zB

= − V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz ,

(2.1)

0

0

where dt > 0 along the integration path and dy > 0 along the curves L1 and L2. To estimate the sign of the integral (2.1), we ﬁrst introduce two lemmas. They can be found in [12] and [13]. For

convenience to read, we give the proof of Lemma 2.2

Lemma 2.1. Let 0 c < d zA (or zB ). If Fi(d) − Fi(z) y¯i(z)) dz 0 (or 0, respectively).

0 (or

0) for c < z < d, then

d c

V

(Fi

(z),

yi

(z),

Lemma 2.2. Consider systems (E1) and (E2), if there exist 0 p1 < u < z1, 0 p2 < z < z2 such that

(1) F1(p1) = F2(p2), F1(z1) = F2(z2); (2) F1(u) < 0 if p1 < u < z1, and F2(z) < 0 if p2 < z < z2; (3) F1(u) < F2(z) if F1(u) = F2(z).

Let y¯i(·) and yi(·) be solutions of (Ei), i = 1, 2, which satisfy the following inequalities: φ y1(u) < F1(u) < φ y¯1(u) , as p1 < u < z1; φ y2(z) < F2(z) < φ y¯2(z) , as p2 < z < z2; y¯1(p1) y¯2(p2) and y1(p1) y2(p2).

D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

303

Then there exist two functions W¯ 2(z) and W2(z) in the interval (p2, z2) such that

W¯ 2(z) = y¯1 F1−1 F2(z) , W2(z) = y1 F1−1 F2(z) ,

and

z1

z2

V F1(u), y1(u), y¯1(u) du > V F2(z), y2(z), y¯2(z) dz.

p1

p2

Proof. When p1 < u < z1 and p2 < z < z2, we let F1(u) = F2(z), which deﬁne a transformation from variables u into z. Consider Eq. (E1) with the following form:

dy = 1 . (2.2) du F1(u) − φ(y) Then (2.2) can be transformed to

dy = 1

F2(z) ,

(E3)

dz F2(z) − φ(y) F1(F1−1(F2(z)))

and solutions y¯1(u) and y1(u) of (2.2) are transformed to

W¯ 2(z) = y¯1 F1−1 F2(z) and W2(z) = y1 F1−1 F2(z) , respectively, where W¯ 2(z) and W2(z) are solutions of Eq. (E3).

It is clear that W¯ 2(p2) = y¯1(p1) and W2(p2) = y1(p1). From y¯1(p1)

y¯2(p2) and y1(p1)

y2(p2), we have

W2(p2) y2(p2), W¯ 2(p2) y¯2(p2).

(2.3)

Next we prove the inequality of integral. Since

F2(z)

−1

< 1,

F1(F1 (F2(z)))

1

1

<

F2(z) ,

F2(z) − φ(y¯2(z)) F2(z) − φ(y¯2(z)) F1(F1−1(F2(z)))

1

1

>

F2(z) .

F2(z) − φ(y2(z)) F2(z) − φ(y2(z)) F1(F1−1(F2(z)))

Comparing Eqs. (E2) and (E3), and note the initial conditions (2.3), we obtain that

W¯ 2(z) y¯2(z) and W2(z) y2(z)

as z p2 by differential inequality theory. Note that φ (y) > 0. Hence,

φ W¯ 2(z) φ y¯2(z) , φ W2(z) φ y2(z) ,

and

F2(z)

+

F2(z)

>

F2(z)

+

F2(z)

.

F2(z) − φ(W2(z)) φ(W¯ 2(z)) − F2(z) F2(z) − φ(y2(z)) φ(y¯2(z)) − F2(z)

Therefore, we have

z1

z2

V F1(u), y1(u), y¯1(u) du − V F2(z), y2(z), y¯2(z) dz

p1

p2

z2

z2

= V F2(z), W2(z), W¯ 2(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

p2

p2

The proof is completed. 2

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Now we are in a position to prove the uniqueness of limit cycles if it exists. We ﬁrst prove the case (ii) and divide

two cases of yA to discuss. (I) If yA 0, then F1(zA) − F1(z) 0 as 0 < z < zA and F2(zB ) − F2(z) 0 as 0 < z < zB . From Lemma 2.1,

we have

zA

zB

V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

0

0

Hence, the integral of divergent of (1.2) along L is negative, which leads that the closed orbit L is stable. Therefore,

system (1.2) has at most one limit cycle. (II) If yA < 0, then we can take points D(zD, yD) and C(zC, yC) in φ(y) = F1(z) and φ(y) = F2(z), respectively,

such that F1(zD) = F2(zC ) = F1(zA). Note that F1(zA) − F1(z) 0 as 0 < zD < z < zA, F2(G(b)) − F2(z) 0 as 0 < z < G(b) and F2(zB ) − F2(z) 0 as 0 < zC < z < zB . From Lemma 2.1, we have

zA

V F1(z), y1(z), y¯1(z) dz 0,

zD zB

V F2(z), y2(z), y¯2(z) dz 0,

zC

G(b)

V F2(z), y2(z), y¯2(z) dz 0.

(2.4)

0

On the other hand, F1(0) = F2(G(b)) = 0 and F1(zD) = F2(zC). From hypothesis (H5) and condition (ii) of Theorem 1.1, and utilizing Lemma 2.2 we obtain

zD

zC

V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

0

G(b)

(2.5)

Summarizing (2.4) and (2.5), we obtain that

zA

zB

V F1(z), y1(z), y¯1(z) dz − V F2(z), y2(z), y¯2(z) dz > 0.

0

0

Hence, the integral of divergent of (1.2) along L is negative, which implies that the closed orbit L is stable. Thus system (1.2) has at most one limit cycle. We complete the proof of conclusion (R1) in Theorem 1.1 for case (ii).

Let y = −Y . Then system (Ei) is transformed to

dz

= −Fi(z) + φ(−Y ),

(Fi )

dY

and the case (i) in Theorem 1.1 is transformed to the case (ii) for system (Fi). The similar arguments can be applied to the case. We can obtain the uniqueness of limit cycles in Theorem 1.1 for case (i). Therefore, we ﬁnish the proof of conclusion (R1) of Theorem 1.1.

We next prove conclusion (R2) of Theorem 1.1. From conclusion (R1), we only need to prove the existence of limit cycles for system (1.2) if we further assume that G(b1) = G(a1) = +∞ and Φ(±∞) = ±∞. The following lemma implies the existence of limit cycles for system (1.2).

Lemma 2.3. Assume that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1), (H2) and (H5). If G(b1) = G(a1) = +∞ and Φ(±∞) = ±∞, then system (1.2) has at least a limit cycle.

D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

305

Fig. 3. The annular region of positive invariant set of system (1.2) and the vertical isocline φ(y) − F (x) = 0.

Proof. Let

V (x, y) = G(x) + Φ(y), where G(x) = 0x g(s) ds and Φ(y) = 0y φ(s) ds. Then

dV (x, y) = −g(x)F (x). dt (1.2)

Taking 0 < r1 min{a, −b}, we have

dV (x, y) > 0,

dt (1.2)

as V (x, y) = r1.

Thus, the graph of V (x, y) = r1 becomes an intra-boundary of an annular region and the unique equilibrium (0, 0) of

system (1.2) is unstable.

On the other hand, since Φ(±∞) = ±∞, respectively, and G(b1) = G(a1) = +∞, any trajectory of system (1.2) in the region {(x, y): V (x, y) > r1} spirals as t → +∞.

From (H2) and (H5), there exists 0, 0 < 0 1 such that F (a + 0) = k1 > 0, F (b − 0) = k2 < 0, and F (x) k1 as a + 0 x < a1 and F (x) k2 as b1 < x b − 0. Let

l1: x = a + 0, l2: x = b − 0.

Taking a point P (a + 0, y0) on line l1 with y0 < 0, we consider that as time t increases the trajectory Γ (P , t) of system (1.2) with the initial point P intersects l2 at Q(b − 0, yQ) and R(b − 0, yR), respectively, and intersects l1 again at S(a + 0, yS) and T (a + 0, yT ), respectively (see Fig. 3). We claim that y0 < yT < 0 as y0 → −∞. Let

y

x

V1(x, y) = φ(s) − k2 ds + g(s) ds.

0

0

We consider the increment of the function V1(x, y) along the segment of trajectory P QRST of system (1.2)

dV1 = F (x) − k2, dy (1.2)

dV1 = −g(x)(F (x) − k2) .

dx (1.2)

φ(y) − F (x)

(2.6) (2.7)

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D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

Note that F (x) k1 as a + 0 x < a1 and F (x) k2 as b1 < x b − 0. From (2.6), we have

yS

V1(S) − V1(T ) = F (x) − k2 dy > (k1 − k2)(yS − yT ) > 0,

yT

yQ

yR

V1(Q) − V1(R) = F (x) − k2 dy = k2 − F (x) dy > 0.

yR

yQ

Since the functions g(x) and F (x) are bounded for x ∈ (b − 0, a + 0), for arbitrary small > 0 there exists a sufﬁciently large M > 0, when −y0 > M,

dV1 < dx (1.2)

by (2.7). Hence,

V1(P ) − V1(Q) < (a − b + 2 0), Note that

V1(R) − V1(S) < (a − b + 2 0).

V1(P ) − V1(T ) = V1(P ) − V1(Q) + V1(Q) − V1(R) + V1(R) − V1(S) + V1(S) − V1(T ).

Thus,

1 V1(P ) − V1(T ) > (k1 − k2)(yS − yT ) > 0,

2 which implies that y0 < yT < 0 as −y0 is sufﬁciently large.

Therefore, the closed curve by the segment of trajectory P QRST of system (1.2) and the segment T P on line l1 becomes an outer-boundary of an annular region (see Fig. 3). The annular region is a bounded positive invariant set for system (1.2), which does not include any equilibria of system (1.2). Therefore, the existence of limit cycles follows directly from the Poincaré–Bendixson theorem. We ﬁnished the proof of the existence of limit cycle of system (1.2). 2

Remark 2.1. Let us compare Theorem 1 in [13] with Theorem 1.1 here in case G(b) > G(a). Using the notations in this paper, Theorem 1 in [13] can be stated as follows:

Theorem 1. (See [13].) Suppose that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1)–(H4), then system (1.2) has at most a limit cycle, and it is stable if it exists.

The differences between Theorem 1 in [13] and Theorem 1.1 here are as follows.

(1) When z > G(a), in Theorem 1, hypothesis (H3) required, that is, F1(z)F1(z) is nondecreasing. However, in Theorem 1.1, hypothesis (H5) is required, that is, F1(z) is nondecreasing for z > G(a).

(2) Hypothesis (H4) in Theorem 1 requires that F1(z) F2(u) if F1(z) = F2(u) for G(a) < z < u. That implies that they need to compare the slopes of F1(z) and F2(u) outside the interval (0, G(a)); but condition (i) in Theorem 1.1 here requires that F1(z) F2(u) if F1(z) = F2(u) for 0 < u < z and G(a) < z < z . That implies that we need to compare the slopes of F1(z) and F2(u) inside the interval (0, z ). Hence, roughly speaking, our uniqueness

conditions of limit cycles are different from that in [13], and these conditions on the interval (0, G(b)) in case (i)

and on interval (0, G(a)) in case (ii) have strongly geometric intuition.

In general case, the condition (i) or (ii) in Theorem 1.1 is not easy to be veriﬁed since the inverse function of G(x) is hardly expressed explicitly for some continuous functions g(x). Hence, in order to apply Theorem 1.1 more conveniently, one hopes to represent conditions in this theorem directly by means of the original functions of system (1.2). We provide an alternative statement of conclusion (R1) of Theorem 1.1 as follows.

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307

Corollary 2.4. Assume that system (1.2) with (A1) and (A2) satisﬁes the hypotheses (H1), (H2) and (H5).

(i) If ba g(x) dx = 0, then system (1.2) has at most one limit cycle (cf. [11, Theorem 2.2]). (ii) If ba g(x) dx = 0, then there exists only a pair of a0 and b0, b1 < b0 < 0 < a0 < a1 such that ba00 g(x) dx = 0

and F (b0) = F (a0). System (1.2) has at most one limit cycle if one of the following conditions hold. (ii.1) If ba g(x) dx < 0 (i.e. G(b) > G(a)), then fg((xx22)) > fg((xx11)) for any pair of (x1, x2) satisfying a < x1 < a0,

b0 < x2 < 0 and F (x1) = F (x2). (ii.2) If ba g(x) dx > 0 (i.e. G(a) > G(b)), then fg((xx22)) > fg((xx11)) for any pair of (x1, x2) satisfying 0 < x1 < a0,

b0 < x2 < b and F (x1) = F (x2).

3. Example and discussion

The existence and uniqueness of limit cycle is one of the most delicate problem in studying of mathematical models. There are various techniques to establish the uniqueness of limit cycles of ecological systems (cf. [3,6,7,11] and references therein). It is the main technique to transform an ecological system into a generalized Liénard system. For example, a general predator–prey system in [11] can be transferred into the generalized Liénard system (1.2) in suitable region of phase plane. And the geometric shape of prey isocline of predator–prey systems can be kept in that of F (x) = φ(y) of the generalized Liénard system (1.2). Some researchers think that the geometric property of prey isocline plays important role in understanding both the globally asymptotically stable of a unique equilibrium and the uniqueness of limit cycles for predator–prey systems. Hwang in [4] showed that local asymptotic stability of a unique positive equilibrium together with the existence of a concave down prey isocline implied that this equilibrium was globally asymptotically stable for certain predator–prey systems. Kuang in [5] pointed out that Freeman conjectured that the existence of a unique unstable equilibrium together with existence of humps of prey isocline implied the existence and uniqueness of the limit cycle for predator–prey systems. The role of our main results (Theorem 1.1 or Corollary 2.4) is to give information about the geometry of F (x) = φ(y) to ensure existence and uniqueness of limit cycles.

We now give an example to show the application of Corollary 2.4. In [7], Ruan and Xiao studied global dynamics of a predator–prey system with nonmonotonic functional response

x˙ = rx 1 − x − xy , K c + x2

y˙ = y μx − D (3.1) c + x2

in the closed ﬁrst quadrant of R2, where r, c, μ, K and D are positive parameters. They gave the conditions that system (3.1) has a unique limit cycle (cf. [7, Theorems 2.4–2.5]). However, there is a gap in the proof of Theorem 2.5. We will point out it in the following discussion of the uniqueness of limit cycle for system (3.1).

It is clear that system (3.1) has a unique positive equilibrium (x1, y1) if

μ2 > 16 cD2, x2 > K > x3,

(3.2)

3

where

μ − μ2 − 4cD2

x1 =

,

2D

y1 = r 1 − x1 K

c + x12 ,

μ + μ2 − 4cD2

x2 =

,

2D

2μ − μ2 − 4cD2

x3 =

.

2D

The nontrivial periodic orbits of system (3.1) must be in the domain E1 if it exists, here

E1 = (x, y): 0 < x < K, 0 < y < +∞ .

To study the uniqueness of periodic orbits of system (3.1), we transfer system (3.1) in E1 into a Liénard system. Let x − x1 = −X, y − y1 = y1(eY − 1) and x dt = (c + x2) dT . Then system (3.1) can be written as

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D. Xiao, Z. Zhang / J. Math. Anal. Appl. 343 (2008) 299–309

dX = φ(Y ) − F (X), dT dY = −g(X), (3.3) dT

where φ(Y ) = y1(eY − 1), F (X) = rKX (X2 + (K − 3x1)X + c + 3x12 − 2Kx1), and g(X) = DX(Xx1−−xX1+x2) , here x1 − K < X < x1 and −∞ < Y < +∞.

It is easy to check the following facts:

(a1) Xg(X) > 0 for X ∈ (x1 − K

0) ∪ (0

x1), G(x1) =

x1 0

g(s) ds

=

+∞

and

G(x1

−

K)

=

D2 ((K − x1)(x1 + 2x2 − K) − 2x1x2 ln xK1 ) > 0.

(a2) φ(0) = 0, φ (Y ) > 0 for −∞ < Y < +∞, and φ(−∞) = −y1 and φ(+∞) = +∞.

0x1−K g(s) ds =

If inequalities (3.2) hold, then we have

K − 3x1 > 0, c + 3x12 − 2Kx1 = x1(x2 + 3x1 − 2K) < 0. Let = (K − 3x1)2 − 4(c + 3x12 − 2Kx1) = K2 + 2x1K − 3x12 − 4c. Hence, there exist a and b, 0 < a < x1 and K − x1 < b < 0 such that

F (b) = F (0) = F (a) = 0,

where √

a = −(K − 3x1) + , 2

√ b = −(K − 3x1) − .

2

f (X) d=ef F (X) = 0 has only two roots in the interval (x1 − K x1), which implies that the prey isocline φ(y) = F (x) of system (3.1) has only two humps in the range

E2 = (x, y): x1 − K < x < x1, −∞ < y < +∞ ,

namely a local maximum and a local minimum. Therefore, F (X) satisﬁes hypotheses (H1), (H2) and (H5). In the proof of Theorem 2.5 in [7], authors only proved the uniqueness of limit cycles of system (3.1) if the limit

cycles enclose two points (a, 0) and (b, 0) inside, and did not prove the uniqueness of limit cycles if the limit cycles enclose only one point of (a, 0) and (b, 0) inside.

To apply Corollary 2.4 for system (3.1), we can modify Theorem 2.5 in [7] to the following conclusion.

Theorem 3.1. Suppose that μ2 > 136 aD2 and x2 > K > x3. Then system (3.1) has at most one limit cycle in the interior of the ﬁrst quadrant if one of the following conditions holds.

(i) ba g(X) dX = 0. (ii) If ba g(X) dX < 0 (i.e. G(b) > G(a)), then fg((XX22)) > fg((XX11)) for any pair (X1, X2) satisfying a < X1 < a0, b0 <

X2 < 0 and F (X1) = F (X2), where a0 and b0 with x1 − K < b0 < 0 < a0 < x1 are solutions of ba00 g(X) dX = 0 and F (b0) = F (a0). (iii) If ba g(X) dX > 0 (i.e. G(a) > G(b)), then fg((XX22)) > fg((XX11)) for any pair (X1, X2) satisfying 0 < X1 < a0, b0 < X2 < b and F (X1) = F (X2), where a0 and b0 with x1 − K < b0 < 0 < a0 < x1 are solutions of ba00 g(X) dX = 0 and F (b0) = F (a0).

Acknowledgments

We would like to thank referees for their helpful comments and suggestions.