Global Analysis of a Lienard System with Quadratic Damping.

1. Introduction and Main Results

Lionard equations have a very wide application in many areas, such as mechanics, electronic technology, and modern biology; see [1-4]. People are strongly interested in the solution existence, vibration, and periodic solutions of Lionard equations, which promote the research of Lionard equations more and more deeply, as shown in [5-9]. All kinds of problems about Lieonard equations are always the focus of the theory of differential equations. In 2016, Llibre  studied the centers of the analytic differential systems and analyzed the focus-center problem. H. Chen and X. Chen  investigated the dynamical behaviour of a cubic Lionard system with global parameters, analyzing the qualitative properties of all the equilibria and judging the existence of limit cycles and homoclinic loops for the whole parameter plane. They gave positive answers to Wang Kooij's  two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness.

In 1977, Lins, de Melo, and Pugh studied the Lionard equations

dx/dt = y - F(x), dy/dt = -x, (1)

where F is a polynomial of degree n + 1, or equivalently,

[mathematical expression not reproducible], (2)

with f(x) = F'(x). They proposed the following result.

Conjecture 1. If f(x) has degree n, then (1) has at most [[pi]/2] limit cycles ([[pi]/2] is the integer part of [pi]/2, n [greater than or equal to] 2).

n = 2 is proved by ; n = 3 is proved by . The problem for n>3is still open. In 1988, Lloyd and Lynch  considered the similar problem for generalized Lienard equations

dx/dt = y - F(x), dy/dt = -g, (1), (3)

where F is a polynomial of degree n+1 and g(x) is a polynomial of degree m. In most cases, they gave an upper bound for the number of small amplitude limit cycles that can bifurcate out of a single nondegenerate singularity. If we denote by N(m, n) the uniform upper bound for the number of limit cycles (admitting a priori that N(m, n) could be infinite), then the results in  give a lower bound for N(m,n). In 1988 Coppel  proved that N(2,1) = 1. In [19-22], it was proved that N(3,1) = 1. Up to now, as far as we know, only these three cases have been completely investigated.

Consider the Lienard equations

dx/dt = y - F(x), dy/dt = -[x.sup.2m+1], (4)

where F(x) = a[x.sup.3] + b[x.sup.2] + cx, a [not equal to] 0 and m [member of] N. We only discuss a > 0, because the case a < 0 can be derived from the case a > 0 by using the transformation x [right arrow] -x, y [right arrow] -y, and a [right arrow] -a. From the above two motivations, we shall give a complete classification for all the global phase portraits of the Lienard system (4).

We give the following theorem.

Theorem 2. All phase portraits of system (4) can be given, as shown in Figures 1 and 2.

The classifications of global phase portraits are explained in Section 2 and the infinite and finite critical points are discussed in Sections 3 and 4.

The paper is organized as follows. Section 2 explains the classification for all kinds of Lienard system (4). The infinite and finite critical points are discussed in Sections 3 and 4, respectively. Section 5 provides the sufficient and necessary condition for Lienard system (4) to have closed orbits.

2. Explanation of Global Dynamics

The bifurcation diagram and global phase portraits of system (4) for parameters a, b, c, m in all cases are shown in Figure 1.

For example, as shown in Figure 1 (k, l), if b > 0, the elliptic sector lies in the negative y-axis; if b < 0, the elliptic sector lies in the positive y-axis.

(A) Global phase portraits of m = 0: there exist infinite critical points A and B.

(1) Suppose c < 0 and a > 0. A unique stable limit cycle appears around the equilibrium O of system (4). If c [less than or equal to] -2,0 is an unstable node, and the global phase portrait is shown in Figure 1(a); if -2 < c < 0, O is an unstable focus, and the global phase portrait is shown in Figure 1(b).

(2) Suppose c >0 and a >0. There are no closed orbits in system (4). If 0 [less than or equal to] c < 2 and a > 0, O is a stable focus, and the global phase portrait is shown in Figure 1(c); if c [greater than or equal to] 2 and a > 0, O is a stable node, and the global phase portrait is shown in Figure 1(d).

(3) Suppose c < 0 and a < 0 There are no closed orbits in system (4). If c [less than or equal to] -2, O is an unstable node, and the global phase portrait is shown in Figure 2(a); if -2 < c < 0, O is an unstable focus, and the global phase portrait is shown in Figure 2(b).

(4) Suppose c > 0 and a < 0. A unique unstable limit cycle appears around the equilibrium O of system (4). If 0 [less than or equal to] c < 2 and a > 0, O is a stable focus, and the global phase portrait is shown in Figure 2(c); if c [greater than or equal to] 2 and a > 0, O is a stable node, and the global phase portrait is shown in Figure 2([theta]).

(B) Global phase portraits of m = 1: there exist infinite critical points [A.sub.1] and B.

(1) Suppose c > 0 or c = b = 0, and a > 0. There are no closed orbits in system (4). O is a stable degenerate node, and the global phase portrait is shown in Figure 1(e).

(2) Suppose c > 0 and a < 0. A unique unstable limit cycle appears around the stable degenerate node O of system (4), and the global phase portrait is shown in Figure 2(e).

(3) Suppose c < 0 or c = b = 0, and a < 0. There are no closed orbits in system (4). O is an unstable degenerate node, and the global phase portrait is shown in Figure 2(f).

(4) Suppose c < 0 and a > 0 A unique stable limit cycle appears around the unstable degenerate node O of system (4), and the global phase portrait is shown in Figure 1(f).

(5) Suppose c = 0 and b [not equal to] 0. There are no closed orbits in system (4). If b > 0, the elliptic sector lies in the positive y-axis, and the global phase portraits are shown in Figures 1(g) and 2(g); if b <0, the elliptic sector lies in the negative y-axis, and the global phase portraits are shown in Figures 1(h) and 2(h).

(C) Global phase portraits of m [greater than or equal to] 2: there exists a unique infinite critical point B.

(1) Suppose c > 0 or c = b = 0, and a > 0. There are no closed orbits in system (4). O is a stable degenerate node, and the global phase portrait is shown in Figure 1(i).

(2) Suppose c > 0 and a < 0. A unique unstable limit cycle appears around the stable degenerate node O of system (4), and the global phase portrait is shown in Figure 2(i).

(3) Suppose c <0 and a >0. A unique stable limit cycle appears around the unstable degenerate node O of system (4), and the global phase portrait is shown in Figure 1(().

(4) Suppose c < 0 or c = b = 0, and a < 0. There are no closed orbits in system (4). O is an unstable degenerate node, and the global phase portrait is shown in Figure 2(().

(5) Suppose c = 0 and b [not equal to] 0. There are no closed orbits in system (4). If b > 0, the elliptic sector lies in the positive y-axis, and the global phase portraits are shown in Figure 1(k) and the picture (k) in Figure 2; if b < 0, the elliptic sector lies in the negative y-axis, and the global phase portraits are shown in Figures 1(l) and 2(l).

3. Analysis of Equilibria

Clearly, system (4) has a unique equilibrium O : (0,0).

Lemma 3. The type of equilibrium O in system (4) is shown as Table 1.

Proof. Now we consider the case m=0. The Jacobian matrix at O is

[mathematical expression not reproducible], (5)

from which we obtain that D = det J = 1, T = traceJ = -c. Further, O is a focus when [DELTA] = [T.sup.2] - 4D = [c.sup.2] - 4 < 0 and anode when [DELTA] < 0. Clearly, [DELTA] = 0 if and only if [c.sup.2] = 4. Therefore, O is a stable focus when 0 < c < 2, an unstable focus when -2 < c <0, a stable node when c [greater than or equal to] 2, and an unstable node when c [less than or equal to] -2.

For the case that c = 0, we consider the case that the linear part of system (4) around O has eigenvalues [alpha](c) [+ or -] i[beta](c) for c near 0, in which [alpha]([[eta].sup.2]) = -c/2. Obviously, [alpha](0) = 0 and [beta](0) [not equal to] 0. Clearly d[alpha](c)/dc = -1/2.

Now, we need to compute the coefficients of Hopf bifurcation of order 1. According to the Hopf bifurcation theory , we obtain the following results for b outside the interval (0,1).By( P.152), we can compute the coefficients of Hopf bifurcation of (4)

Re[C.sub.1] (a) = - a/8. (6)

We can get Re[C.sub.1] (a) > 0 for a <0; and we can get Re[C.sub.1] (a) < 0 for a > 0.

We need to compute the sign of cRe[C.sub.1]d[alpha](c)/dc = ac/16. When ac [greater than or equal to] 0, we can get cRe[C.sub.1]d[alpha](c)/dc [greater than or equal to] 0; and when ac > 0, we can get cRe[C.sub.1]d[alpha](c)/dc > 0.

Therefore, we obtain the following lemma.

Lemma 4. When a > 0 and c < 0, the equilibrium O of system (4) is an unstable weak focus with multiplicity 1, and there is a unique stable limit cycle bifurcating from O; when a < 0 and c > 0, the equilibrium O of system (4) is a stable weak focus with multiplicity 1, and there is a unique unstable limit cycle bifurcating from O; when a > 0 and c > 0, the equilibrium O of system (4) is an unstable weak focus with multiplicity 1, and there are no closed orbits near O; when a < 0 and c [less than or equal to] 0, the equilibrium O of system (4) is a stable weak focus with multiplicity 1, and there are no closed orbits near O.

3.1. Degenerate Bogdanov-Takens Bifurcation. In another case m [greater than or equal to] 1 and c [not equal to] 0, only one eigenvalue of linearization of system (4) at O equals zero. In fact, by a reversible transformation

[mathematical expression not reproducible], (7)

which changes the linearization of system (4) into Jordan canonical form near O, when m = 1, we get

[mathematical expression not reproducible]. (8)

Let the second equation of (8) equal zero, and we solve that [mathematical expression not reproducible] by the Implicit Function Theorem. Substituting y of the first equation of (8) by [Y.sub.1]([??]), we obtain that

[mathematical expression not reproducible]. (9)

When c > 0, O is a stable degenerate node; when c < 0, O is an unstable degenerate node.

In the remaining case that m [greater than or equal to] 1 and c = 0, the two eigenvalues of the linearization of system (4) at O are both zero but the linear part does not equal zero identically. System (4) is equivalent to this system

[mathematical expression not reproducible]. (10)

By Theorem 7.2 of [24, Chapter 2], when b = 0 and a > 0, O is a stable degenerate node; when b = 0 and a < 0, O is an unstable degenerate node.

When b [not equal to] 0, we can get that an elliptic sector and a hyperbolic sector consist of the field of the O by Theorem 7.2 of [24, Chapter 2].

Lemma 5. Suppose m = 1, b = 0, and c = 0, then there is a neighborhood V of the point (0, 0) in R such that system (16) displays a degenerate Bogdanov-Takens bifurcation near 0(0, 0) when ([epsilon], c) varies in V. More concretely, there exist six curves

(a) [R.sup.+] = {([epsilon], c) | [epsilon] = 0, c < 0},

(b) [R.sup.-] = {([epsilon], c) | [epsilon] = 0, c < 0}

(c) [H.sub.1] = {([epsilon], c) | c = 0, [epsilon] < 0},

(d) [H.sub.2] = {([epsilon], c) | c = [epsilon] + O([[epsilon].sup.2]), [epsilon] > 0},

(e) HL = {([epsilon], c) | c = -(5/4) [epsilon] + O([[epsilon].sup.2/3]), [epsilon] > 0},

(f ) B = {([epsilon], c) | c = -[c.sub.0] [epsilon] + O([[epsilon].sup.2/3]), [epsilon] > 0, [c.sub.0] [approximately equal to] 0.752}.

When a < 0, system (16) displays a bifurcation of equilibria, a Hopf bifurcation, a homoclinic bifurcation, and a double limit cycle bifurcation near O when ([epsilon], c) pass through the curves [R.sup.+] [union] [R.sup.-], [H.sub.1] [union] [H.sub.2], HL, and B. c will be replaced with -c when a > 0.

Proof. When a <0, being the standard form of degenerate Bogdanov-Takens system as shown in , the equilibrium O of system (16) is a stable degenerate node. Thus, equilibrium O of system (4) is a stable degenerate node and a degenerate Bogdanov-Takens bifurcation of codimension-2 will occur near the stable degenerate node when parameter c crosses c = 0, respectively, with b = 0 and m = 1. By , we know the following two-parameter family provides a universal unfolding of (16).

[mathematical expression not reproducible]. (11)

The bifurcation diagrams and phase portraits of (17) are shown in Figure 3.

When a > 0, with the transformation y [right arrow] -y and dt = -dt, we can know the following two-parameter family provides a universal unfolding of (16)

[mathematical expression not reproducible]. (12)

Therefore c will be replaced with -c when a > 0.

Lemma 6. Suppose that m = 1, b [not equal to] 0, and c = 0, then there is a neighborhood [V.sub.2] of the point (0,0, 0) in R such that system (16) displays a codimension-3 Bogdanov-Takens bifurcation near 0(0,0) when ([[mu].sub.1], [[mu].sub.2], c) varies in [V.sub.2].

4. Equilibria at Infinity

In this section, we discuss the qualitative properties of the equilibria at infinity, which reflect the tendencies of x, y as going up by a large amount. With a Poincaree transformation x = 1/z, y = u/z, system (4) can be rewritten as

[mathematical expression not reproducible], (13)

where d[tau] = dt/[z.sup.2] and m = 0.

[mathematical expression not reproducible], (14)

where dx = dt/[z.sup.2m] and m [greater than or equal to] 1. System (13) has an equilibrium A : (0,0) on the u-axis, and system (14) has an equilibrium [A.sub.1] : (1/a,0) when m = 1 and no equilibria when m > 1 on the M-axis, which corresponds to an equilibrium [I.sub.A] at infinity on the x-axis. With another Poincare transformation x = v/z, y = 1/z, system (4) is changed into

[mathematical expression not reproducible], (15)

where d[tau] = dt/[z.sup.2] and m = 0.

[mathematical expression not reproducible], (16)

where d[tau = dt/[z.sup.2]m and m [greater than or equal to] 1. We only need to study the equilibrium B : (0, 0) of systems (15) and (16), which corresponds to an equilibrium [I.sub.B] of system (4) at infinity on the y-axis.

Lemma 7. Equilibria A and [A.sub.1] are unstable nodes when a > 0 and stable nodes when a < 0.

System (16) provides an interesting example for highly degenerate equilibria when m is greater than 1. As m is unspecified, the lowest degree of nonzero terms in (16) is 2m. One could not use the blowing-up methods as done in  2m times to decompose the equilibrium B into simple ones. So a natural idea is to study the system with normal sectors, as in . We will see that the method of normal sectors does not work in some cases, while we show how to apply the method of generalized normal sectors  (GNS for short).

Lemma 8. For system (16), when m =0,1 and a > 0, there are infinite orbits approaching B : (0, 0) in two directions [theta] = [pi], there is a unique orbit approaching B : (0,0) in two directions [theta] = 0, and there are infinite orbits leaving B : (0,0) in two directions [theta] = 0; when m =0,1 and a <0, there are infinite orbits leavingB : (0,0) in two directions [theta] = 0, there is a unique orbit leaving B : (0, 0) in two directions [theta] = n, and there are infinite orbits approaching B : (0, 0) in two directions [theta] = [pi]; when m [greater than or equal to] 2, there are infinite orbits leaving B : (0,0) in two directions [theta] = 0, and there are infinite orbits approaching B : (0, 0) in two directions [theta] = [pi].

Proof. It is equivalent to consider the equilibrium A of system (13). By Theorem II.3.1 in , we only need to discuss the orbits in exceptional directions, as seen in Frommer . With the substitution u = r cos [theta], v = r sin [theta], system (13) can be written as

dr/rd[theta] = [H.sub.1] ([theta]) + o(1)/[G.sub.1]([theta]) + o(1), as r [right arrow] 0, (17)

where [G.sub.1]([theta]) = -[sin.sup.3][theta], [H.sub.1]([theta]) = cos [theta][sin.sup.2][theta] when m = 0, [G.sub.1]([theta]) = -[sin.sup.2m+1] [theta], [H.sub.1]([theta]) = cos [theta] [sin.sup.2m] [theta] when m [greater than or equal to] 1. A necessary condition for [theta] = [[theta].sub.0] to be an exceptional direction is that [G.sub.1]([theta]) = 0. Obviously, [G.sub.1]([theta]) has two roots 0 and [pi]. As in , except in these exceptional directions, no orbits connect B.

When m = 0, using the Briot-Bouquet transformation  v = v, z = [z.sub.1]v, which desingularizes the degenerate equilibrium D : (0, 0) of system (15) in the directions of [z.sub.1]-axis, we reduce (15) to the following form (18):

[mathematical expression not reproducible], (18)

where d[sigma] = vd[tau]. We need to investigate the origin of (18) which is a degenerate equilibrium of system (18). In polar coordinates v = r cos [theta] and [z.sub.1] = r sin [theta], we have

G([theta]) = 2a sin [theta][cos.sup.2][theta], H([theta]) = a cos [theta] ([sin.sup.2] [theta] - [cos.sup.2] [theta]), (19)

for system (18). The equation G([theta]) = 0 has exactly four real roots 0, [pi]/2, n, and 3[pi]/2, and we can check that

G'(0)H(0) = G' ([pi]) H ([pi]) = -2[a.sup.2] < 0. (20)

By Theorem 3.7 of [24, Chapter 2], system (18) has a unique orbit approaching the origin in the direction [theta] = 0, a unique orbit leaving the origin in [theta] = [pi] as t [right arrow] +[infinity], which are exactly the positive v-axis and the negative v-axis, respectively. And for [theta] = [pi]/2 and [theta] = 3[pi]/2, we can check that H([pi]/2) = H(3[pi]/2) = 0.

Applying the Briot-Bouquet transformation v = [v.sub.2][z.sub.1], [z.sub.1] = [z.sub.1], we can change system (18) into the following form:

[mathematical expression not reproducible], (21)

where ds = [z.sub.1]d[sigma]. We need to investigate the origin of system (21) which is degenerate. In polar coordinates [v.sub.2] = r cos [theta] and [z.sub.1] = r sin [theta], we have

[mathematical expression not reproducible], (22)

for system (21). The equation G([theta]) = 0 has exactly six real roots 0, arctana, [pi]/2, n, n + arctan a, and 3[pi]/2 when a > 0, 0,[pi]--arctan(-a), [pi]/2, [pi], 2[pi] - arctan(-a), and 3[pi]/2 when a < 0, and we can check that

[mathematical expression not reproducible]. (23)

By Theorem 3.7 of [24, Chapter 2] system (21) has a unique orbit approaching the origin in the direction [theta] = 0, a unique orbit leaving the origin in [theta] = n, a unique orbit approaching the origin in [theta] = [pi]/2, and a unique orbit leaving the origin in [theta] = 3[pi]/2 as r [right arrow] +[infinity], which are exactly the positive [v.sub.2]-axis, the negative [v.sub.2]-axis, the positive [z.sub.1]-axis, and the negative [z.sub.1] axis, respectively. And for [theta] = arctan a and [theta] = [pi] + arctan a when a > 0 or [theta] = [pi]-arctan(-a) and [theta] = 2[pi] - arctan(-a) when a > 0, we can check that H([theta]) = 0.

Applying the Briot-Bouquet transformation [v.sub.2] = [v.sub.2], [z.sub.1] = [z.sub.3][v.sub.2], we can change system (21) into the following form:

[mathematical expression not reproducible], (24)

where d[s.sub.1] = [v.sub.2]ds. One can check that system (24) has exactly two equilibria (0,0) and (0, a) on the [z.sub.3]-axis, and we only need to investigate the qualitative properties of (0, a) which corresponds to the directions [theta] = arctan a and [theta] = [pi] + arctan a when a > 0 or [theta] = [pi] - arctan(-a) and [theta] = 2[pi] - arctan(-fl) when a > 0, of system (21). Applying the transformation [[bar.v].sub.2] = [v.sub.2], [[bar.z].sub.3] = [z.sub.3] - a, which translates the equilibrium (0, a) to the origin, for simplicity, we denote [[bar.v].sub.2] and [[bar.z].sub.3] by [v.sub.2] and [z.sub.3], respectively, and system (24) can be written into the form

[mathematical expression not reproducible], (25)

and we only need to analyze the qualitative properties of the origin of system (25).

Applying the transformation [v'.sub.2] = [v.sub.2], [z'.sub.3] = ab[v.sub.2] - [z.sub.3], and d[s.sub.2] = -d[s.sub.1], for simplicity, we denote [v'.sub.2] and [z'.sub.3] by [v.sub.2] and [z.sub.3], respectively, and system (25) can be written as

[mathematical expression not reproducible], (26)

and we only need to analyze the qualitative properties of the origin of system (26).

When a[b.sup.2] + [a.sup.2]c [not equal to] 0, there exists a function

[z.sub.3] = [X.sub.2] ([v.sub.2]) = - (a[b.sup.2] + [a.sup.2]c) [v.sup.2.sub.2] + h.o.t (27)

which can be derived from the second equation of system (26). Substitute the function (27) into the first equation of system (26), and we obtain that

d[v.sub.2]/d[s.sub.2] = [a.sup.3] (1 + 2[c.sup.2]) [v.sub.2.sup.5] + h.o.t. (28)

By Theorem 7.1 in [24, Chapter 2], we obtain that when a > 0, the origin of system (26) is an unstable node; we obtain that when a < 0, the origin of system (26) is a stable node. So, according to the method of the Briot-Bouquet transformation, the theorem of m =0 is proved. Based on the proof of m =0, we can also use the same method to get the same result of m = 1.

When m > 1, some difficulties are caused when we discuss orbits in the directions [theta] = 0, [pi], because [G'.sub.1](0) = [H.sub.1](0) = 0, which does not match any conditions of the theorems in references, e.g., . However, in what follows, we construct GNSes or some related open quasi-sectors which allow curves and orbits to be their boundaries, to determine how many orbits connect A in [theta] =0, [pi].

From dz/dt = [theta] in (16), two horizontal isoclines are determined near [theta] = 0,[pi]: one is [V.sub.1] := {v [member of] [R.sub.+] : z = 0} and the other is [V.sub.2] := {v [member of] [R.sub.-] : z = 0}. Furthermore, let

[mathematical expression not reproducible], (29)

where [[sigma].sub.1] > 0 and [[sigma].sub.1] is closed to zero.

Case 1. a < 0. Notice that there are no vertical isoclines near [theta] = 0 in (16). We claim that the open sector [mathematical expression not reproducible] is a GNS in class I. In fact, we have dz/dt > 0 between [[Laplace].sub.1] and [V.sub.1] and dz/dt < 0 between [[Laplace].sub.2] and [V.sub.1]. So dr/dt > 0 in the closure [mathematical expression not reproducible]. Therefore, what we claim is proved by the definition of GNS. Lemma 1 in  guarantees that system (16) has infinitely many orbits in connection with (actually leaving from) B in [mathematical expression not reproducible]. If m =2 and -2 [less than or equal to] a < 0, we notice that there are no vertical isoclines near [theta] = n in (16). Hence in [mathematical expression not reproducible], we have dv/dz < 0 and dv/dz > 0, respectively, implying that infinitely many orbits connect B in the two sectors by Lemma 1 in . If m =2 and a < -2, or m > 2, from dv/d[tau] = 0 in (16), we obtain vertical isoclines [mathematical expression not reproducible] is a sufficiently small constant. Obviously, [H.sub.i](i = 1,2) is tangent to v-axis at B; hence in [mathematical expression not reproducible], we have dv/dz > 0 and dv/dz < 0, respectively, implying that infinitely many orbits connect B in the two sectors by Lemma 1 in .

Case 2. a > 0. Based on the proof of a < 0, we can also use the same method to get the same result of a > 0. We can give the three cases as shown in Figure 3.

5. Nonexistence and Uniqueness of Closed Orbits

Let us consider the Lienard system

dx/dt = y - F(x), dy/dt = -g, (1), (30)

in which F(x) and g(x) are continuous functions on R satisfying locally Lipschitz condition. We assume that

F(0) = 0, xg(x) > 0 if x [not equal to] 0. (31)

Then the origin is the only critical point. Let M = min{[[integral].sup.[infinity].sub.0] g(x)dx, [[integral].sup.-[infinity].sub.0] g(x)dx} (M may be [infinity]) and let

w = G(x) = [[integral].sup.x.sub.0] [absolute value of g(x)] dx. (32)

Then by (31), G(x) is strictly increasing. We denote the inverse function of G(x) by [G.sup.-1](w).

In article , Sugie and Hara gave the following condition on F(x) and g(x) under which system (30) has no periodic solutions except the origin.

Lemma 9 (see ). Suppose that

F ([G.sup.-1] (w)) [not equal to] F ([G.sup.-1] (-w)) for 0 < w < M. (33)

Then system (30) has no nonconstant periodic solutions.

Let [x.sub.i](z) be the inverse function of z = G(x) and [(-1).sup.i+1]x [greater than or equal to] 0, where i = 1,2; (30) will be equations ([E.sub.1]) and ([E.sub.2]) in domains x [greater than or equal to] 0 and x [less than or equal to] 0, respectively.

dz/dy = [F.sub.i] (z) - y, [([E.sub.i]).sub.i=1,2], (34)

where [F.sub.i](z) = F([x.sub.i](z)).

Lemma 10 (see ). Assume f(x) and g(x) are continuous functions in (-[infinity], +[infinity]), xg(x) > 0 for x [not equal to] 0, G([+ or -][infinity]) = +[infinity], and verify

(1) [there exists]3 [greater than or equal to] 0, [F.sub.1](z) [less than or equal to] 0 [less than or equal to] [F.sub.2](z) for 0 < z < a, [F.sub.1](z) [??] [F.sub.2](z), [F.sub.1](z) > 0 for z > a,

(2) [F.sub.2](z) [less than or equal to] 0 for [F.sub.2](z) < 0,

(3) [F.sub.1](z)[F.sub.1]'(z) is non-decrease for z > a,

(4) when [F.sub.1](z) = [F.sub.2](u) for u [greater than or equal to] z > a, we have [F.sub.1]'(z) [greater than or equal to] [F.sub.2]'(u). Then system (30) has at most one limit cycle in (-[infinity], +[infinity]); if it exists, it must be simple and stable.

Lemma 11. When ac [greater than or equal to] 0, system (4) has no closed orbits; when ac < 0, system (4) has a unique closed orbit.

Proof. We can easily compute

[mathematical expression not reproducible] (35)

Obviously,

[mathematical expression not reproducible]. (36)

When ac [greater than or equal to] 0, F([G.sup.-1](w)) [not equal to] F([G.sup.-1](-w)) for w > 0. Therefore, system (4) has no closed orbits by Theorem 4.5 of [24, Chapter 2] 5 when ac [greater than or equal to] 0.

When ac < 0, we only discuss a > 0, since the proof of the case a < 0 is reduced to that of the case a > 0 by the transformations y [right arrow] -y and t [right arrow] -t.

(1) b >0. The equation [mathematical expression not reproducible]. So, system (30) has at most one limit cycle in (-[infinity], +[infinity]); if it exists, it must be simple and stable.

(2) b < 0. The proof of the case (2) is reduced to that of the case (1) by the transformations y [right arrow] -y and x [right arrow] -x.

The existence of limit cycles can be proved by Theorem 1.3 in [24, Chapter 2]. Thus, system (30) has a unique stable limit cycle.

https://doi.org/10.1155/2018/1249620

Conflicts of Interest

The author declares that he has no conflicts of interest.

Acknowledgments

The author would like to thank Professor Hebai Chen for the help and valuable guidance on his paper.

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Feng Guo (iD)

Department of Basic Courses, Shandong Women's University, Jinan 250300, China

Correspondence should be addressed to Feng Guo; sdwugf@163.com

Received 5 October 2017; Revised 4 February 2018; Accepted 9 August 2018; Published 2 September 2018

Academic Editor: Miguel Angel Lopez

Caption: Figure 1: The global phase portraits of system (4) as the parameter a > 0.

Caption: Figure 2: The global phase portraits of system (4) as the parameter a < 0.

Caption: Figure 3: The qualitative properties of (0,0) of the system (4).
Table 1: Qualitative properties of equilibria O.

Possibilities a, b, c            Type and stability

m = 0             c [less than or equal to] -2        unstable node
-2 < c < 0                         unstable focus
c = 0                            unstable weak focus
0 < c < 2                           stable focus
c [greater than or equal to] 2       stable node

m [greater than   c < 0                            unstable degenerate
or equal to] 1    c > 0                                   node
b [not equal to] 0   stable degenerate
c = 0                    b = 0       node cusp

c <0 c >0 m>1 b = 0 c = 0 b = 0  stable degenerate
node
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