Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 1249620, 9 pages
https://doi.org/10.1155/2018/1249620

Research Article

Global Analysis of a Liénard System with Quadratic Damping

Feng Guo

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 Ángel López

Copyright © 2018 Feng Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

1. Introduction and Main Results

Liénard 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 Liénard equations, which promote the research of Liénard equations more and more deeply, as shown in [5–9]. All kinds of problems about Liénard equations are always the focus of the theory of differential equations. In 2016, Llibre [10] studied the centers of the analytic differential systems and analyzed the focus-center problem. H. Chen and X. Chen [11–13] investigated the dynamical behaviour of a cubic Liénard 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 [14] two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness.

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

where F is a polynomial of degree , or equivalently,

with . They proposed the following result.

Conjecture 1. If has degree , then (1) has at most limit cycles ( is the integer part of , ).
is proved by [15]; is proved by [16]. The problem for is still open. In 1988, Lloyd and Lynch [17] considered the similar problem for generalized Liénard equations

where F is a polynomial of degree and is a polynomial of degree . 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 the uniform upper bound for the number of limit cycles (admitting a priori that could be infinite), then the results in [17] give a lower bound for . In 1988 Coppel [18] proved that . In [19–22], it was proved that . Up to now, as far as we know, only these three cases have been completely investigated.

Consider the Liénard equations

where , and . We only discuss , because the case can be derived from the case by using the transformation , , and . From the above two motivations, we shall give a complete classification for all the global phase portraits of the Liénard 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.

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

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

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 Liénard 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 Liénard system (4) to have closed orbits.

2. Explanation of Global Dynamics

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

For example, as shown in Figure 1 (), if , the elliptic sector lies in the negative -axis; if , the elliptic sector lies in the positive -axis.

(A)Global phase portraits of : there exist infinite critical points and .

(1)Suppose and . A unique stable limit cycle appears around the equilibrium of system (4). If , is an unstable node, and the global phase portrait is shown in Figure 1(a); if , is an unstable focus, and the global phase portrait is shown in Figure 1(b).(2) Suppose and . There are no closed orbits in system (4). If and , is a stable focus, and the global phase portrait is shown in Figure 1(c); if and , is a stable node, and the global phase portrait is shown in Figure 1(d).(3) Suppose and . There are no closed orbits in system (4). If , is an unstable node, and the global phase portrait is shown in Figure 2(a); if , is an unstable focus, and the global phase portrait is shown in Figure 2(b).(4) Suppose and . A unique unstable limit cycle appears around the equilibrium of system (4). If and , is a stable focus, and the global phase portrait is shown in Figure 2(c); if and , is a stable node, and the global phase portrait is shown in Figure 2(d).

(B)Global phase portraits of : there exist infinite critical points and .

(1)Suppose or , and . There are no closed orbits in system (4). is a stable degenerate node, and the global phase portrait is shown in Figure 1(e).(2)Suppose and . A unique unstable limit cycle appears around the stable degenerate node of system (4), and the global phase portrait is shown in Figure 2(e).(3) Suppose or , and . There are no closed orbits in system (4). is an unstable degenerate node, and the global phase portrait is shown in Figure 2(f).(4) Suppose and . A unique stable limit cycle appears around the unstable degenerate node of system (4), and the global phase portrait is shown in Figure 1(f).(5) Suppose and . There are no closed orbits in system (4). If , the elliptic sector lies in the positive -axis, and the global phase portraits are shown in Figures 1(g) and 2(g); if , the elliptic sector lies in the negative -axis, and the global phase portraits are shown in Figures 1(h) and 2(h).

(C)Global phase portraits of : there exists a unique infinite critical point .

(1)Suppose or , and . There are no closed orbits in system (4). is a stable degenerate node, and the global phase portrait is shown in Figure 1(i).(2)Suppose and . A unique unstable limit cycle appears around the stable degenerate node of system (4), and the global phase portrait is shown in Figure 2(i).(3) Suppose and . A unique stable limit cycle appears around the unstable degenerate node of system (4), and the global phase portrait is shown in Figure 1(j).(4) Suppose or , and . There are no closed orbits in system (4). is an unstable degenerate node, and the global phase portrait is shown in Figure 2(j).(5) Suppose and . There are no closed orbits in system (4). If , the elliptic sector lies in the positive -axis, and the global phase portraits are shown in Figure 1(k) and the picture  (k) in Figure 2; if , the elliptic sector lies in the negative -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 .

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

Table 1: Qualitative properties of equilibria .

Possibilities
Type and stability

unstable node

unstable focus

unstable weak focus

stable focus

stable node

unstable degenerate node

stable degenerate node

cusp

stable degenerate node

Proof. Now we consider the case . The Jacobian matrix at is

from which we obtain that , . Further, is a focus when and a node when . Clearly, if and only if . Therefore, is a stable focus when , an unstable focus when , a stable node when , and an unstable node when .
For the case that , we consider the case that the linear part of system (4) around has eigenvalues for near , in which . Obviously, and . Clearly .
Now, we need to compute the coefficients of Hopf bifurcation of order 1. According to the Hopf bifurcation theory [23], we obtain the following results for outside the interval . By ([23] P.152), we can compute the coefficients of Hopf bifurcation of (4)

We can get for ; and we can get for .
We need to compute the sign of . When , we can get ; and when , we can get .

Therefore, we obtain the following lemma.

Lemma 4. When and , the equilibrium of system (4) is an unstable weak focus with multiplicity 1, and there is a unique stable limit cycle bifurcating from ; when and , the equilibrium of system (4) is a stable weak focus with multiplicity 1, and there is a unique unstable limit cycle bifurcating from ; when and , the equilibrium of system (4) is an unstable weak focus with multiplicity 1, and there are no closed orbits near ; when and , the equilibrium of system (4) is a stable weak focus with multiplicity 1, and there are no closed orbits near .

3.1. Degenerate Bogdanov-Takens Bifurcation

In another case and , only one eigenvalue of linearization of system (4) at equals zero. In fact, by a reversible transformation

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

Let the second equation of (8) equal zero, and we solve that by the Implicit Function Theorem. Substituting of the first equation of (8) by , we obtain that

When , is a stable degenerate node; when , is an unstable degenerate node.

In the remaining case that and , the two eigenvalues of the linearization of system (4) at are both zero but the linear part does not equal zero identically. System (4) is equivalent to this system

By Theorem 7.2 of [24, Chapter 2], when and , is a stable degenerate node; when and , is an unstable degenerate node.

When , we can get that an elliptic sector and a hyperbolic sector consist of the field of the by Theorem 7.2 of [24, Chapter 2].

Lemma 5. Suppose , and , then there is a neighborhood of the point in such that system (16) displays a degenerate Bogdanov-Takens bifurcation near when varies in . More concretely, there exist six curves

(a),(b),(c),(d),(e),(f).

When , system (16) displays a bifurcation of equilibria, a Hopf bifurcation, a homoclinic bifurcation, and a double limit cycle bifurcation near when pass through the curves , and . will be replaced with when .

Proof. When , being the standard form of degenerate Bogdanov-Takens system as shown in [1], the equilibrium of system (16) is a stable degenerate node. Thus, equilibrium 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 crosses , respectively, with and . By [16], we know the following two-parameter family provides a universal unfolding of (16).

The bifurcation diagrams and phase portraits of (17) are shown in Figure 3.
When , with the transformation and , we can know the following two-parameter family provides a universal unfolding of (16)

Therefore will be replaced with when .

(a)

(b) and , or and

(c) and , or and

Figure 3: The qualitative properties of (0,0) of the system (4).

Lemma 6. Suppose that , and , then there is a neighborhood of the point in such that system (16) displays a codimension-3 Bogdanov-Takens bifurcation near when varies in .

4. Equilibria at Infinity

In this section, we discuss the qualitative properties of the equilibria at infinity, which reflect the tendencies of , as going up by a large amount. With a Poincaré transformation , , system (4) can be rewritten as

where and .

where and . System (13) has an equilibrium on the –axis, and system (14) has an equilibrium when and no equilibria when on the –axis, which corresponds to an equilibrium at infinity on the –axis. With another Poincaré transformation , , system (4) is changed into

where and .

where and . We only need to study the equilibrium of systems (15) and (16), which corresponds to an equilibrium of system (4) at infinity on the –axis.

Lemma 7. Equilibria and are unstable nodes when and stable nodes when .

System (16) provides an interesting example for highly degenerate equilibria when is greater than . As is unspecified, the lowest degree of nonzero terms in (16) is . One could not use the blowing-up methods as done in [24] times to decompose the equilibrium into simple ones. So a natural idea is to study the system with normal sectors, as in [24]. 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 [24] (GNS for short).

Lemma 8. For system (16), when and , there are infinite orbits approaching in two directions , there is a unique orbit approaching in two directions , and there are infinite orbits leaving in two directions ; when and , there are infinite orbits leaving in two directions , there is a unique orbit leaving in two directions , and there are infinite orbits approaching in two directions ; when , there are infinite orbits leaving in two directions , and there are infinite orbits approaching in two directions .

Proof. It is equivalent to consider the equilibrium of system (13). By Theorem II.3.1 in [24], we only need to discuss the orbits in exceptional directions, as seen in Frommer [25]. With the substitution , , system (13) can be written as

where , when , , when . A necessary condition for to be an exceptional direction is that . Obviously, has two roots and . As in [24], except in these exceptional directions, no orbits connect .
When , using the Briot-Bouquet transformation [24] , , which desingularizes the degenerate equilibrium of system (15) in the directions of -axis, we reduce (15) to the following form (18):

where . We need to investigate the origin of (18) which is a degenerate equilibrium of system (18). In polar coordinates and , we have

for system (18). The equation has exactly four real roots , , , and , and we can check that

By Theorem 3.7 of [24, Chapter 2], system (18) has a unique orbit approaching the origin in the direction , a unique orbit leaving the origin in as , which are exactly the positive -axis and the negative -axis, respectively. And for and , we can check that .
Applying the Briot-Bouquet transformation , , we can change system (18) into the following form:

where . We need to investigate the origin of system (21) which is degenerate. In polar coordinates and , we have

for system (21). The equation has exactly six real roots , , , , , and when , ,, , , , and when , and we can check that

By Theorem 3.7 of [24, Chapter 2] system (21) has a unique orbit approaching the origin in the direction , a unique orbit leaving the origin in , a unique orbit approaching the origin in , and a unique orbit leaving the origin in as , which are exactly the positive -axis, the negative -axis, the positive -axis, and the negative -axis, respectively. And for and when or and when , we can check that .
Applying the Briot-Bouquet transformation , , we can change system (21) into the following form:

where . One can check that system (24) has exactly two equilibria and on the -axis, and we only need to investigate the qualitative properties of which corresponds to the directions and when or and when , of system (21). Applying the transformation , , which translates the equilibrium to the origin, for simplicity, we denote and by and , respectively, and system (24) can be written into the form

and we only need to analyze the qualitative properties of the origin of system (25).
Applying the transformation , , and , for simplicity, we denote and by and , respectively, and system (25) can be written as

and we only need to analyze the qualitative properties of the origin of system (26).
When , there exists a function

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

By Theorem 7.1 in [24, Chapter 2], we obtain that when , the origin of system (26) is an unstable node; we obtain that when , the origin of system (26) is a stable node. So, according to the method of the Briot-Bouquet transformation, the theorem of is proved. Based on the proof of , we can also use the same method to get the same result of .
When , some difficulties are caused when we discuss orbits in the directions , because , which does not match any conditions of the theorems in references, e.g., [24]. 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 in .
From in (16), two horizontal isoclines are determined near : one is and the other is . Furthermore, let

where and is closed to zero.
Case 1. . Notice that there are no vertical isoclines near in (16). We claim that the open sector is a GNS in class I. In fact, we have between and and between and . So in the closure cl. Therefore, what we claim is proved by the definition of GNS. Lemma 1 in [26] guarantees that system (16) has infinitely many orbits in connection with (actually leaving from) in . If and , we notice that there are no vertical isoclines near in (16). Hence in and , we have and , respectively, implying that infinitely many orbits connect in the two sectors by Lemma 1 in [26]. If and , or , from in (16), we obtain vertical isoclines and , where is a sufficiently small constant. Obviously, is tangent to –axis at ; hence in and , we have and , respectively, implying that infinitely many orbits connect in the two sectors by Lemma 1 in [26].
Case 2. . Based on the proof of , we can also use the same method to get the same result of . We can give the three cases as shown in Figure 3.

5. Nonexistence and Uniqueness of Closed Orbits

Let us consider the Liénard system

in which and are continuous functions on satisfying locally Lipschitz condition. We assume that

Then the origin is the only critical point. Let ( may be ) and let

Then by (31), is strictly increasing. We denote the inverse function of by .

In article [27], Sugie and Hara gave the following condition on and under which system (30) has no periodic solutions except the origin.

Lemma 9 (see [27]). Suppose that

Then system (30) has no nonconstant periodic solutions.

Let be the inverse function of and , where ; (30) will be equations and in domains and , respectively.

where .

Lemma 10 (see [28]). Assume and are continuous functions in , for , , and verify

(1), for , , for ,(2) for ,(3) is non-decrease for ,(4)when for , we have . Then system (30) has at most one limit cycle in ; if it exists, it must be simple and stable.

Lemma 11. When , system (4) has no closed orbits; when , system (4) has a unique closed orbit.

Proof. We can easily compute

Obviously,

When , for . Therefore, system (4) has no closed orbits by Theorem 4.5 of [24, Chapter 2] 5 when .
When , we only discuss , since the proof of the case is reduced to that of the case by the transformations and .

(1). The equation has three roots , where , . We can get . Therefore, for and for . Because for , we can easily compute . When , we can get and is an increase function, and is also an increase function. Therefore, is non-decrease for . When for , we can get . Therefore, . So, system (30) has at most one limit cycle in ; if it exists, it must be simple and stable.(2). The proof of the case is reduced to that of the case by the transformations and .

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.

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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