Global Analysis of a Liénard System with Quadratic Damping

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 systemwith closed orbits are obtained.The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.


Introduction and Main Results
Liénard equations have a very wide application in many areas, such as mechanics, electronic technology, and modern biology; see [1][2][3][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][6][7][8][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][12][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.
Conjecture 1.If () has degree , then (1) has at most [/2] limit cycles ([ /2] is the integer part of /2,  ≥ 2). = 2 is proved by [15];  = 3 is proved by [16].The problem for  > 3 is still open.In 1988, Lloyd and Lynch [17] considered the similar problem for generalized Liénard equations where F is a polynomial of degree +1 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 (2, 1) = 1.In [19][20][21][22], it was proved that (3, 1) = 1.
Up to now, as far as we know, only these three cases have been completely investigated.
Consider the Liénard equations where () =  3 +  2 + ,  ̸ = 0 and  ∈ N. We only discuss  > 0, because the case  < 0 can be derived from the case  > 0 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.
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.

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  > 0, the elliptic sector lies in the negative -axis; if  < 0, the elliptic sector lies in the positive -axis.
(A) Global phase portraits of  = 0: there exist infinite critical points  and .
(1) Suppose  < 0 and  > 0. A unique stable limit cycle appears around the equilibrium  of system (4).If  ≤ −2,  is an unstable node, and the global phase portrait is shown in Figure 1(a); if −2 <  < 0,  is an unstable focus, and the global phase portrait is shown in Figure 1(b).(2) Suppose  > 0 and  > 0. There are no closed orbits in system (4).If 0 ≤  < 2 and  > 0,  is a stable focus, and the global phase portrait is shown in Figure 1(c); if  ≥ 2 and  > 0,  is a stable node, and the global phase portrait is shown in Figure 1(d).
(3) Suppose  < 0 and  < 0. There are no closed orbits in system (4).If  ≤ −2,  is an unstable node, and the global phase portrait is shown in Figure 2(a); if −2 <  < 0,  is an unstable focus, and the global phase portrait is shown in Figure 2(b).(4) Suppose  > 0 and  < 0. A unique unstable limit cycle appears around the equilibrium  of system (4).If 0 ≤  < 2 and  > 0,  is a stable focus, and the global phase portrait is shown in Figure 2(c); if  ≥ 2 and  > 0,  is a stable node, and the global phase portrait is shown in Figure 2(d).
(B) Global phase portraits of  = 1: there exist infinite critical points  1 and .
(1) Suppose  > 0 or  =  = 0, and  > 0. 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  > 0 and  < 0. 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  < 0 or  =  = 0, and  < 0. 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  < 0 and  > 0. 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  = 0 and  ̸ = 0.There are no closed orbits in system (4).If  > 0, the elliptic sector lies in the positive -axis, and the global phase portraits are shown in Figures 1(g) and 2(g); if  < 0, 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  ≥ 2: there exists a unique infinite critical point .
(1) Suppose  > 0 or  =  = 0, and  > 0. 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  > 0 and  < 0. 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  < 0 and  > 0. 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  < 0 or  =  = 0, and  < 0. 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  = 0 and  ̸ = 0.There are no closed orbits in system (4).If  > 0, 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  < 0, the elliptic sector lies in the negative -axis, and the global phase portraits are shown in Figures 1(l) and 2(l).
Figure 1: The global phase portraits of system (4) as the parameter  > 0.
The global phase portraits of system (4) as the parameter  < 0.

Lemma 3.
The type of equilibrium  in system ( 4) is shown as Table 1.
Therefore, we obtain the following lemma.Lemma 4. When  > 0 and  < 0, the equilibrium  of system ( 4) is an unstable weak focus with multiplicity 1, and there is a unique stable limit cycle bifurcating from ; when  < 0 and  > 0, the equilibrium  of system ( 4) is a stable weak focus with multiplicity 1, and there is a unique unstable limit cycle bifurcating from ; when  > 0 and  > 0, the equilibrium  of system ( 4) is an unstable weak focus with multiplicity 1, and there are no closed orbits near ; when  < 0 and  ≤ 0, the equilibrium  of system ( 4) is a stable weak focus with multiplicity 1, and there are no closed orbits near .

Degenerate Bogdanov-Takens Bifurcation.
In another case  ≥ 1 and  ̸ = 0, 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  = 1, we get Let the second equation of (8) equal zero, and we solve that ) by the Implicit Function Theorem.Substituting ỹ of the first equation of ( 8) by  1 ( x), we obtain that When  > 0,  is a stable degenerate node; when  < 0,  is an unstable degenerate node.
In the remaining case that  ≥ 1 and  = 0, 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  = 0 and  > 0,  is a stable degenerate node; when  = 0 and  < 0,  is an unstable degenerate node.
When  ̸ = 0, 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].
Proof.When  < 0, 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  = 0, respectively, with  = 0 and  = 1.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.

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  = 1/,  = /, system (4) can be rewritten as where  = / 2 and  = 0.
where  = / 2 and  ≥ 1.We only need to study the equilibrium  : (0, 0) of systems ( 15) and ( 16), which corresponds to an equilibrium   of system (4) at infinity on the -axis.
Lemma 7. Equilibria  and  1 are unstable nodes when  > 0 and stable nodes when  < 0.
System (16) provides an interesting example for highly degenerate equilibria when  is greater than 1.As  is unspecified, the lowest degree of nonzero terms in ( 16) is 2.One could not use the blowing-up methods as done in [24] 2 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  = 0, 1 and  > 0, there are infinite orbits approaching  : (0, 0) in two directions  = , there is a unique orbit approaching  : (0, 0) in two directions  = 0, and there are infinite orbits leaving  : (0, 0) in two directions  = 0; when  = 0, 1 and  < 0, there are infinite orbits leaving  : (0, 0) in two directions  = 0, there is a unique orbit leaving  : (0, 0) in two directions  = , and there are infinite orbits approaching  : (0, 0) in two directions  = ; when  ≥ 2, there are infinite orbits leaving  : (0, 0) in two directions  = 0, and there are infinite orbits approaching  : (0, 0) in two directions  = .
As in [24], except in these exceptional directions, no orbits connect .
Applying the Briot-Bouquet transformation V 2 = V 2 ,  1 =  3 V 2 , we can change system (21) into the following form: where  1 = V 2 .One can check that system (24) has exactly two equilibria (0, 0) and (0, ) on the  3 -axis, and we only need to investigate the qualitative properties of (0, ) which corresponds to the directions  = arctan  and  =  + arctan  when  > 0 or  =  − arctan(−) and  = 2 − arctan(−) when  > 0, of system (21).Applying the transformation V 2 = V 2 ,  3 =  3 − , which translates the equilibrium (0, ) to the origin, for simplicity, we denote V 2 and  3 by V 2 and  3 , 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 V  2 = V 2 ,   3 = V 2 −  3 , and  2 = − 1 , for simplicity, we denote V  2 and   3 by V 2 and  3 , respectively, and system (25) can be written as and we only need to analyze the qualitative properties of the origin of system (26).When  2 +  2  ̸ = 0, 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  > 0, the origin of system ( 26) is an unstable node; we obtain that when  < 0, the origin of system ( 26) is a stable node.So, according to the method of the Briot-Bouquet transformation, the theorem of  = 0 is proved.Based on the proof of  = 0, we can also use the same method to get the same result of  = 1.
When  > 1, some difficulties are caused when we discuss orbits in the directions  = 0, , because   1 (0) =  1 (0) = 0, 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  = 0, .
Case 2.  > 0. Based on the proof of  < 0, we can also use the same method to get the same result of  > 0. We can give the three cases as shown in Figure 3.

Nonexistence and Uniqueness of Closed Orbits
Let us consider the Liénard system Then by (31), () is strictly increasing.We denote the inverse function of () by  −1 ().

Table 1 :
Qualitative properties of equilibria .