# Existence of Limit Cycles for a Class of Quartic Polynomial Differential System Depending on Parameters

Authors: Sarah Abdullah Qadha; Muneera Abdullah Qadha; Haibo Chen
Journal: Journal of Mathematics
 (2021)

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## Abstract

We studied the existence of limit cycles for the quartic polynomial differential systems depending on parameters. To prove that, first, we used the formal series method based on Poincare’ ideas to determine the center-focus. Then, by the Hopf bifurcation theory, we obtained the sufficient condition for the existence of the limit cycles. Finally, we provided some numerical examples for illustration.


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## Body

## 1. Introduction

In the twentieth century, progress in applied electronics has been rapid. Physicists invented the triode vacuum tube, making stable self-excited oscillations of constant amplitude. Thus, propagating sound and pictures through electronics became possible. However, it was not possible to describe this oscillation phenomenon by linear differential equations. In [1] 1926, van der Pol first obtained a differential equation to describe oscillations of the constant amplitude of a triode vacuum tube, as follows:(1)x¨+μx2−1x˙+x=0.This was the main reason for the appearance of the nonlinear equations and qualitative theory. One of the main issues in analysing the qualitative theory of differential equations is the limit cycle. The limit cycle was discovered by Poincare’ (1881–1886). Recently, the problem of limit cycles has become increasingly critical and has attracted the attention of many pure and applied mathematicians, see for instance [1–10]. The necessary problems in the qualitative theory of differential equation include the existence of the limit cycle.Existence, nonexistence, uniqueness, and other properties of limit cycles have been studied extensively by mathematicians and physicists. Then, since the 1950s, many mathematical models in physics, engineering, chemistry, biology, and economics have been displayed as autonomous plane systems with limit cycles. Many researchers have studied the existence of limit cycles for the quadratic and cubic systems, and numerous research achievements have been obtained, see for instance [3, 5, 7, 10–18].For examples, consider the following system:(2)dxdt=Px,y,dydt=Qx,y,where P (x, y) and Q (x, y) denote homogeneous polynomial of degree n and real variable x, y and some parameter.Consider the following system:(3)dxdt=−y+Pnx,y,dydt=x+Qnx,y,where Pnx,y and Qnx,y are polynomials of degree n, with real coefficients without constant terms. System (3) always has either a center or fine (weak) focus at the origin. Distinguishing between the center and a focus at the origin is the main problem.We know that linear differential systemdx/dt=−y,dy/dt=x can have the center. However, the perturbation of this center inside the class of linear differential systems cannot produce a limit cycle because the linear differential system cannot have an isolated periodic solution in the set of all periodic solutions:(1)
Whenn = 2, system (3) is the quadratic polynomial differential system whose centers have been completely classified. Many papers study how many limit cycles can bifurcate from the periodic orbits of these centers. As an example, consider the following system:(4)dxdt=−y1+x+αy,dydt=x+λ+β+γy+ax2+α+β+γxy+cγy2.In [19], the authors obtained a quadratic system that has at most four limit cycles and only in the (3 : 1) distribution when a=1/2,c=−1.(2)
Whenn = 3, system (3) is a cubic system.In [20], Malkin obtained the necessary and sufficient conditions for system (3) for the origin to be the center. In [21], the authors proved that these conditions were satisfactory for the integrability of the system. In [22], Liu derived the formulas of the first five focus values of the origin for system (3) and proved that the origin is a center if the five focus values are all zero. This finding implies that a central symmetric cubic system with respect to the origin has at most five small amplitude limit cycles around the origin (see [23]).Consider the following system:(5)dxdt=−y1−ax+1−bx+a1x+a2x2+a3x3,dydt=x1−cx1−dx.In [24], Quan obtained the sufficient and necessary condition for the existence of a limit cycle in the case a2=0,a=c,andb=d. The authors of [25] studied the condition for the existence of limit cycles in the system when a=0 and b=d. One article [26] showed that the case a=candb=d=0 has no limit cycle.In this paper, we will study the existence of limit cycles for the following quartic polynomial differential systems depending on parameters:(6)dxdt=−y+kx2+lxy,dydt=x+mx2y+ny4,where k,l,m,andn are parameters, in which the linear system is the center. If the singular point for nonlinear is the center, quartic system (6) does not have a limit cycle. If the singular point for nonlinear systems is the focus or node, then quartic system (6) has the limit cycle. System (6) only has one singular point on the origin. It does not have any singular point on the neighborhood; so, the origin (the singular point) is isolated.In the first section of this paper, we presented some necessary definitions and theorem. In the second section, we used a formal series method based on Poincare’ ideas to determine the center-focus. By using the Hopf bifurcation theory, we obtained the sufficient condition for the existence of limit cycle for quartic system (6), which bifurcated from the equilibrium point (singular point). We also obtained sufficient conditions for the existence of the limit cycle according to the change analysis of the stability of the focus when the parameters change. Finally, we provided some examples for illustration.


## 2. Some Preliminary Results

In this section, we introduce some definitions and notations that we will need for the existence of the quartic differential system (6).Theorem 1 (see [1]).
Closed trajectories of different vector fields of a complete family do not intersect.Theorem 2 (Bendixson’s criterion) (see [27]).
IfPx,Qy is continuous in R which is simply connected and divF⟶=∂P/∂x+∂Q/∂y≠0 at any point of R, then the system has no closed trajectories inside R.Definition 1.
(Hopf bifurcation of orderk) (see [28], page 385). Consider the following system:(7)dxdt=Px,y,η,dydt=Qx,y,η,where x,y∈ℝ1,η∈ℝn,andP,Q∈C∞, when two eigenvalues λ1,2=aη±ibη of the linear part of the system above (7) cross the imaginary axis. We say that system (7) has a Hopf bifurcation of order k (k > 1) at the origin if(1)
a0=0(2)
b0=b0≠0(3)
ReC1=ReC2=⋯=ReCk−1=0,ReCk≠0where C1,…,Ck are the coefficients of Poincaré–Lyapunov constants.
In this case, we also say that the origin is a weak focus of orderk.
Note: In two dimensions, when the focus Point switches from stable to unstable (or vice versa), then a periodic solution appears and a Hopf bifurcation occurs. [29].Theorem 3 (see [28], page 385).
If we let system (2) be a C∞ with an equilibrium (0, 0), that is, a weak focus of order k, then(1)
Ifn≥k, and 5η=0=2, then there is δ>0 and a neighborhood Γ of x,y=0,0 exists. η<δ,5 has at most k limit cycles in neighborhood Γ(2)
For any integeri, 1≤i≤k and a neighborhood Γ∗⊂Γ of x,y=0,0, there exists a system of the form 5η=0 with 5η=0=2, and a number δ∗>0, such that 5η=0 has exactly i limit cycles in Γ∗ for η∈S, where S is an open subset of (η/0<η<δ∗) and 0∈S¯Remark 1.
(see [27]). The linear part of system (6) has closed trajectories because the linear system has TrA=PxA+QyA=0 and detA=PxAQyA−PyAQxA=1>0 where A is a singular point for system. λ1andλ2 are two conjugate pure imaginary roots. Then, the linear system (6) can be transformed into the following:(8)rr˙=xx˙+yy˙,thenr˙=0,r2θ˙=xy˙−yx˙,thenθ˙=1,whose solution is as follows:(9)r=r0≥0,θ=t+θ0.
The orbit form has a family of closed curves around the origin A. The singular point is called a center. The simulation is presented in Figure1.Figure 1
Center.


## 3. Main Results

### 3.1. Determine the Center-Focus and It Is Stable

Theorem 4.
For system (6), we have the following:(1)
If2kl−m≠0,kl≠0,m+kl/4<0,thenA0,0 is an unstable first-order weak focus(2)
If2kl−m≠0,kl≠0,m+kl/4>0,thenA0,0 is a stable first-order weak focus(3)
If2kl−m=0,kl=0,then the pointA0,0 is the center point(4)
Ifkl=0, 2kl−m>0,then  the pointA0,0 is an unstable second-order weak focus(5)
Ifkl=02kl−m<0,then  the pointA0,0 is a stable second-order weak focus(6)
If2kl−m=0,kl<0,then the pointA0,0 is an unstable third-order weak focus(7)
If2kl−m=0,kl>0,then the pointA0,0 is a stable third-order weak focusProof.
By using the formal series method based on Poincare’ ideas, we can easily obtain these results of this theorem. The details are omitted here.


### 3.2. The Nonexistence of Limit Cycle

Theorem 5.
If2kl−m=0,kl=0, then system (6) has no limit cycle in the whole plane.Proof.
When the conditions above hold,then the pointA0,0 is the center point. And, when kl=0,m=0, then P−x,y=Px,y,Q−x,y=−Qx,y. The vector field Px,y,Qx,y is symmetrical with respect to y-axis. So, there is no limit cycle in the neighborhood of point A0,0 when the conditions of Theorem 5 hold. The simulation is presented in Figure 2.Figure 2
Nonexistence of limit cycle for system (6).


### 3.3. Existence of Limit Cycles

Theorem 6.
When one of the following conditions holds, system (6) has at least one limit cycle around A0,0:(1)
kl+m/4<0,kl≠0,2kl−m≠0, −1≪m<0(2)
kl+m/4>0,kl≠0, 2kl−m≠0,0<m≪1(3)
2kl−m>0,kl=0, −1≪m<0(4)
2kl−m<0,kl=0,0<m≪1(5)
m−2kl=0,kl<0, −1≪m<0(6)
m−2kl=0,kl>0,0<m≪1Proof.
For conditions (1), (3), and (5), the singular point A0,0 is an unstable weak focus point of system (6). When m decreases from zero, the singular point A0,0 of system (6) changes from an unstable weak focus to a stable weak focus. According to the Hopf bifurcation theory, these parameters change because when the stability changes, system (6) generates at least one unstable limit cycle in the neighborhood of A0,0.
For condition (2), (4), and (6), the singular point A0,0 is the stable weak focus point of system (6). When m increases from zero, the singular point A0,0 of system (6) changes from a stable weak focus to an unstable weak focus, and according to the Hopf bifurcation theory, under these parameters, system (6) generates at least one stable limit cycle in the neighborhood of A0,0.Theorem 7.
System (6) has at least three limit cycles bifurcating from the origin A0,0Proof.
From Theorem4, we determine that system (6) has three weak focuses. We denote the number of weak focus as k, and k=3. System (6) is a class quartic; thus, we denote n as n=4 According to Theorem 3, we determine that system (6) has three limit cycles around A0,0


### 3.4. Numerical Solution

Example 1.
In system (6), if we take the values of the parameters as k=1, l=1, m=1, and n=1, then a limit cycle exists. The simulation is presented in Figure 3.Figure 3
Existence of limit cycle for system (6), when k = 1, l = 1, m = 1, n = 1.Example 2.
(the existence). In system (6), if we take values of the parameters as k=0.2,l=−0.2,m=−0.06,andn=1, then by Theorem 6 Case 1, there exists a limit cycle around A0,0. The simulation is presented in Figure 4.
In system (6), if we take values of the parameters as k=−1,l=1,m=1,andn=1, then by Theorem 6 Case 2, there exists a limit cycle around A0,0. The simulation is presented in Figure 5.
In system (6), if we take values of the parameters as k=1,l=0,m=−0.2,andn=1, then by Theorem 6 Case 3, there exists a limit cycle around A0,0. The simulation is presented in Figure 6.
In system (6), if we take values of the parameters as k=−1,l=0,m=1,andn=1, then by Theorem 6 Case 4, there exists a limit cycle around A0,0. The simulation is presented in Figure 7.
In system (6), if we take values of the parameters as k=0.25,l=−0.25,m=−0.125andn=1, then by Theorem 6 Case 5, there exists a limit cycle around A0,0. The simulation is presented in Figure 8.
In system (6), if we take values of parameters as k=−0.25,l=−0.25,m=0.125,andn=1, then by Theorem 6 Case 6, there exists a limit cycle around A0,0. The simulation is presented in Figure 9.Figure 4
Existence of limit cycle of system (6), when k=0.2,l=−0.2,m=−0.06,andn=1.Figure 5
Existence of limit cycle of system (6), when k=−1,l=1,m=1,andn=1.Figure 6
Existence of limit cycle of system (6), when k=1,l=0,m=−0.2,andn=1.Figure 7
Existence of limit cycle of system (6), when k=−1,l=0,m=1,andn=1.Figure 8
Existence of limit cycle of system (6), when k=0.25,l=−0.25,m=−0.125,andn=1.Figure 9
Existence of limit cycle of system (6), when k=−0.25,l=−0.25,m=0.125,andn=1.


## 3.1. Determine the Center-Focus and It Is Stable

Theorem 4.
For system (6), we have the following:(1)
If2kl−m≠0,kl≠0,m+kl/4<0,thenA0,0 is an unstable first-order weak focus(2)
If2kl−m≠0,kl≠0,m+kl/4>0,thenA0,0 is a stable first-order weak focus(3)
If2kl−m=0,kl=0,then the pointA0,0 is the center point(4)
Ifkl=0, 2kl−m>0,then  the pointA0,0 is an unstable second-order weak focus(5)
Ifkl=02kl−m<0,then  the pointA0,0 is a stable second-order weak focus(6)
If2kl−m=0,kl<0,then the pointA0,0 is an unstable third-order weak focus(7)
If2kl−m=0,kl>0,then the pointA0,0 is a stable third-order weak focusProof.
By using the formal series method based on Poincare’ ideas, we can easily obtain these results of this theorem. The details are omitted here.


## 3.2. The Nonexistence of Limit Cycle

Theorem 5.
If2kl−m=0,kl=0, then system (6) has no limit cycle in the whole plane.Proof.
When the conditions above hold,then the pointA0,0 is the center point. And, when kl=0,m=0, then P−x,y=Px,y,Q−x,y=−Qx,y. The vector field Px,y,Qx,y is symmetrical with respect to y-axis. So, there is no limit cycle in the neighborhood of point A0,0 when the conditions of Theorem 5 hold. The simulation is presented in Figure 2.Figure 2
Nonexistence of limit cycle for system (6).


## 3.3. Existence of Limit Cycles

Theorem 6.
When one of the following conditions holds, system (6) has at least one limit cycle around A0,0:(1)
kl+m/4<0,kl≠0,2kl−m≠0, −1≪m<0(2)
kl+m/4>0,kl≠0, 2kl−m≠0,0<m≪1(3)
2kl−m>0,kl=0, −1≪m<0(4)
2kl−m<0,kl=0,0<m≪1(5)
m−2kl=0,kl<0, −1≪m<0(6)
m−2kl=0,kl>0,0<m≪1Proof.
For conditions (1), (3), and (5), the singular point A0,0 is an unstable weak focus point of system (6). When m decreases from zero, the singular point A0,0 of system (6) changes from an unstable weak focus to a stable weak focus. According to the Hopf bifurcation theory, these parameters change because when the stability changes, system (6) generates at least one unstable limit cycle in the neighborhood of A0,0.
For condition (2), (4), and (6), the singular point A0,0 is the stable weak focus point of system (6). When m increases from zero, the singular point A0,0 of system (6) changes from a stable weak focus to an unstable weak focus, and according to the Hopf bifurcation theory, under these parameters, system (6) generates at least one stable limit cycle in the neighborhood of A0,0.Theorem 7.
System (6) has at least three limit cycles bifurcating from the origin A0,0Proof.
From Theorem4, we determine that system (6) has three weak focuses. We denote the number of weak focus as k, and k=3. System (6) is a class quartic; thus, we denote n as n=4 According to Theorem 3, we determine that system (6) has three limit cycles around A0,0


## 3.4. Numerical Solution

Example 1.
In system (6), if we take the values of the parameters as k=1, l=1, m=1, and n=1, then a limit cycle exists. The simulation is presented in Figure 3.Figure 3
Existence of limit cycle for system (6), when k = 1, l = 1, m = 1, n = 1.Example 2.
(the existence). In system (6), if we take values of the parameters as k=0.2,l=−0.2,m=−0.06,andn=1, then by Theorem 6 Case 1, there exists a limit cycle around A0,0. The simulation is presented in Figure 4.
In system (6), if we take values of the parameters as k=−1,l=1,m=1,andn=1, then by Theorem 6 Case 2, there exists a limit cycle around A0,0. The simulation is presented in Figure 5.
In system (6), if we take values of the parameters as k=1,l=0,m=−0.2,andn=1, then by Theorem 6 Case 3, there exists a limit cycle around A0,0. The simulation is presented in Figure 6.
In system (6), if we take values of the parameters as k=−1,l=0,m=1,andn=1, then by Theorem 6 Case 4, there exists a limit cycle around A0,0. The simulation is presented in Figure 7.
In system (6), if we take values of the parameters as k=0.25,l=−0.25,m=−0.125andn=1, then by Theorem 6 Case 5, there exists a limit cycle around A0,0. The simulation is presented in Figure 8.
In system (6), if we take values of parameters as k=−0.25,l=−0.25,m=0.125,andn=1, then by Theorem 6 Case 6, there exists a limit cycle around A0,0. The simulation is presented in Figure 9.Figure 4
Existence of limit cycle of system (6), when k=0.2,l=−0.2,m=−0.06,andn=1.Figure 5
Existence of limit cycle of system (6), when k=−1,l=1,m=1,andn=1.Figure 6
Existence of limit cycle of system (6), when k=1,l=0,m=−0.2,andn=1.Figure 7
Existence of limit cycle of system (6), when k=−1,l=0,m=1,andn=1.Figure 8
Existence of limit cycle of system (6), when k=0.25,l=−0.25,m=−0.125,andn=1.Figure 9
Existence of limit cycle of system (6), when k=−0.25,l=−0.25,m=0.125,andn=1.


## 4. Conclusions

In this paper, we studied the existence of the limit cycle and used the formal series method to determine the center-focus point. By the Hopf bifurcation theory, we obtained the sufficient condition for the existence of the limit cycles for this system, which bifurcate from the equilibrium point. We provided some examples for illustration.



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