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

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.


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. us, 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: is 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. e limit cycle was discovered by Poincare' (1881)(1882)(1883)(1884)(1885)(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][2][3][4][5][6][7][8][9][10]. e 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. en, 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][11][12][13][14][15][16][17][18].
For examples, consider the following system: dx dt � P(x, 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: dx dt � −y + P n (x, y), where P n (x, y) and Q n (x, 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 system (dx/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) When n � 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: 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.
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. is 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: In [24], Quan obtained the sufficient and necessary condition for the existence of a limit cycle in the case e 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 � c and b � 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: dx dt � −y + kx 2 + lxy, where k, l, m, and n 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.

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]). If P x , Q y is continuous in R which is simply connected and div F → � (zP/zx) + (zQ/zy) ≠ 0 at any point of R, then the system has no closed trajectories inside R. Definition 1 (Hopf bifurcation of order k) (see [28], page 385). Consider the following system: where x, y ∈ R 1 , η ∈ R n , and P, Q ∈ C ∞ , when two eigenvalues λ 1,2 � a(η) ± ib(η) 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 where C 1 , . . . , C k are the coefficients of Poincaré-Lyapunov constants.
In this case, we also say that the origin is a weak focus of order k.
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].

Journal of Mathematics
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) If n ≥ k, and (5) η�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 integer i, 1 ≤ i ≤ k and a neighborhood Γ * ⊂ Γ of (x, y) � (0, 0), there exists a system of the form (5) η�0 with (5) η�0 � (2), and a number δ * > 0, such that (5) η�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]). e linear part of system (6) has closed trajectories because the linear system has Tr where A is a singular point for system. λ 1 and λ 2 are two conjugate pure imaginary roots. en, the linear system (6) can be transformed into the following: whose solution is as follows: e orbit form has a family of closed curves around the origin A. e singular point is called a center. e simulation is presented in Figure 1.

Determine the Center-Focus and It Is Stable
Theorem 4. For system (6), we have the following: is an unstable first-order weak focus is a stable first-order weak focus

Existence of Limit Cycles
Theorem 6. When one of the following conditions holds, system (6) has at least one limit cycle around A(0, 0): A(0, 0) is an unstable weak focus point of system (6). When m decreases from zero, the singular point A(0, 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 A(0, 0).

Proof. For conditions (1), (3), and (5), the singular point
For condition (2), (4), and (6), the singular point A(0, 0) is the stable weak focus point of system (6). When m increases from zero, the singular point A(0, 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 A(0, 0). □ Theorem 7. System (6) has at least three limit cycles bifurcating from the origin A(0, 0)

Journal of Mathematics
Proof. From eorem 4, 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 eorem 3, we determine that system (6) has three limit cycles around A(0, 0) □

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. e simulation is presented in Figure 3.
In system (6), if we take values of the parameters as k � −1, l � 1, m � 1, and n � 1, then by eorem 6 Case 2, there exists a limit cycle around A(0, 0). e simulation is presented in Figure 5.
In system (6), if we take values of the parameters as k � 1, l � 0, m � −0.2, and n � 1, then by eorem 6 Case 3, there exists a limit cycle around A(0, 0). e simulation is presented in Figure 6.
In system (6), if we take values of the parameters as k � −1, l � 0, m � 1, and n � 1, then by eorem 6 Case 4, there exists a limit cycle around A(0, 0). e simulation is presented in Figure 7.
In system (6), if we take values of the parameters as k � 0.25, l � −0.25, m � −0.125 and n � 1, then by eorem 6 Case 5, there exists a limit cycle around A(0, 0). e simulation is presented in Figure 8.

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.

Data Availability
All data required for this paper are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.   Journal of Mathematics 5