Periodic Solutions for a Four-Dimensional Coupled Polynomial System with N-Degree Homogeneous Nonlinearities

This paper studies the periodic solutions of a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities of which the unperturbed linear system has a center singular point in generalization resonance n : 1 at the origin. Considering arbitrary positive integers n and N with N n ≤ and 2 ≥ N , the new explicit expression of displacement function for the four-dimensional system is detected by introducing the technique on power trigonometric integrals. Then some precise and detailed results in comparison with the existing works, including the existence condition, the exact number, and the parameter control conditions of periodic solutions, are obtained, which can provide a new theoretical description and mechanism explanation for the phenomena of emergence and disappearance of periodic solutions. Results obtained in this paper improve certain existing results under some parameter conditions and can be extensively used in engineering applications. To verify the applicability and availability of the new theoretical results, as an application, the periodic solutions of a circular mesh antenna model are obtained by theoretical method and numerical simulations.


Introduction
Many problems in the fields of engineering and science can be described by nonlinear polynomial systems. Due to interaction between different variables, these systems often exhibit complicated dynamic characteristics and bifurcation behaviors [1][2][3]. One of the important ingredients for describing the dynamic behaviors of nonlinear systems is the bifurcation of periodic solutions, which is closely related to the second part of Hilbert's 16th problem [4]. Many scholars have done a lot of work in recent years, and some meaningful results for one-dimensional and planar systems have been obtained [5][6][7][8][9]. With the development of science and technology, the study of one-dimensional and planar systems cannot satisfy the need of practical applications. Hence, it is urgent to study the bifurcation of periodic solutions for high-dimensional polynomial systems. However, due to the complexity of geometric structure and numerical calculation of high-dimensional nonlinear polynomial systems, research on the bifurcation of periodic solutions is much more sophisticated than the one-dimensional and planar systems.
Up to now, some contributions have been made in the bifurcation theory of periodic solutions of high-dimensional nonlinear polynomial systems. Various classical and effective methods, such as the Poincaré map [10], the Melnikov method [11], the harmonic balance method [12], and the averaging method [13], were proposed to detect the periodic solutions. Further study on the bifurcation of

Preliminaries
Consider a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities of which the unperturbed linear system has a center singular point in generalization resonance 1 : n as follows, where ε > 0 is a small parameter, A, A ∈ R 4×4 , A k , A k ∈ R 2×2 , A k = 0 −n k−1 n k−1 0 , A k = (a 2k−1 , a 2k ) T , a k = (a k 1000 , a k 0100 ), a j = (a j 0010 , a j 0001 ), k = 1, 2, j = 3, 4, F = (F 1 , F 2 , F 3 , F 4 ) T , F i (x) = and [·] denotes the integer part. We are concerned with the existence condition, exact number, and parameter control conditions of periodic solutions of System (1). We all know that when N = 1, System (1) has no periodic solution. Hence, the degree of the nonlinear terms of System (1), N ≥ 2, will be considered in this paper.
In this section, we will present some important lemmas on the transformations of System (1) and the exact formulas of power trigonometric integrals as preliminaries.
Rescaling System (1), the following lemma can be obtained: By introducing the scale transformation System (1) can be rewritten as where G(x) = Ax + F(x) ∈ R 4 .

Proof.
Rescaling the variables of System (1) by Transformation (3), we obtain which can be reduced to . x = (A + ε A)x + εF(x).
This proof is completed.
Considering θ as a new independent variable, we obtain a non-autonomous system with the form dy dθ = εH(θ, y) This proof is completed.
(i) The expression J K 1 (s). Considering the introduced notation J K 1 (s), together with the above analysis, we obtain The exact expression of J K 1 (s) will be discussed based on the parity of λ 2 .

(a)
When λ 2 is odd, When λ 2 is even, This proof is completed.

Periodic Solutions of a Four-Dimensional Nonlinear System
In this section, the existence, exact number, and parameter control conditions of the periodic solutions of System (1) are investigated by detecting the new explicit expression of the displacement function based on the Poincaré map.

Displacement Function
System (1) is transformed into System (6) in Section 2, which implies that the periodic solutions of System (1) can be obtained by considering a Poincaré map for System (6). Denote by y(θ, z, ε) the solution of System (6) with the initial condition y(0, z, ε) = z, where θ ∈ S 1 , z = (r 0 , ρ 0 , s 0 ) T ∈ R 2 ×S 1 . Define a global cross section to vector field (6) by Note that H(θ, y), shown in System (6), is continuous and 2π periodic with respect to variable θ. and define the displacement function as In what follows, we will detect the expression of displacement function.
We note that a zero solution of the displacement function corresponds to a periodic solution of System (6), which implies that studying the exact expression of Φ(z, ε) is crucial for our investigation.

The Expression of h(z)
In this subsection, we study the important term, h(z), of the displacement function Φ(z). Considering System (6) and Lemma 3, the following lemma can be obtained.

Proof.
Writing Next, we discuss the exact expressions of h 1 i (z) and h N i (z).
It is remarkable that the explicit expression h(z) of System (1) for arbitrary positive integers n and N with n ≤ N and N ≥ 2 is obtained by introducing the technique on the exact formulas of power trigonometric integrals, which is new and plays an important role in detecting the exact number and parameter control conditions of the periodic solutions of System (1). Note that when n = N, we obtain µ 23 = µ 11 , µ 21 = −µ 13 based on Lemma 5, then the exact expression ϕ 3 (z) is of the form which shows that the form of expression h(z) in this case is in agreement with certain existing result in the previous literature [27], but we obtain all the coefficients in this paper. The result obtained in this section can help to provide more detailed bifurcation information about System (1) by considering the complex coefficients.

Periodic Solutions
In this subsection, we study the existence condition, exact number, and the parameter control conditions of the periodic solutions of System (1) by supposing b 1 b 2 0. Theorem 1. Based on the expression of h(z), the following statements hold for ε > 0 sufficiently small.
System (1) has a periodic solution (2) The number of periodic solutions of System (1) can be provided by the number of solutions of h(z) = 0 that satisfy statement (1) and s * 0 ∈ (0, 2π].
Proof. We prove Theorem 1 by the following two steps.
there exists a unique vector function (z * (ε), ε) in the neighborhood of (z * , 0) such that Φ(z * (ε), ε) = 0 for ε > 0 sufficiently small by the implicit function theorem. Hence, based on the definition of the Poincaré map, System (6) has a periodic solution in the neighborhood of z * .
(2) A solution of h(z) = 0, which satisfies Statement (1), provides a periodic solution of System (1). Hence, the number of periodic solutions of System (1) can be obtained by discussing the number of solutions of h(z) = 0 that satisfy Statement (1) and s * 0 ∈ (0, 2π].
This proof is completed.
Theorem 1 provides a sufficient condition for analyzing the existence and number of periodic solutions of System (1). Next, we discuss the exact number of periodic solutions and the parameter control conditions based on Theorem 1. (1), there is no periodic solution when N + n is odd and b 1 b 2 0 based on the displacement function of order ε 0 .

Theorem 2. For System
Proof. When N + n is odd, based on Lemma 5, we have Since b 1 b 2 0, there is no solution that satisfies (r 0 ) 2 + (ρ 0 ) 2 0 for h(z) = 0. Hence, there is no periodic solution for System (1) in this case and Theorem 2 holds. This proof is completed.
Next, we discuss the number and parameter control conditions of the periodic solutions of System (1) when N + n is even. For convenience, we introduce some notations: and some important sets:  Denote 1 (U i ) as the elements number of set U i (i = 1, 2) and write Denoting as the number of periodic solutions of System (1), the following result can be obtained.

Theorem 3.
Considering System (1) and supposing P Q 3 , when N + n is even and b 1 b 2 0, the following statements hold for ε > 0 sufficiently small based on the displacement function of order ε 0 .
Proof. When N + n is even, the exact expression of h(z) can be rewritten as: where z = (r 0 , ρ 0 , u, v), u = sin(ns 0 ), v = cos(ns 0 ). Next, we discuss the number of periodic solutions of System (1) by considering the real solutions of h( z) = 0 based on the following two cases, where (1) If µ 11 = µ 21 = 0, i.e., P ∈ P 31 , the equation h( z) = 0 has no solution with respect to z since b 1 b 2 0, and we obtain = 0 in this case. If µ 12 = µ 22 = 0, i.e., P ∈ P 32 , now P ∈ Q 3 , and we will not discuss the periodic solutions of System (1) in this case.
(2) If P P 31 ∪ P 32 , equation h( z) = 0 has real solutions with respect to z only in the case of r 0 ρ 0 0.
Hence, based on Equation (18), the following equations can be obtained: and where the expressions of B 1 , B 2 , and B 3 are shown in (15). Based on Equation (19), the relationship between u and v can be obtained, which reveals the value of = n if the Jacobian of h(z) is nonzero at the solution. Next, we discuss the periodic solutions of system (1) by considering the solutions of h( z) = 0 based on the following two cases: Case 1. P ∈ P 0 ∪ P 11 ∪ P 12 ∪ P 21 . The periodic solutions of System (1) will be discussed by the following three subcases: (1) If µ 12 µ 21 − µ 11 µ 22 = 0, which only exists for P ∈ P 0 , then .
The number of solutions of h( z) = 0 that satisfy r 0 , ρ 0 > 0 is one for l 1 > 0 and zero for l 1 < 0.
In summary, together with the above cases, we can obtain the fact that (P 22 ∪ P 31 ) ∩ W 11 = ∅, which implies Γ 53 = Γ 5 and Γ 61 = Γ 6 . Hence, the exact number of the periodic solutions of System (1) and the parameter control conditions can be obtained as shown in Theorem 3.
This proof is completed.
In fact, the upper bounds of the number of periodic solutions for a four-dimensional system with all the N-degree homogeneous nonlinearities in the cases of n = N and n = 0 have been obtained [27,29]. However, if some coefficients of the N-degree terms are zero and the system can be reduced to Systems (1) or (3), some more precise and detailed results, including the existence condition, exact number, and parameter control conditions of the periodic solutions are obtained for n ≤ N, N ≥ 2. Considering a special case, n = N, for System (1), the exact number of periodic solutions obtained in Theorem 3 verifies and improves the upper bound of the number of periodic solutions obtained in the previous literature. Results shown in Theorems 1-3 present a new theoretical description and mechanism explanation for the phenomena of the emergence and disappearance of periodic solutions, which can be widely and directly applied to engineering applications of form (1) and provides engineers the parameter method for vibration control.

Application
To demonstrate the applicability and effectiveness of our theoretical results, the periodic breath vibrations of a two-degree-of-freedom mechanical model of the circular mesh antenna subjected to the thermal excitation are investigated. An equivalent circular cylindrical shell model is regarded as a simplified model of the circular mesh antenna, as shown in Figure 1 [30]. and zero for 0 has no solution.
(II) When 222 P P  , we obtain with respect to z is one for 0 and zero for 0 Hence, based on cases (I)-(II), we obtain n   for In summary, together with the above cases, we can obtain the fact that     (1) and the parameter control conditions can be obtained as shown in Theorem 3. This proof is completed.
In fact, the upper bounds of the number of periodic solutions for a four-dimensional system with all the N-degree homogeneous nonlinearities in the cases of N n  and 0  n have been obtained [27,29]. However, if some coefficients of the N-degree terms are zero and the system can be reduced to Systems (1) or (3), some more precise and detailed results, including the existence condition, exact number, and parameter control conditions of the periodic solutions are obtained for . Considering a special case, N n  , for System (1), the exact number of periodic solutions obtained in Theorem 3 verifies and improves the upper bound of the number of periodic solutions obtained in the previous literature. Results shown in Theorems 1-3 present a new theoretical description and mechanism explanation for the phenomena of the emergence and disappearance of periodic solutions, which can be widely and directly applied to engineering applications of form (1) and provides engineers the parameter method for vibration control.

Application
To demonstrate the applicability and effectiveness of our theoretical results, the periodic breath vibrations of a two-degree-of-freedom mechanical model of the circular mesh antenna subjected to the thermal excitation are investigated. An equivalent circular cylindrical shell model is regarded as a simplified model of the circular mesh antenna, as shown in Figure 1  Based on Reddy's first-order shear deformation theorem, Hamilton's principle, and the Galerkin procedure, a two-degree-of-freedom dynamic system of the circular cylindrical shell is obtained as follows [30]: ..
where γ 10 and γ 20 are frictional coefficients, f T is thermal excitation, and ω 1 and ω 2 are two linear frequencies.
Considering the case of primary parameter resonance and 1:1 internal resonance, we have the relations the averaged equations are obtained based on normal form theory and the method of multiple scales: .
By introducing the transformations if σ 1 0, System (27) can be rewritten as Now System (28) is in the form of System (1) and the degree of its homogeneous terms is three, which shows that N = 3. Supposing γ 10 γ 20 γ 14 γ 26 σ 1 0, we discuss the periodic solutions of System (28) by considering the cases of n = 1, n = 2, and n = 3, where: n = σ 2 σ 1 .
(2) n = 2. Now N + n is odd, and System (28) has no periodic solution based on Theorem 2.
To verify the effectiveness of our results, we detect the phase portraits of the periodic solutions of System (28)   The following two statements hold for System (28) (2) 2 = n . Now n N + is odd, and System (28) has no periodic solution based on Theorem 2.

Conclusions
In this paper, the periodic solutions for a four-dimensional coupled polynomial system with N-degree homogeneous nonlinearities are investigated. When the unperturbed linear part of the four-dimensional system has a center singular point in generalization resonance 1 : n at the origin for n ∈ Z + , n ≤ N, N ≥ 2, the new explicit expression of the displacement function of order ε 0 is obtained by introducing the exact formulas on power trigonometric integrals, which is in agreement with certain existing results, but we obtain the more detailed expression by including the exact coefficients. By considering the zero solutions of h(z) with complex coefficient relations, the results on the existence condition, exact number, and parameter control conditions of the periodic solutions of System (1) are obtained, which shows that the parity of N + n has a great influence on the periodic solutions. Results obtained in this paper provide more precise and detailed bifurcation information for System (1) than the existing results in the case of n = 0, N, which can be widely applied to various engineering applications and help engineers better analyze the complex periodic vibrations exhibited in reality. Theorems 1-3 verify and improve the upper bound of the number of periodic solutions obtained in the previous literature for System (1), which presents the effects of important parameters on the phenomena of the emergence and disappearance of periodic solutions and shows a new theoretical explanation on vibration mechanisms.
As an application, the periodic solutions of a two-degree-of-freedom circular mesh antenna model subjected to the thermal excitation are investigated by theoretical and numerical methods. The results on the exact number, parameter control conditions, and the relative positions of the periodic solutions are obtained, which verify the applicability and validity of the theoretical results and show the complicated periodic motions of circular mesh antenna. Periodic vibrations with high amplitudes may lead to serious damage to the device, and the theoretical results obtained in this paper may provide a parameter method for vibration control of circular mesh antenna.
It is remarkable that the method for detecting the existence of periodic solutions of System (1) can also be extended to the four-dimensional system with m-degree nonhomogeneous terms: where F i (x) = Hence, we can obtain a result similar to Theorem 1. However, due to the complex coefficients of h(z), research on the exact number and parameter control conditions of periodic solutions of System (29) is still a difficult topic that will be investigated in a further study. In addition, some higher dimensional (odd or even) systems have arisen more and more frequently in the fields of science and engineering in recent years, so we will try to study their periodic solutions as well as the stability in later work.