The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

Abstract

We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold.

1. Introduction

About dynamical behavior of the trajectories of three-dimensional system, bifurcation of limit cycles is one of major concerns; particularly, for Hopf bifurcation of a nondegenerate equilibrium with a pair of pure imaginary roots and a negative one, many investigations have been carried out in the past decades, for example, [1–4] for the three-dimensional chaotic systems, [5–8] for the three-dimensional Lotka-Volterra systems, and [9] for general three-dimensional systems.

However, up till now, study on bifurcation of limit cycles from the degenerate singularity for high-dimensional nonlinear dynamical systems is hardly seen in published references. In this paper, we investigate the following three-dimensional systems: d𝑥d𝑡=𝑦+𝜇𝑥²+^∞_{𝑘+2𝑗+2𝑙=3}𝐴_𝑘𝑗𝑙𝑥^𝑘𝑦^𝑗𝑢^𝑙=𝑋𝑥,𝑦,𝑢,d𝑦d𝑡=−2𝑥³+2𝜇𝑥𝑦+^∞_{𝑘+2𝑗+2𝑙=4}𝐵_𝑘𝑗𝑙𝑥^𝑘𝑦^𝑗𝑢^𝑙=𝑌𝑥,𝑦,𝑢,d𝑢d𝑡=−𝑑𝑢+^∞_{𝑘+𝑗+𝑙=2}𝑑_𝑘𝑗𝑙𝑥^𝑘𝑦^𝑗𝑢^𝑙=𝑈𝑥,𝑦,𝑢, (1) where 𝑥,𝑦,𝑢,𝑡,𝑑,𝐴_𝑘𝑗𝑙,𝐵_𝑘𝑗𝑙,𝑑_𝑘𝑗𝑙∈ℝ(𝑑>0,𝑘,𝑗,𝑙∈ℕ). Obviously system (1) has the Jacobian matrix at the origin as follows: 𝐴𝐴=¹00−𝑑,where𝐴₁=.0100 (2) One notices that the origin has two zero eigenvalues and one negative eigenvalue and the block matrix 𝐴₁ is nilpotent. From the center manifold theorem, for the system (1), there exists the center manifold: 𝑢=𝑢(𝑥,𝑦) with 𝑢(0,0)=0, D𝑢(0,0)=0, and more the flow on the center manifold is governed by 𝑎/𝑎𝑥̇𝑦̇=𝐴₁𝑥𝑦+𝑋𝑌,𝑥,𝑦,𝑢𝑥,𝑦−𝑦𝑥,𝑦,𝑢𝑥,𝑦 (3) namely, d𝑥d𝑡=𝑦+𝜇𝑥²+^∞_𝑘+2𝑗=3𝑎_𝑘𝑗𝑥^𝑘𝑦^𝑗=𝑋𝑥,𝑦,d𝑦d𝑡=−2𝑥³+2𝜇𝑥𝑦+^∞_𝑘+2𝑗=4𝑏_𝑘𝑗𝑥^𝑘𝑦^𝑗=𝑌𝑥,𝑦, (4) where 𝑥,𝑦,𝑡,𝑎_𝑘𝑗,𝑏_𝑘𝑗∈ℝ(𝑘,𝑗∈ℕ). It is usually called the reduced system and system (1) is topological equivalent to system (4) in the vicinity of the origin [10]. Thus by investigating system (4), we can solve Hopf bifurcation of the origin on center manifold of the three-dimensional system (1). At the same time, we also discuss the center-focus problem for the flow restricted to the center manifold, which closely relates to the maximum number of limit cycles bifurcating from the origin.

As far as investigation about limit cycles bifurcation of a nilpotent critical point in planar system is concerned, it is similar to nondegenerate case that detecting nilpotent center and calculating the focal value are needed [11]. There exist some available and classical ways, for instance, Poincare return map [12], Lyapunov function [13], and the normal form theory [14]. At the same time, some good results on the cyclicity were obtained [11, 12, 15, 16]. Recently in [17, 18], the authors gave an inverse integral factor method of calculating the quasi-Lyapunov constants of the three-order nilpotent critical point; it is convenient to compute the higher order focal values and solve the center-focus problem of the equilibrium. Here we extend this method's application to the three-dimensional system (1) and consider its specific example as follows: d𝑥d𝑡=𝑦+𝑎₀𝑢𝑥−2𝑥𝑦=𝑋,d𝑦d𝑡=𝑦²−2𝑥³+𝑏₀𝑢𝑦=𝑌,d𝑢d𝑡=−𝑢+𝑑₁𝑥𝑦=𝑈, (5) where 𝑎₀,𝑏₀,𝑑₁∈ℝ.

The rest of this paper is organized as follows. In Section 2, the corresponding quasi-Lyapunov constants are computed and the center conditions on the center manifold are determined. In Section 3, the multiple local bifurcations at the origin for system (5) are investigated, three limit cycles from the origin are obtained, and it is proved at most three small limit cycles from the origin via local bifurcation. In this work, the system and problem are all considered for the first time.

2. Quasi-Lyapunov Constants and Center Conditions

In this part, we firstly investigate the singular point quantities of the origin. For the center manifold of system (5), one can determine the formal expression 𝑢=𝑢(𝑥,𝑦) and obtain the same form as system (4): d𝑥d𝑡=𝑦+𝑎₀𝑢𝑥−2𝑥𝑦=𝑋,d𝑦d𝑡=𝑦²−2𝑥³+𝑏₀𝑢𝑦=𝑌. (6) Furthermore, we give the definition of quasi-Lyapunov constants for system (6) and the way of computing them; more details can be found in [17–19].

Lemma 1 (Theorems 8.7.1 and 8.7.2 in [17]). For system (6), any positive integer 𝑠 and a given number sequence {𝑐_0𝛽},𝛽⩾3, we can derive successively and uniquely the terms of the following formal series with the coefficients 𝑐_𝛼𝛽 satisfying 𝛼≠0, 𝑀𝑥,𝑦=𝑦²+^∞_𝛼+𝛽=3𝑐_𝛼𝛽𝑥^𝛼𝑦^𝛽 (7) such that 𝜕𝑋+𝜕𝑥𝜕𝑌(𝜕𝑦𝑀−𝑠+1)𝜕𝑀𝜕𝑥𝑋+𝜕𝑀𝑌=𝜕𝑦^∞_𝑚=6𝜔_𝑚𝑥^𝑚. (8) And, if 𝛼≠0, 𝑐_𝛼𝛽 is determined by the following recursive formula: 𝑐_𝛼𝛽=1(𝑠+1)𝛼^{𝛼+𝛽−1}_𝑘+𝑗=2[𝑘−(𝑠+1)(𝛼−𝑘)]𝑎_𝑘𝑗𝑐_{𝛼−𝑘,𝛽−𝑗+1}+𝑗−(𝑠+1)𝛽−𝑗+2×𝑏_𝑘𝑗𝑐_{𝛼−𝑘−1,𝛽−𝑗+2} (9) and, for any positive integer 𝑚⩾6, 𝜔_𝑚 is determined by the following recursive formula: 𝜔_𝑚=^𝑚−1_𝑘+𝑗=2(𝑘−(𝑠+1)(𝛼−𝑘+1))𝑎_𝑘𝑗𝑐_{𝑚−𝑘+1,−𝑗}+𝑗−(𝑠+1)1−𝑗𝑏_𝑘𝑗𝑐_{𝑚−𝑘,1−𝑗} (10) and, when 𝛼<0 or 𝛽<0, we have let 𝑐_𝛼𝛽=0.
Particularly, by choosing appropriate 𝑠 and number sequence {𝑐_0𝛽}(𝛽⩾3), we can make 𝜔_2𝑖+1(𝑠)=0 and then let 𝜔_2𝑖+4(𝑠)=(2𝑖−4𝑠−1)𝜆_𝑖. (11)

Definition 2. The 𝜆_𝑖 in (11) is called the 𝑖th quasi-Lyapunov constant of the origin of system (6), 𝑖=1,2,…, and more if all the quasi-Lyapunov constants vanish, that is, 𝜆_𝑚=0, 𝑚=1,2,…, then the origin of system (5) is a center on the local center manifold at the origin.

Lemma 3. For system (6), the 𝑚th focal value 𝑣_2𝑚(−2𝜋) at the origin of system (6) is algebraic equivalent to 𝜎_𝑚𝜆_𝑚; that is, for any positive integer 𝑚=2,3,…, if 𝑣₂=𝑣₄=⋯=𝑣_2𝑚−2=0 and 𝜆₁=𝜆₂=⋯=𝜆_𝑚−1=0 hold, then 𝑣_2𝑚=𝜎_𝑚𝜆_𝑚, where 𝜎_𝑚 is a defined constant (which is given in [17, 18]).

Remark 4. According to Lemma 1, one sees that each 𝜔_𝑚 in (10) is related to only the coefficients of the first 𝑚−1 degree terms of system (6). Here we determine the above 𝑢(𝑥,𝑦) only up to a tenth degree polynomial with respect to 𝑥, 𝑦 as follows: 𝑢𝑥,𝑦=𝑑₁𝑦𝑥−𝑦+𝑑₁𝑦²𝑥+𝑦+𝑢₄+𝑢₅+⋯+𝑢₁₀, (12) where 𝑢_𝑖,𝑖=4,…,10 is a homogeneous polynomial in 𝑥, 𝑦 of degree 𝑖, respectively, (which can be seen in the Appendix); thus 𝑋 and 𝑌 in system (6) are two polynomials with degree 11. And more all 𝑎_𝑘𝑗, 𝑏_𝑘𝑗 of 𝑐_𝛼𝛽 in (9) are given definitely by the coefficients 𝑎₀, 𝑏₀, 𝑑₁ (the specific 𝑐_𝛼𝛽 is available in Email address of the corresponding author); the 𝜔_𝑚 in (10) is given by the following specific form: 𝜔_𝑚=−32𝑑₁686−94𝑎₀𝑑₁−31𝑏₀𝑑₁×21𝑎₀+𝑏₀−𝑎₀𝑚+10𝑎₀𝑠−𝑎₀[]𝑚𝑠×𝑐𝑚−10,0−2208𝑑₁19𝑎₀+𝑏₀−𝑎₀𝑚+9𝑎₀𝑠−𝑎₀[]𝑚𝑠×𝑐𝑚−9,0+4𝑑₁144−11𝑎₀𝑑₁−3𝑏₀𝑑₁×17𝑎₀+𝑏₀−𝑎₀𝑚+8𝑎₀𝑠−𝑎₀[]𝑚𝑠×𝑐𝑚−8,0+40𝑑₁15𝑎₀+𝑏₀−𝑎₀𝑚+7𝑎₀𝑠−𝑎₀[]𝑚𝑠×𝑐𝑚−7,0−24𝑑₁13𝑎₀+𝑏₀−𝑎₀𝑚+6𝑎₀𝑠−𝑎₀[]𝑚𝑠×𝑐𝑚−6,0+2𝑑₁9𝑎₀+𝑏₀−𝑎₀𝑚+4𝑎₀𝑠−𝑎₀[]()𝑐[].𝑚𝑠×𝑐𝑚−4,0+21+𝑠𝑚−3,0 (13)

Applying the powerful symbolic computation function of Mathematica system and the recursive formulas in Lemma 1, and from Remark 4, we obtain the first 7 quantities as follows: 𝜔₆𝜔=0,₇=3(1+𝑠)𝑐₀₃,𝜔₈=45𝑑₁2𝑎₀−𝑏₀(),𝜔3−4𝑠₉=23𝑎₀𝑑₁(),𝜔1−𝑠₁₀=27𝑎₀𝑑₁1−6𝑎₀𝑑₁,𝜔₁₁=111273𝑎₀𝑑₁−704𝑎²₀𝑑²₁+840𝑐₀₅,𝜔₁₂=−397𝑎105⁰𝑑₁. (14) In the above expression of each 𝜔_𝑘, 𝑘=7,…,12, we have already let 𝜔₆=⋯=𝜔_𝑘−1=0.

Particularly, in order to make 𝜔_2𝑖+1=0, 𝑖=1,…,5, we let 𝑠=1 and choose 𝑐₀₃=0,𝑐₀₅=1840704𝑎²₀𝑑²₁−73𝑎₀𝑑₁. (15)

Thus from the expression (11) in Lemma 1, we have the following.

Theorem 5. For the flow on center manifold of system (5), we get the first 4 quasi-Lyapunov constants of the origin as follows: 𝜆₁𝜆=0,₂=45𝑑₁2𝑎₀−𝑏₀,𝜆₃=27𝑎₀𝑑₁1−6𝑎₀𝑑₁,𝜆₄=−397𝑎315⁰𝑑₁. (16)

Theorem 6. For system (5), the origin is a center on the local center manifold if and only if the following condition is satisfied: 𝑑₁=0𝑜𝑟𝑎₀=𝑏₀=0. (17)

Proof. The proof of the necessity is easy; then we omit it. Now we prove the sufficient condition; this technique derives from the method of Darboux (also see [20–23]). Obviously, if 𝑑₁=0 in the conditions (17) holds, then system (5) has the corresponding form as follows: d𝑥d𝑡=𝑦+𝑎₀𝑢𝑥−2𝑥𝑦,d𝑦d𝑡=𝑦²−2𝑥³+𝑏₀𝑢𝑦,d𝑢d𝑡=−𝑢. (18) And more we can figure out easily one algebraic invariant surface for system (18): 𝐹(𝑥,𝑦,𝑢)=𝑢; in fact, there exists a polynomial 𝐾(𝑥,𝑦,𝑢)=−1, the cofactor of 𝐹(𝑥,𝑦,𝑢), such that d𝐹/d𝑡|₍₁₇₎=𝐾𝐹. One can observe that 𝐹(𝑥,𝑦,𝑢)=0 is tangent to the center eigenspace, the (𝑥,𝑦)-plane, at the origin. Thus it forms a local center manifold in a neighborhood of the origin. From 𝐹(𝑥,𝑦,𝑢)=𝑢=0, we substitute it into the first and second equations of the system defined by system (18) and we have the differential equations d𝑥d𝑡=𝑦−2𝑥𝑦,d𝑦d𝑡=𝑦²−2𝑥³ (19) which is a Hamiltonian system with Hamiltonian function: 𝐻(𝑥,𝑦)=𝑥⁴−2𝑥𝑦²+𝑦². Therefore the origin is a center for systems (18) or (19) as shown in Figure 1(a).
And if 𝑎₀=𝑏₀=0 in the conditions (17) holds, then system (5) has the corresponding form as follows: d𝑥d𝑡=𝑦−2𝑥𝑦,d𝑦d𝑡=𝑦²−2𝑥³,d𝑢d𝑡=−𝑢+𝑑₁𝑥𝑦. (20) Clearly, the above 𝐻=𝑥⁴−2𝑥𝑦²+𝑦² is also a first integral of system (20); thus the origin is a center for the flow of system (5) restricted to a center manifold.

(a)

(b)

Figure 1 

(a) The phase portrait of system (19) on the (𝑥,𝑦)-plane which shows that it is integrable at the origin; (b) the stable limit cycle bifurcated from the origin of system (24) with 𝛿=0.001, 𝑑1=1, 𝑎0=1/6, 𝑏0=1/3+0.001.

According to Lemma 3 and Theorem 6, then we have the following.

Theorem 7. For the origin of system (5) as a weak focus on center manifold, the highest is the fourth order and its first 4 focal values are as follows: 𝑣₂(−2𝜋)=𝜎₁𝜆₁,𝑣₄(−2𝜋)=𝜎₂𝜆₂,𝑣₆(−2𝜋)=𝜎₃𝜆₃,𝑣₈(−2𝜋)=𝜎₄𝜆₄, (21) where, for the expression of 𝑣₄, we have let 𝑣₂=0, and 𝑣₂=𝑣₄=0 for 𝑣₆, 𝑣₂=𝑣₄=𝑣₆=0 for 𝑣₈.

3. Multiple Limit Cycle Bifurcation of the System

In this section, we apply the focal values obtained in last section to discuss multiple bifurcation of the equilibrium and demonstrate there exist 3 and at most 3 limit cycles.

From Theorems 5 and 7, we have

Theorem 8. For the flow on center manifold of (5), the origin is a 4th-order fine focus; that is, 𝜆₁=𝜆₂=𝜆₃=0, 𝜆₄≠0 if and only if 𝑏₀=2𝑎₀,6𝑎₀𝑑₁=1𝑜𝑟𝑏₀=13𝑑₁,𝑎₀=16𝑑₁𝑑₁≠0. (22)

Proof. Firstly, we prove the necessity, from the expressions (16) in Theorem 5; we let 𝜆₁=𝜆₂=𝜆₃=0 and consider 𝜆₄≠0 and then only 𝑏₀=2𝑎₀ and 6𝑎₀𝑑₁=1 are obtained at the same time. Then for the proof of the sufficiency, by substituting the above conditions into 𝜆_𝑖,𝑖=1,2,3,4, one can get the conclusion easily. Furthermore we can transform the above conditions into the following form: 𝑏₀=1/3𝑑₁, 𝑎₀=(1/6𝑑₁)(𝑑₁≠0). Thus the proof is completed.

Remark 9. For the coefficients of system (5) or (6), when 𝑑₁≠0, there always exists a group of critical values: 𝑎₀=1/6𝑑₁, 𝑏₀=1/3𝑑₁ such that the conditions (22) hold.

From Theorem 5, one calculates easily the Jacobian determinant with respect to the functions 𝜆₂, 𝜆₃ and variables 𝑎₀, 𝑏₀, 𝜕𝜆𝐽=²,𝜆₃𝜕𝑎₀𝑏₀=8𝑑35²¹1−12𝑎₀𝑑₁. (23) Considering the conditions (22) and when 𝑑₁≠0 holds, we obtain 𝐽=−(8/35)𝑑²₁≠0. At the same time, one can get that 𝜆₄=−(397/1890)≠0 holds.

Thus we take some appropriate perturbations for the coefficients of system (6) and get the perturbed system with the following form: d𝑥d𝑡=𝛿𝑥+𝑦+𝑎₀𝑢𝑥−2𝑥𝑦,d𝑦d𝑡=2𝛿𝑦+𝑦²−2𝑥³+𝑏₀𝑢𝑦, (24) where 0<𝛿≪1. According to Theorem 2.2 in [18], system (6) has 3 limit cycles in the neighborhood of the origin; that is, system (5) has 3 limit cycles on the center manifold necessarily. One can refer to [9, 17, 18] for more details about constructing of limit cycles. And using numerical method, we get an example of one stable limit cycle for system (24) as shown in Figure 1(b).

Therefore, from the above discussion and Theorem 7, we have the following.

Theorem 10. For system (5), 3 and at most 3 small amplitude limit cycles can be bifurcated from the origin on the center manifold.

4. Conclusions

In summary, based on precise symbolic computation, we have investigated deeply multiple limit cycles bifurcation in the vicinity of the degenerate equilibrium for a three-dimensional system. Firstly by computing its quasi-Lyapunov constants, we solved its center-focus problem on center manifold; thus the center conditions are found and as weak foci the highest order is proved to be the fourth. Then we obtained that the system can generate 3 small limit cycles from nilpotent critical point and at most 3 small limit cycles from the origin via local bifurcation. For the flow of a high-dimensional system restricted to the center manifold, only one interesting example of multiple limit cycles bifurcating from its nilpotent singularity was given here, but we believe that more outcomes on this problem will be shown in the near future.

Appendix

Consider𝑢₄=−𝑑₁−2𝑥⁴+12𝑥³𝑦−36𝑥²𝑦²+𝑎₀𝑑₁𝑥²𝑦²+𝑏₀𝑑₁𝑥²𝑦²+72𝑥𝑦³−3𝑎₀𝑑₁𝑥𝑦³−5𝑏₀𝑑₁𝑥𝑦³−69𝑦⁴+3𝑎₀𝑑₁𝑦⁴+7𝑏₀𝑑₁𝑦⁴,𝑢₅=𝑑₁𝑦20𝑥⁴−134𝑥³𝑦+474𝑥²𝑦²−4𝑎₀𝑑₁𝑥²𝑦²−5𝑏₀𝑑₁𝑥²𝑦²−876𝑥𝑦³+5𝑎₀𝑑₁𝑥𝑦³+5𝑏₀𝑑₁𝑥𝑦³+600𝑦⁴+7𝑎₀𝑑₁𝑦⁴+28𝑏₀𝑑₁𝑦⁴,𝑢₆=−𝑑₁24𝑥⁶−288𝑥⁵𝑦+14𝑎₀𝑑₁𝑥⁵𝑦+6𝑏₀𝑑₁𝑥⁵𝑦+1732𝑥⁴𝑦²−144𝑎₀𝑑₁𝑥⁴𝑦²−88𝑏₀𝑑₁𝑥⁴𝑦²−6944𝑥³𝑦³+744𝑎₀𝑑₁𝑥³𝑦³+552𝑏₀𝑑₁𝑥³𝑦³−3𝑎²₀𝑑²₁𝑥³𝑦³−6𝑎₀𝑏₀𝑑²₁𝑥³𝑦³−3𝑏²₀𝑑²₁𝑥³𝑦³+20358𝑥²𝑦⁴−2443𝑎₀𝑑₁𝑥²𝑦⁴−2081𝑏₀𝑑₁𝑥²𝑦⁴+17𝑎²₀𝑑²₁𝑥²𝑦⁴+46𝑎₀𝑏₀𝑑²₁𝑥²𝑦⁴+33𝑏²₀𝑑²₁𝑥²𝑦⁴−42468𝑥𝑦⁵+5038𝑎₀𝑑₁𝑥𝑦⁵+4882𝑏₀𝑑₁𝑥𝑦⁵−40𝑎²₀𝑑²₁𝑥𝑦⁵−134𝑎₀𝑏₀𝑑²₁𝑥𝑦⁵−126𝑏²₀𝑑²₁𝑥𝑦⁵+45468𝑦⁶−5003𝑎₀𝑑₁𝑦⁶−5153𝑏₀𝑑₁𝑦⁶+40𝑎²₀𝑑²₁𝑦⁶+152𝑎₀𝑏₀𝑑²₁𝑦⁶+168𝑏²₀𝑑²₁𝑦⁶,𝑢₇=𝑑₁40𝑥⁷−1104𝑥⁶𝑦+12060𝑥⁵𝑦²−260𝑎₀𝑑₁𝑥⁵𝑦²−128𝑏₀𝑑₁𝑥⁵𝑦²−77700𝑥⁴𝑦³+2860𝑎₀𝑑₁𝑥⁴𝑦³+1700𝑏₀𝑑₁𝑥⁴𝑦³+337632𝑥³𝑦⁴−15498𝑎₀𝑑₁𝑥³𝑦⁴−10704𝑏₀𝑑₁𝑥³𝑦⁴+24𝑎²₀𝑑²₁𝑥³𝑦⁴+56𝑎₀𝑏₀𝑑²₁𝑥³𝑦⁴+33𝑏²₀𝑑²₁𝑥³𝑦⁴−1012896𝑥²𝑦⁵+49266𝑎₀𝑑₁𝑥²𝑦⁵+39042𝑏₀𝑑₁𝑥²𝑦⁵−94𝑎²₀𝑑²₁𝑥²𝑦⁵−251𝑎₀𝑏₀𝑑²₁𝑥²𝑦⁵−169𝑏²₀𝑑²₁𝑥²𝑦⁵+1898388𝑥𝑦⁶−84891𝑎₀𝑑₁𝑥𝑦⁶−72276𝑏₀𝑑₁𝑥𝑦⁶+66𝑎²₀𝑑²₁𝑥𝑦⁶+67𝑎₀𝑏₀𝑑²₁𝑥𝑦⁶−166𝑏²₀𝑑²₁𝑥𝑦⁶−1625580𝑦⁷+54873𝑎₀𝑑₁𝑦⁷+45075𝑏₀𝑑₁𝑦⁷+174𝑎²₀𝑑²₁𝑦⁷+915𝑎₀𝑏₀𝑑²₁𝑦⁷+1419𝑏²₀𝑑²₁𝑦⁷,𝑢₈=−𝑑₁−576𝑥⁸+44𝑎₀𝑑₁𝑥⁸+12𝑏₀𝑑₁𝑥⁸+10976𝑥⁷𝑦−1264𝑎₀𝑑₁𝑥⁷𝑦−496𝑏₀𝑑₁𝑥⁷𝑦−106352𝑥⁶𝑦²+16048𝑎₀𝑑₁𝑥⁶𝑦²+7696𝑏₀𝑑₁𝑥⁶𝑦²−114𝑎²₀𝑑²₁𝑥⁶𝑦²−116𝑎₀𝑏₀𝑑²₁𝑥⁶𝑦²−34𝑏²₀𝑑²₁𝑥⁶𝑦²+704496𝑥⁵𝑦³−126632𝑎₀𝑑₁𝑥⁵𝑦³−69528𝑏₀𝑑₁𝑥⁵𝑦³+1700𝑎²₀𝑑²₁𝑥⁵𝑦³+2172𝑎₀𝑏₀𝑑²₁𝑥⁵𝑦³+816𝑏²₀𝑑²₁𝑥⁵𝑦³−3558660𝑥⁴𝑦⁴+710088𝑎₀𝑑₁𝑥⁴𝑦⁴+428808𝑏₀𝑑₁𝑥⁴𝑦⁴−12764𝑎²₀𝑑²₁𝑥⁴𝑦⁴−19064𝑎₀𝑏₀𝑑²₁𝑥⁴𝑦⁴−8340𝑏²₀𝑑²₁𝑥⁴𝑦⁴+14𝑎³₀𝑑³₁𝑥⁴𝑦⁴+42𝑎²₀𝑏₀𝑑³₁𝑥⁴𝑦⁴+42𝑎₀𝑏²₀𝑑³₁𝑥⁴𝑦⁴+14𝑏³₀𝑑³₁𝑥⁴𝑦⁴[phone]𝑥³𝑦⁵−2960538𝑎₀𝑑₁𝑥³𝑦⁵−1907570𝑏₀𝑑₁𝑥³𝑦⁵+61685𝑎²₀𝑑²₁𝑥³𝑦⁵+103855𝑎₀𝑏₀𝑑²₁𝑥³𝑦⁵+50130𝑏²₀𝑑²₁𝑥³𝑦⁵−125𝑎³₀𝑑³₁𝑥³𝑦⁵−463𝑎²₀𝑏₀𝑑³₁𝑥³𝑦⁵−591𝑎₀𝑏²₀𝑑³₁𝑥³𝑦⁵−261𝑏³₀𝑑³₁𝑥³𝑦⁵−43327872𝑥²𝑦⁶+9065880𝑎₀𝑑₁𝑥²𝑦⁶+6166752𝑏₀𝑑₁𝑥²𝑦⁶−200895𝑎²₀𝑑²₁𝑥²𝑦⁶−374342𝑎₀𝑏₀𝑑²₁𝑥²𝑦⁶−196423𝑏²₀𝑑²₁𝑥²𝑦⁶+504𝑎³₀𝑑³₁𝑥²𝑦⁶+2186𝑎²₀𝑏₀𝑑³₁𝑥²𝑦⁶+3360𝑎₀𝑏²₀𝑑³₁𝑥²𝑦⁶+1854𝑏³₀𝑑³₁𝑥²𝑦⁶[phone]𝑥𝑦⁷−18564504𝑎₀𝑑₁𝑥𝑦⁷−13274868𝑏₀𝑑₁𝑥𝑦⁷+412530𝑎²₀𝑑²₁𝑥𝑦⁷+833201𝑎₀𝑏₀𝑑²₁𝑥𝑦⁷+468601𝑏²₀𝑑²₁𝑥𝑦⁷−1097𝑎³₀𝑑³₁𝑥𝑦⁷−5317𝑎²₀𝑏₀𝑑³₁𝑥𝑦⁷−9303𝑎₀𝑏²₀𝑑³₁𝑥𝑦⁷−6011𝑏³₀𝑑³₁𝑥𝑦⁷−105628356𝑦⁸[phone]𝑎₀𝑑₁𝑦⁸[phone]𝑏₀𝑑₁𝑦⁸−411312𝑎²₀𝑑²₁𝑦⁸−868420𝑎₀𝑏₀𝑑²₁𝑦⁸−503532𝑏²₀𝑑²₁𝑦⁸+1097𝑎³₀𝑑³₁𝑦⁸+5673𝑎²₀𝑏₀𝑑³₁𝑦⁸+10687𝑎₀𝑏²₀𝑑³₁𝑦⁸+7551𝑏³₀𝑑³₁𝑦⁸,𝑢₉=𝑑₁−2208𝑥⁹+77328𝑥⁸𝑦−2384𝑎₀𝑑₁𝑥⁸𝑦−784𝑏₀𝑑₁𝑥⁸𝑦−1227512𝑥⁷𝑦²+64396𝑎₀𝑑₁𝑥⁷𝑦²+26272𝑏₀𝑑₁𝑥⁷𝑦²[phone]𝑥⁶𝑦³−835460𝑎₀𝑑₁𝑥⁶𝑦³−396608𝑏₀𝑑₁𝑥⁶𝑦³+3144𝑎²₀𝑑²₁𝑥⁶𝑦³+3498𝑎₀𝑏₀𝑑²₁𝑥⁶𝑦³+1096𝑏²₀𝑑²₁𝑥⁶𝑦³−89203392𝑥⁵𝑦⁴+6916384𝑎₀𝑑₁𝑥⁵𝑦⁴+3676396𝑏₀𝑑₁𝑥⁵𝑦⁴−49214𝑎²₀𝑑²₁𝑥⁵𝑦⁴−63780𝑎₀𝑏₀𝑑²₁𝑥⁵𝑦⁴−22834𝑏²₀𝑑²₁𝑥⁵𝑦⁴[phone]𝑥⁴𝑦⁵−40282356𝑎₀𝑑₁𝑥⁴𝑦⁵−23266264𝑏₀𝑑₁𝑥⁴𝑦⁵+378254𝑎²₀𝑑²₁𝑥⁴𝑦⁵+553724𝑎₀𝑏₀𝑑²₁𝑥⁴𝑦⁵+217218𝑏²₀𝑑²₁𝑥⁴𝑦⁵−180𝑎³₀𝑑³₁𝑥⁴𝑦⁵−607𝑎²₀𝑏₀𝑑³₁𝑥⁴𝑦⁵−687𝑎₀𝑏²₀𝑑³₁𝑥⁴𝑦⁵−261𝑏³₀𝑑³₁𝑥⁴𝑦⁵−1968992136𝑥³𝑦⁶[phone]𝑎₀𝑑₁𝑥³𝑦⁶[phone]𝑏₀𝑑₁𝑥³𝑦⁶−1780906𝑎²₀𝑑²₁𝑥³𝑦⁶−2890212𝑎₀𝑏₀𝑑²₁𝑥³𝑦⁶−1233114𝑏²₀𝑑²₁𝑥³𝑦⁶+1292𝑎³₀𝑑³₁𝑥³𝑦⁶+4961𝑎²₀𝑏₀𝑑³₁𝑥³𝑦⁶+6477𝑎₀𝑏²₀𝑑³₁𝑥³𝑦⁶+2877𝑏³₀𝑑³₁𝑥³𝑦⁶[phone]𝑥²𝑦⁷−495558804𝑎₀𝑑₁𝑥²𝑦⁷−321861864𝑏₀𝑑₁𝑥²𝑦⁷+5224500𝑎²₀𝑑²₁𝑥²𝑦⁷+9158682𝑎₀𝑏₀𝑑²₁𝑥²𝑦⁷+4161294𝑏²₀𝑑²₁𝑥²𝑦⁷−3274𝑎³₀𝑑³₁𝑥²𝑦⁷−12840𝑎²₀𝑏₀𝑑³₁𝑥²𝑦⁷−16145𝑎₀𝑏²₀𝑑³₁𝑥²𝑦⁷−5819𝑏³₀𝑑³₁𝑥²𝑦⁷−11169394848𝑥𝑦⁸[phone]𝑎₀𝑑₁𝑥𝑦⁸[phone]𝑏₀𝑑₁𝑥𝑦⁸−8530348𝑎²₀𝑑²₁𝑥𝑦⁸−15428426𝑎₀𝑏₀𝑑²₁𝑥𝑦⁸−6980247𝑏²₀𝑑²₁𝑥𝑦⁸+949𝑎³₀𝑑³₁𝑥𝑦⁸−2716𝑎²₀𝑏₀𝑑³₁𝑥𝑦⁸−23065𝑎₀𝑏²₀𝑑³₁𝑥𝑦⁸−31601𝑏³₀𝑑³₁𝑥𝑦⁸[phone]𝑦⁹−750336780𝑎₀𝑑₁𝑦⁹−509164044𝑏₀𝑑₁𝑦⁹+5239852𝑎²₀𝑑²₁𝑦⁹+8941749𝑎₀𝑏₀𝑑²₁𝑦⁹+3331701𝑏²₀𝑑²₁𝑦⁹+7827𝑎³₀𝑑³₁𝑦⁹+50215𝑎²₀𝑏₀𝑑³₁𝑦⁹+119361𝑎₀𝑏²₀𝑑³₁𝑦⁹+108056𝑏³₀𝑑³₁𝑦⁹,𝑢₁₀=−𝑑₁21952𝑥¹⁰−3008𝑎₀𝑑₁𝑥¹⁰−992𝑏₀𝑑₁𝑥¹⁰−605184𝑥⁹𝑦+107232𝑎₀𝑑₁𝑥⁹𝑦+42144𝑏₀𝑑₁𝑥⁹𝑦−1104𝑎²₀𝑑²₁𝑥⁹𝑦−752𝑎₀𝑏₀𝑑²₁𝑥⁹𝑦−160𝑏²₀𝑑²₁𝑥⁹𝑦+8513712𝑥⁸𝑦²−1841888𝑎₀𝑑₁𝑥⁸𝑦²−822896𝑏₀𝑑₁𝑥⁸𝑦²+34440𝑎²₀𝑑²₁𝑥⁸𝑦²+29616𝑎₀𝑏₀𝑑²₁𝑥⁸𝑦²+7896𝑏²₀𝑑²₁𝑥⁸𝑦²−81848832𝑥⁷𝑦³[phone]𝑎₀𝑑₁𝑥⁷𝑦³[phone]𝑏₀𝑑₁𝑥⁷𝑦³−527376𝑎²₀𝑑²₁𝑥⁷𝑦³−530448𝑎₀𝑏₀𝑑²₁𝑥⁷𝑦³−162816𝑏²₀𝑑²₁𝑥⁷𝑦³+1050𝑎³₀𝑑³₁𝑥⁷𝑦³+1774𝑎²₀𝑏₀𝑑³₁𝑥⁷𝑦³+1102𝑎₀𝑏²₀𝑑³₁𝑥⁷𝑦³+250𝑏³₀𝑑³₁𝑥⁷𝑦³[phone]𝑥⁶𝑦⁴−173386212𝑎₀𝑑₁𝑥⁶𝑦⁴−90723084𝑏₀𝑑₁𝑥⁶𝑦⁴+5270270𝑎²₀𝑑²₁𝑥⁶𝑦⁴+5948592𝑎₀𝑏₀𝑑²₁𝑥⁶𝑦⁴+2010036𝑏²₀𝑑²₁𝑥⁶𝑦⁴−21020𝑎³₀𝑑³₁𝑥⁶𝑦⁴−41960𝑎²₀𝑏₀𝑑³₁𝑥⁶𝑦⁴−31132𝑎₀𝑏²₀𝑑³₁𝑥⁶𝑦⁴−8640𝑏³₀𝑑³₁𝑥⁶𝑦⁴−3601377936𝑥⁵𝑦⁵[phone]𝑎₀𝑑₁𝑥⁵𝑦⁵[phone]𝑏₀𝑑₁𝑥⁵𝑦⁵−38357066𝑎²₀𝑑²₁𝑥⁵𝑦⁵−47295826𝑎₀𝑏₀𝑑²₁𝑥⁵𝑦⁵−17155148𝑏²₀𝑑²₁𝑥⁵𝑦⁵+212538𝑎³₀𝑑³₁𝑥⁵𝑦⁵+481702𝑎²₀𝑏₀𝑑³₁𝑥⁵𝑦⁵+405182𝑎₀𝑏²₀𝑑³₁𝑥⁵𝑦⁵+128818𝑏³₀𝑑³₁𝑥⁵𝑦⁵−85𝑎⁴₀𝑑⁴₁𝑥⁵𝑦⁵−340𝑎³₀𝑏₀𝑑⁴₁𝑥⁵𝑦⁵−510𝑎²₀𝑏²₀𝑑⁴₁𝑥⁵𝑦⁵−340𝑎₀𝑏³₀𝑑⁴₁𝑥⁵𝑦⁵−85𝑏⁴₀𝑑⁴₁𝑥⁵𝑦⁵[phone]𝑥⁴𝑦⁶−5874570084𝑎₀𝑑₁𝑥⁴𝑦⁶−3333151404𝑏₀𝑑₁𝑥⁴𝑦⁶[phone]𝑎²₀𝑑²₁𝑥⁴𝑦⁶[phone]𝑎₀𝑏₀𝑑²₁𝑥⁴𝑦⁶[phone]𝑏²₀𝑑²₁𝑥⁴𝑦⁶−1401156𝑎³₀𝑑³₁𝑥⁴𝑦⁶−3518659𝑎²₀𝑏₀𝑑³₁𝑥⁴𝑦⁶−3250715𝑎₀𝑏²₀𝑑³₁𝑥⁴𝑦⁶−1130262𝑏³₀𝑑³₁𝑥⁴𝑦⁶+1085𝑎⁴₀𝑑⁴₁𝑥⁴𝑦⁶+5094𝑎³₀𝑏₀𝑑⁴₁𝑥⁴𝑦⁶+9160𝑎²₀𝑏²₀𝑑⁴₁𝑥⁴𝑦⁶+7498𝑎₀𝑏³₀𝑑⁴₁𝑥⁴𝑦⁶+2363𝑏⁴₀𝑑⁴₁𝑥⁴𝑦⁶−73199185920𝑥³𝑦⁷[phone]𝑎₀𝑑₁𝑥³𝑦⁷[phone]𝑏₀𝑑₁𝑥³𝑦⁷−892808908𝑎²₀𝑑²₁𝑥³𝑦⁷−1249355132𝑎₀𝑏₀𝑑²₁𝑥³𝑦⁷−497385264𝑏²₀𝑑²₁𝑥³𝑦⁷+6463496𝑎³₀𝑑³₁𝑥³𝑦⁷[phone]𝑎²₀𝑏₀𝑑³₁𝑥³𝑦⁷[phone]𝑎₀𝑏²₀𝑑³₁𝑥³𝑦⁷+6564610𝑏³₀𝑑³₁𝑥³𝑦⁷−6527𝑎⁴₀𝑑⁴₁𝑥³𝑦⁷−34734𝑎³₀𝑏₀𝑑⁴₁𝑥³𝑦⁷−72008𝑎²₀𝑏²₀𝑑⁴₁𝑥³𝑦⁷−69410𝑎₀𝑏³₀𝑑⁴₁𝑥³𝑦⁷−26441𝑏⁴₀𝑑⁴₁𝑥³𝑦⁷[phone]𝑥²𝑦⁸−73717828776𝑎₀𝑑₁𝑥²𝑦⁸−44037236592𝑏₀𝑑₁𝑥²𝑦⁸[phone]𝑎²₀𝑑²₁𝑥²𝑦⁸[phone]𝑎₀𝑏₀𝑑²₁𝑥²𝑦⁸[phone]𝑏²₀𝑑²₁𝑥²𝑦⁸−20693125𝑎³₀𝑑³₁𝑥²𝑦⁸−60906347𝑎²₀𝑏₀𝑑³₁𝑥²𝑦⁸−64886985𝑎₀𝑏²₀𝑑³₁𝑥²𝑦⁸−25527395𝑏³₀𝑑³₁𝑥²𝑦⁸+23205𝑎⁴₀𝑑⁴₁𝑥²𝑦⁸+135806𝑎³₀𝑏₀𝑑⁴₁𝑥²𝑦⁸+313360𝑎²₀𝑏²₀𝑑⁴₁𝑥²𝑦⁸+341634𝑎₀𝑏³₀𝑑⁴₁𝑥²𝑦⁸+150475𝑏⁴₀𝑑⁴₁𝑥²𝑦⁸−541133408592𝑥𝑦⁹[phone]𝑎₀𝑑₁𝑥𝑦⁹[phone]𝑏₀𝑑₁𝑥𝑦⁹−5592047067𝑎²₀𝑑²₁𝑥𝑦⁹−8557415979𝑎₀𝑏₀𝑑²₁𝑥𝑦⁹−3635890530𝑏²₀𝑑²₁𝑥𝑦⁹[phone]𝑎³₀𝑑³₁𝑥𝑦⁹[phone]𝑎²₀𝑏₀𝑑³₁𝑥𝑦⁹[phone]𝑎₀𝑏²₀𝑑³₁𝑥𝑦⁹[phone]𝑏³₀𝑑³₁𝑥𝑦⁹−48844𝑎⁴₀𝑑⁴₁𝑥𝑦⁹−305846𝑎³₀𝑏₀𝑑⁴₁𝑥𝑦⁹−760242𝑎²₀𝑏²₀𝑑⁴₁𝑥𝑦⁹−901682𝑎₀𝑏³₀𝑑⁴₁𝑥𝑦⁹−438506𝑏⁴₀𝑑⁴₁𝑥𝑦⁹[phone]𝑦¹⁰−159806071620𝑎₀𝑑₁𝑦¹⁰−99249887196𝑏₀𝑑₁𝑦¹⁰[phone]𝑎²₀𝑑²₁𝑦¹⁰[phone]𝑎₀𝑏₀𝑑²₁𝑦¹⁰[phone]𝑏²₀𝑑²₁𝑦¹⁰−42181381𝑎³₀𝑑³₁𝑦¹⁰−135185562𝑎²₀𝑏₀𝑑³₁𝑦¹⁰−155218542𝑎₀𝑏²₀𝑑³₁𝑦¹⁰−64874815𝑏³₀𝑑³₁𝑦¹⁰+48844𝑎⁴₀𝑑⁴₁𝑦¹⁰+318016𝑎³₀𝑏₀𝑑⁴₁𝑦¹⁰+824332𝑎²₀𝑏²₀𝑑⁴₁𝑦¹⁰+1024232𝑎₀𝑏³₀𝑑⁴₁𝑦¹⁰+525776𝑏⁴₀𝑑⁴₁𝑦¹⁰. (A.1)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by Scientific Research Foundation of Guangxi Education Department (no. ZD2014131), Research Foundation of Hezhou University (no. HZUBS201302), and Natural Science Foundation of China grants nos. 11261013 and 11371373.