Limit Cycles and Isochronous Centers in a Class of Ninth Degree System

Abstract

A class of ninth degree system is studied and the conditions ensuring that its five singular points can be centers and isochronous centers (or linearizable centers) at the same time by exact calculation and strict proof are obtained. What is more, the expressions of Lyapunov constants and periodic constants are simplified, and 21 limit circles could be bifurcated at least.

1. Introduction

As we all know, it is difficult for planar polynomial differential systems to characterize their centers and isochronous centers. Hence, up to now the sufficient and necessary condition determining an isochronous center can only be found by making some appropriate analytic changes of coordinates which let the original system be reduced to a linear system. This kind of appropriate analytic change is very difficult to obtain, so only a handful of isochronous systems have been investigated. Several classes of known isochronous systems studied are as follows: quadratic isochronous centers (see [1]); isochronous centers of a linear center perturbed by third, fourth and fifth degree homogeneous polynomials (see [2–4]); complex polynomial systems (see [5]); reversible systems (see [6, 7]); and isochronous centers of cubic systems with degenerate infinity (see [8]).

In [9], a class of ninth degree system with four isochronous centers was investigated. In this paper, we consider the following ninth degree system: 𝑑𝑥𝑥𝑑𝑡=−𝜆𝑦²+𝑦²⁴+𝑦𝑝₄₀𝑥⁴+𝑝₃₁𝑥³𝑦+𝑝₂₂𝑥²𝑦²+𝑝₀₄𝑦⁴+𝑦𝑝₆₀𝑥⁶+𝑝₄₂𝑥⁴𝑦²+𝑝₂₄𝑥²𝑦⁴+𝑝₀₆𝑦⁶,𝑑𝑦𝑥𝑑𝑡=𝜆𝑥²+𝑦²⁴+𝑥𝑞₄₀𝑥⁴+𝑞₂₂𝑥²𝑦²+𝑞₁₃𝑥𝑦³+𝑞₀₄𝑦⁴+𝑥𝑞₆₀𝑥⁶+𝑞₄₂𝑥⁴𝑦²+𝑞₂₄𝑥²𝑦⁴+𝑞₀₆𝑦⁶, (1) where 𝑝₄₀=−𝑞₀₄=12−10𝜆+4𝐴₃₂+8𝐴₅₀,𝑝−5𝛽₃₁=−𝑞₁₃=−8𝐵₅₀,𝑝₂₂=−𝑞₂₂=10𝜆−6𝐴₃₂−4𝐴₅₀𝑝+5𝛽,₀₄=−𝑞₄₀=12,𝑝−2𝜆−𝛽₆₀=−𝑞₀₆=1212𝜆−4𝐴₃₂−8𝐴₅₀,𝑝+7𝛽₄₂=−𝑞₂₄=12−4𝜆+8𝐴₃₂+16𝐴₅₀,𝑝−9𝛽₂₄=−𝑞₄₂3=−24𝜆−4𝐴₃₂−8𝐴₅₀,𝑝+5𝛽₀₆=−𝑞₆₀=124𝜆+𝛽,𝜆𝛽≠0. (2)

The paper is organized as follows. In Section 2, the preliminary methods to calculate the focal values (or Lyapunov constants) and period constants which are necessary in Sections 3 and 5 are presented. In Section 3, two appropriate transformations, changing system (1) to a class of Z4-equivariant fifth degree system in which the focal values are calculated, are provided. Based on it, the condition and proof that the infinity and four elementary singular points of (1) can be centers are gained. The bifurcations of 21 limit circles are the subject of Section 4. At last, in Section 5, the period constants with simpler expressions are given; moreover, the sufficient and necessary condition and proof are given to illustrate that the infinity and four elementary singular points of (1) become isochronous centers.

2. Some Preliminary Results

In [10, 11], the complex system of complex center is 𝑑𝑧𝑑𝑇=𝑧+^∞_𝑘=2𝑍_𝑘(𝑧,𝑤)=𝑍(𝑧,𝑤),𝑑𝑤𝑑𝑇=−𝑤−^∞_𝑘=2𝑊_𝑘(𝑧,𝑤)=−𝑊(𝑧,𝑤), (3) where 𝑍_𝑘(𝑧,𝑤)=_{𝛼+𝛽=𝑘}𝑎_𝛼𝛽𝑧^𝛼𝑤^𝛽,𝑊_𝑘(𝑧,𝑤)=_{𝛼+𝛽=𝑘}𝑏_𝛼𝛽𝑤^𝛼𝑧^𝛽, (4) and the authors gave two recursive algorithms to determine necessary conditions for a center. We now restate the definitions and algorithms.

Theorem 1 (see [11]). For system (3), we can derive successively the terms of the following formal series: 𝑀(𝑧,𝑤)=^∞_𝛼+𝛽=0𝑐_𝛼𝛽𝑧^𝛼𝑤^𝛽, (5) such that 𝜕(𝑀𝑍)−𝜕()𝜕𝑧𝑀𝑊=𝜕𝑤^∞_𝑚=1(𝑚+1)𝜇_𝑚(𝑧𝑤)^𝑚, (6) where 𝑐₀₀=1, for all 𝑐_𝑘𝑘∈𝑅, 𝑘=1,2,…, and for any integer 𝑚, 𝜇_𝑚 is determined by the following recursive formulae: 𝑐₀₀=1;𝑤ℎ𝑒𝑛𝛼=𝛽>0𝑜𝑟𝛼<0,𝑜𝑟𝛽<0,𝑐_𝛼𝛽𝑐=0;𝑒𝑙𝑠𝑒_𝛼𝛽=1𝛽−𝛼^𝛼+𝛽+2_𝑘+𝑗=3(𝛼+1)𝑎_{𝑘,𝑗−1}−𝛽+1𝑏_{𝑗,𝑘−1}𝑐_{𝛼−𝑘+1,𝛽−𝑗+1},𝜇_𝑚=^2𝑚+2_𝑘+𝑗=3𝑎_{𝑘,𝑗−1}−𝑏_{𝑗,𝑘−1}𝑐_{𝑚−𝑘+1,𝑚−𝑗+1}. (7)

Theorem 2 (see [12]). For system (3), we can derive uniquely the following formal series: 𝑓(𝑧,𝑤)=𝑧+^∞_𝑘+𝑗=2𝑐^′_𝑘𝑗𝑧^𝑘𝑤^𝑗,𝑔(𝑧,𝑤)=𝑤+^∞_𝑘+𝑗=2𝑑^′_𝑘𝑗𝑤^𝑘𝑧^𝑗, (8) where 𝑐^′_𝑘+1,𝑘=𝑑^′_𝑘+1,𝑘=0, 𝑘=1,2,…, such that 𝑑𝑓()+𝑑𝑇=𝑓𝑧,𝑤^∞_𝑗=1𝑝^′_𝑗𝑧^𝑗+1𝑤^𝑗,𝑑𝑔()−𝑑𝑇=−𝑔𝑧,𝑤^∞_𝑗=1𝑞^′_𝑗𝑤^𝑗+1𝑧^𝑗, (9) and when 𝑘−𝑗−1≠0, 𝑐^′_𝑘𝑗 and 𝑑^′_𝑘𝑗 are determined by the following recursive formulae: 𝑐^′_𝑘𝑗=1𝑗+1−𝑘^𝑘+𝑗+1_𝛼+𝛽=3(𝑘−𝛼+1)𝑎_{𝛼,𝛽−1}−𝑗−𝛽+1𝑏_{𝛽,𝛼−1}𝑐^′_{𝑘−𝛼+1,𝑗−𝛽+1},𝑑^′_𝑘𝑗=1𝑗+1−𝑘^𝑘+𝑗+1_𝛼+𝛽=3(𝑘−𝛼+1)𝑏_{𝛼,𝛽−1}−𝑗−𝛽+1𝑎_{𝛽,𝛼−1}𝑑^′_{𝑘−𝛼+1,𝑗−𝛽+1}, (10) and for any positive integer 𝑗, 𝑝^′_𝑗 and 𝑞^′_𝑗 are determined by the following recursive formulae: 𝑝^′_𝑗=^2𝑗+2_𝛼+𝛽=3𝑗−𝛼+2𝑎_{𝛼,𝛽−1}−𝑗−𝛽+1𝑏_{𝛽,𝛼−1}𝑐^′_{𝑗−𝛼+2,𝑗−𝛽+1},𝑞^′_𝑗=^2𝑗+2_𝛼+𝛽=3𝑗−𝛼+2𝑏_{𝛼,𝛽−1}−𝑗−𝛽+1𝑎_{𝛽,𝛼−1}𝑑^′_{𝑗−𝛼+2,𝑗−𝛽+1}. (11) In the above expression, we have let 𝑐^′₁₀=𝑑^′₁₀=1, 𝑐^′₀₁=𝑑^′₀₁=0, and if 𝛼<0 or 𝛽<0, let 𝑎_𝛼𝛽=𝑏_𝛼𝛽=𝑐^′_𝛼𝛽=𝑑^′_𝛼𝛽=0.

3. Singular Point Quantities and Center Conditions

It can be easily checked that system (1) have four elementary critical points (±1,0), (0,±1) and the infinity. After introducing the method to calculate the focal values and period constants of system (1), we try to make some appropriate transformations so as to carry out our investigation.

By means of the Bendixson homeomorphous transformation 𝑥𝑢=𝑥²+𝑦²𝑦,𝑣=𝑥²+𝑦², (12) and time transformation 𝑑𝑡=𝑥²+𝑦²⁴𝑑𝜏, (13) system (1) can be transformed into the following real system: 𝑑𝑢𝑑𝜏=−𝜆𝑣+−2𝜆+2𝐴₃₂+4𝐴₅₀−52𝛽𝑢²𝑣+𝛽2𝜆+2𝑣³+3𝜆−2𝐴₃₂−4𝐴₅₀+32𝛽𝑢⁴𝑣+8𝐵₅₀𝑢³𝑣²+2𝜆−2𝐴₃₂+4𝐴₅₀+𝛽𝑢²𝑣³−𝛽𝜆+2𝑣⁵,𝑑𝑣𝛽𝑑𝜏=𝜆𝑢−2𝜆+2𝑢³+2𝜆−2𝐴₃₂−4𝐴₅₀+52𝛽𝑢𝑣²+𝛽𝜆+2𝑢⁵+−2𝜆+2𝐴₃₂−4𝐴₅₀−𝛽𝑢³𝑣²−8𝐵₅₀𝑢²𝑣³+−3𝜆+2𝐴₃₂+4𝐴₅₀−32𝛽𝑢𝑣⁴. (14) Then, the infinity and four elementary focuses (±1,0), (0,±1) of (1) become the origin and (±1,0), (0,±1) of system (14), respectively. Thus, we have the following theorems.

Theorem 3. System (14) is a class of Z4-equivariant differential polynomial system of fifth degree about point (−1,0).

Proof. The system is invariant by the transformation of rotation 𝜋𝑢=𝑥cos2𝜋−𝑦sin2𝜋,𝑣=𝑥cos2𝜋+𝑦sin2. (15) Then system (14) is a Z4-equivariant polynomial system about point (−1,0).

Now, we will consider the (0,0) of system (14). Let 𝑧=𝑥+𝑖𝑦,𝑤=𝑥−𝑖𝑦,𝑇=𝑖𝑡. (16) System (14) can be transformed into the following complex system: 𝑑𝑧𝑑𝑇=𝜆𝑧−𝜆+𝑎₃₂+2𝑎₅₀+14𝛽𝑤𝑧²+𝑎₃₂−𝜆+2𝑎₅₀−3𝛽4𝑤³+𝑎₅₀𝑧⁵+𝜆−𝑎₃₂−𝑎₅₀+12𝛽𝑤⁴𝑧+𝑎₃₂𝑤²𝑧³,𝑑𝑤𝑑𝑇=−𝜆𝑤+𝜆+𝑏₃₂+2𝑏₅₀+14𝛽𝑧𝑤²−𝑏₃₂−𝜆+2𝑏₅₀−3𝛽4𝑧³−𝑏₅₀𝑤⁵−𝜆−𝑏₃₂−𝑏₅₀+12𝛽𝑧⁴𝑤−𝑏₃₂𝑧²𝑤³, (17) where 𝑎₅₀=𝐴₅₀+𝐵₅₀𝑖2,𝑏₅₀=𝐴₅₀−𝐵₅₀𝑖2,𝑎₃₂=𝐴₃₂−2𝐵₅₀𝑖2,𝑏₃₂=𝐴₃₂+2𝐵₅₀𝑖2. (18)

Theorem 4. The first two singular point quantities at the origin of system (17) are as follows: 𝑢₁=0,𝑢₂=2𝑖𝐵₅₀𝜆. (19)

Theorem 5. For system (17), the first two singular point quantities are zero if and only if the following condition holds: 𝐵₅₀=0. (20)

Theorem 6. For system (14), all the singular point quantities at the origin are zero if and only if the first two singular point quantities are zero; that is, the condition in Theorem 5 holds. Correspondingly, the condition in Theorem 5 is the center condition of the origin.

Proof. If condition (20) holds, system (14) becomes 𝑑𝑢𝑑𝜏=−𝜆𝑣+−2𝜆+2𝐴₃₂+4𝐴₅₀−52𝛽𝑢²𝑣+𝛽2𝜆+2𝑣³+3𝜆−2𝐴₃₂−4𝐴₅₀+32𝛽𝑢⁴𝑣+2𝜆−2𝐴₃₂+4𝐴₅₀+𝛽𝑢²𝑣³−𝛽𝜆+2𝑣⁵,𝑑𝑣𝛽𝑑𝜏=𝜆𝑢−2𝜆+2𝑢³+2𝜆−2𝐴₃₂−4𝐴₅₀+52𝛽𝑢𝑣²+𝛽𝜆+2𝑢⁵+−2𝜆+2𝐴₃₂−4𝐴₅₀−𝛽𝑢³𝑣²+−3𝜆+2𝐴₃₂+4𝐴₅₀−32𝛽𝑢𝑣⁴. (21) Remark (21) is symmetric with 𝑣 axis.

System (14) is a class of Z4-equivariant differential polynomial system, so that the four symmetric elementary focus points of system (14) have the same topological structure; then they have the same center conditions and isochronous center conditions and so on. Without loss of generality, we only need to consider the case of (1,0).

Let 𝑥=𝑢−1,𝑦=𝑣. (22) Then, system (14) can be changed into 𝑑𝑥𝑑𝑡=−𝛽𝑦+8𝜆−4𝐴₃₂−8𝐴₅₀++𝛽𝑥𝑦16𝜆−10𝐴₃₂−20𝐴₅₀+132𝛽𝑥²𝑦+12𝜆−8𝐴₃₂−16𝐴₅₀+6𝛽𝑥³𝑦+3𝜆−2𝐴₃₂−4𝐴₅₀+32𝛽𝑥⁴𝑦+8𝐵₅₀𝑦²+8𝐵₅₀𝑥³𝑦²+4𝜆−2𝐴₃₂+4𝐴₅₀+32𝛽𝑦³+4𝜆−4𝐴₃₂+8𝐴₅₀+2𝛽𝑥𝑦³+24𝐵₅₀𝑥𝑦²+24𝐵₅₀𝑥²𝑦²+2𝜆−2𝐴₃₂+4𝐴₅₀+𝛽𝑥²𝑦³−1𝜆+2𝛽𝑦⁵,𝑑𝑦7𝑑𝑡=𝛽𝑥+4𝜆+2𝛽𝑥²+98𝜆+2𝛽𝑥³+55𝜆+2𝛽𝑥⁴+1𝜆+2𝛽𝑥⁵+32𝛽−8𝐴₅₀𝑦²+4𝐴₃₂−4𝜆−16𝐴₅₀−12𝛽𝑥𝑦²+6𝐴₃₂−6𝜆−12𝐴₅₀−3𝛽𝑥²𝑦²+−2𝜆+2𝐴₃₂−4𝐴₅₀−𝛽𝑥³𝑦²+8𝐵₅₀𝑦³+16𝐵₅₀𝑥𝑦³+8𝐵₅₀𝑥²𝑦³+−3𝜆+2𝐴₃₂+4𝐴₅₀−32𝛽𝑦⁴+−3𝜆+2𝐴₃₂+4𝐴₅₀−32𝛽𝑥𝑦⁴. (23) By noting that 𝑧=𝑥+𝑖𝑦,𝑤=𝑥−𝑖𝑦,𝑇=𝑖𝑡 (24) system (23) can be transformed into its complex system: 𝑑𝑧𝑑𝑇=𝛽𝑧+3𝜆−2𝑎₃₂+34𝛽𝑤²+3𝜆−3𝑎₃₂−2𝑎₅₀+54𝛽𝑤³+𝜆−𝑎₃₂−𝑎₅₀+12𝛽𝑤⁴+2𝜆−8𝑎₅₀+52𝛽+𝑤𝑧6𝜆−3𝑎₃₂−6𝑎₅₀+3𝛽𝑤²𝑧+4𝜆−4𝑎₃₂−4𝑎₅₀+2𝛽𝑤³𝑧+𝜆−𝑎₃₂−𝑎₅₀+12𝛽𝑤⁴𝑧+2𝑎₃₂−𝜆+8𝑎₅₀+14𝛽𝑧²+5𝑎₃₂−𝜆−2𝑎₅₀+14𝛽𝑤𝑧²+3𝑎₃₂𝑤²𝑧²+𝑎₃₂+10𝑎₅₀𝑧³+2𝑎₃₂𝑤𝑧³+𝑎₃₂𝑤²𝑧³+5𝑎₅₀𝑧⁴+𝑎₅₀𝑧⁵,𝑑𝑤𝑑𝑇=−𝛽𝑤−3𝜆−2𝑏₃₂+34𝛽𝑧²−3𝜆−3𝑏₃₂−2𝑏₅₀+54𝛽𝑧³−𝜆−𝑏₃₂−𝑏₅₀+12𝛽𝑧⁴−2𝜆−8𝑏₅₀+52𝛽−𝑤𝑧6𝜆−3𝑏₃₂−6𝑏₅₀+3𝛽𝑧²𝑤−4𝜆−4𝑏₃₂−4𝑏₅₀+2𝛽𝑧³𝑤−𝜆−𝑏₃₂−𝑏₅₀+12𝛽𝑧⁴𝑤−2𝑏₃₂−𝜆+8𝑏₅₀+14𝛽𝑤²−5𝑏₃₂−𝜆−2𝑏₅₀+14𝛽𝑧𝑤²−3𝑏₃₂𝑤²𝑧²−𝑏₃₂+10𝑏₅₀𝑤³−2𝑏₃₂𝑧𝑤³−𝑏₃₂𝑧²𝑤³−5𝑏₅₀𝑤⁴−𝑏₅₀𝑤⁵. (25) Through computing, we have the following theorem.

Theorem 7. The first five singular point quantities at the origin of system (25) are 𝑢₁=−4𝑖𝐵₅₀4𝜆−2𝐴₃₂−12𝐴₅₀+5𝛽𝛽²,𝑢₂=8𝑖𝐵₅₀𝛽−4𝐴₅₀2𝐴₃₂−28𝐴₅₀+5𝛽𝛽³,𝑢₃=8𝑖𝐵₅₀𝛽−4𝐴₅₀²−4480𝐴²₅₀+560𝐵²₅₀+1992𝐴₅₀𝛽−215𝛽²𝛽⁵⁻¹𝑢₄=−64𝑖𝐵₅₀𝛽−4𝐴₅₀²𝑓175𝛽⁷,𝑢₅=−16384𝑖𝐴³₅₀𝐵₅₀𝛽−4𝐴₅₀²𝑔1827307548828125𝛽⁹, (26) where 𝑓=67737600𝐴⁴₅₀−63598080𝐴³₅₀𝛽+22017616𝐴²₅₀𝛽²−3335040𝐴₅₀𝛽³+186875𝛽⁴,𝑔=16146722095087449600𝐴³₅₀−9886197819204477680𝐴²₅₀𝛽+2031858784197168536𝐴₅₀𝛽²−142685023001803215𝛽³. (27)

Theorem 8. For system (25), the first five singular point quantities are zero if and only if one of the following conditions holds: 1𝜆=2𝐴₃₂−4𝐴₅₀,𝛽=4𝐴₅₀;𝐵₅₀=0. (28)(29)

Then we can get the following theorem.

Theorem 9. For system (25), all the singular point quantities at the origin are zero if and only if the first five singular point quantities are zero; that is, one of the conditions in Theorem 8 holds. Correspondingly, the conditions in Theorem 8 are the center conditions of the origin.

Proof. Under condition (28), the system (25) becomes 𝑑𝑥𝑑𝑡=−4𝐴₅₀𝑦−20𝐴₅₀𝑥𝑦−2𝐴₃₂+26𝐴₅₀𝑥²𝑦−2𝐴₃₂+16𝐴₅₀𝑥³1𝑦−2𝐴₃₂+4𝐴₅₀𝑥⁴𝑦+8𝐵₅₀𝑦²+24𝐵₅₀𝑥𝑦²+24𝐵₅₀𝑥²𝑦²+8𝐵₅₀𝑥³𝑦²+2𝐴₅₀𝑦³+8𝐴₅₀−2𝐴₃₂𝑥𝑦³+4𝐴₅₀−𝐴₃₂𝑥²𝑦³−12𝐴₃₂𝑦⁵,𝑑𝑦𝑑𝑡=4𝐴₅₀𝑥+2𝐴₃₂+3𝐴₅₀𝑥²+22𝐴₃₂+𝐴₅₀𝑥³+52𝐴₃₂𝑥⁴+12𝐴₃₂𝑥⁵−2𝐴₅₀𝑦²+2𝐴₃₂−5𝐴₅₀𝑥𝑦²+3𝐴₃₂−4𝐴₅₀𝑥²𝑦²+𝐴₃₂−4𝐴₅₀𝑥³𝑦²+8𝐵₅₀𝑦³+16𝐵₅₀𝑥𝑦³+8𝐵₅₀𝑥²𝑦³+12𝐴₃₂+4𝐴₅₀𝑦⁴+12𝐴₃₂+4𝐴₅₀𝑥𝑦⁴. (30) System (30) has an integrating factor 𝑢=1𝑥,𝑦1+2𝑥+𝑥²+𝑦²³. (31) Under condition (29), system (25) becomes 𝑑𝑥𝑑𝑡=−𝛽𝑦+8𝜆−4𝐴₃₂−8𝐴₅₀++𝛽𝑥𝑦16𝜆−10𝐴₃₂−20𝐴₅₀+132𝛽𝑥²𝑦+12𝜆−8𝐴₃₂−16𝐴₅₀+6𝛽𝑥³𝑦+3𝜆−2𝐴₃₂−4𝐴₅₀+32𝛽𝑥⁴𝑦+4𝜆−2𝐴₃₂+4𝐴₅₀+32𝛽𝑦³+4𝜆−4𝐴₃₂+8𝐴₅₀+2𝛽𝑥𝑦³+2𝜆−2𝐴₃₂+4𝐴₅₀+𝛽𝑥²𝑦³−1𝜆+2𝛽𝑦⁵,𝑑𝑦7𝑑𝑡=𝛽𝑥+4𝜆+2𝛽𝑥²+98𝜆+2𝛽𝑥³+55𝜆+2𝛽𝑥⁴+1𝜆+2𝛽𝑥⁵+−8𝐴₅₀+32𝛽𝑦²+−4𝜆+4𝐴₃₂−16𝐴₅₀−12𝛽𝑥𝑦²+−6𝜆+6𝐴₃₂−12𝐴₅₀−3𝛽𝑥²𝑦²+−2𝜆+2𝐴₃₂−4𝐴₅₀−𝛽𝑥³𝑦²+−3𝜆+2𝐴₃₂+4𝐴₅₀−32𝛽𝑦⁴+−3𝜆+2𝐴₃₂+4𝐴₅₀−32𝛽𝑥𝑦⁴, (32) where (32) is symmetric with 𝑥 axis.

So we get the following theorem.

Theorem 10. The system (14) has five centers (0,0), (±1,0), (0,±1) when 𝐵₅₀=0. Namely, the system (1) has five centers the infinity and (±1,0), (0,±1) when 𝐵₅₀=0.
The system (14) has four centers (±1,0), (0,±1) when 𝜆=(1/2)(𝐴₃₂−4𝐴₅₀), 𝛽=4𝐴₅₀, 𝐵₅₀≠0. Namely, the system (1) has four centers (±1,0), (0,±1) when 𝜆=(1/2)(𝐴₃₂−4𝐴₅₀), 𝛽=4𝐴₅₀, and 𝐵₅₀≠0.

4. Bifurcation of Limit Circles

Theorem 11. For system (14), the (1,0) is a fifth order weak focus if and only if 1𝜆=220𝐴₅₀,𝐴−5𝛽₃₂=1228𝐴₅₀,𝐵−5𝛽₂₅₀=4480𝐴²₅₀−1992𝐴₅₀𝛽+215𝛽²,560𝑓=0,𝑔≠0. (33)

Proof. When 𝑢₁=𝑢₂=𝑢₃=𝑢₄=0, we could find suitable 𝛽 and 𝐴₅₀ to satisfy 𝐵²₅₀>0,𝑢₄=0,_{Resultant𝑓,𝑔,𝐴50=16309978866916163079012825599795663295988359261138540000000000000000000000000000000000𝛽12.} (34)(35)
So the (1,0) is a fifth order weak focus.

Theorem 12. Assume that the conditions in Theorem 11 hold. Then the 5 limit cycles can bifurcate from the (1,0) of System (14), which lie in the neighborhood of the origin.

Proof. If the conditions in Theorem 11 hold, we can obtain that the Jacobian of the 𝑢₁,𝑢₂,𝑢₃,𝑢₄ with respect to 𝜆,𝐴₃₂,𝐵₅₀,𝐴₅₀𝐷𝑢𝐽=¹,𝑢₂,𝑢₃,𝑢₄𝐷𝜆,𝐴₃₂,𝐵₅₀,𝐴₅₀=−2097152𝐵⁵₅₀−4𝐴₅₀+𝛽⁴175𝛽¹7×203212800𝐴⁴₅₀−192864000𝐴³₅₀𝛽+67884512𝐴₅₀𝛽²−10506964𝐴₅₀𝛽³+603755𝛽⁴;Resultant𝑓,𝐽,𝐴₅₀=−661345270173473293422286219719475200000000𝛽¹⁶. (36) So 𝐽≠0. Applying the theory in [13], the result is completed.

Theorem 13. Assume that the conditions in Theorem 11 hold. Then 1 limit cycle can bifurcate from the (0,0) of System (14), which lie in the neighborhood of the origin.

Since System (14) is Z4-equivariant, taking into account Theorems 12 and 13, we conclude the following.

Theorem 14. Assume that the conditions in Theorem 11 hold. Then 5 limit cycles can bifurcate from each of the four fine focus points (±1,0), (0,±1) of System (14), and one limit circle could be created from the origin at the same time; namely, 21 limit cycles can bifurcate from the System (14) in all, five of which are located in the neighborhood of each of the four fine focus points (±1,0), (0,±1) and one of which is located in the neighborhood of origin.

In fact, there are 21 limit cycles for system (1); 5 limit cycles can bifurcate from each of the four fine focus points (±1,0), (0,±1) of System (1), and one limit circle could be created from the infinity at the same time; namely, 21 limit cycles can bifurcate from the system (1) in all. It is a very interesting result.

5. Period Constants and Isochronous Center Conditions

In this section, we focus on the calculation of period constants and isochronous center conditions of the origin and (1,0) for system (17) and (25). According to Theorems 2, we begin by computing period constants through recursive formulae. Under the center conditions given in Section 3, we discuss three different relevant cases: (a) about the origin, case condition (20) holds.

Theorem 15. Under condition (20), the origin of system (17) is a complex isochronous center if and only if the following conditions holds 𝐴₃₂=𝐵₅₀=0,𝐴₅₀=−2𝜆,𝛽=−4𝜆. (37)

Proof. If condition (20) holds,substituting 𝐵₅₀=0 into the recursive formulae in Theorem 2, we can easily get the first four period constants 𝜏₁=−4𝜆+2𝐴₃₂+4𝐴₅₀−𝛽,𝜏2𝜆₂=−64𝜆²+16𝜆𝐴₅₀+20𝜆𝛽+3𝛽²8𝜆²,𝜏₃=−4𝜆+𝛽80𝜆²+46𝜆𝛽+9𝛽²32𝜆³,𝜏₄=3𝛽4𝜆+𝛽104𝜆²+40𝜆𝛽+9𝛽²512𝜆⁴. (38) Clearly, 𝜏₁=𝜏₂=𝜏₃=𝜏₄=0 implies that condition (37) holds.
On the other hand, case condition (37) holds, and system (17) is changed into 𝑑𝑧()𝑧𝑑𝑇=−𝑧𝑧+1)(𝑧−1²+1,𝑑𝑤=()𝑤(𝑑𝑇−1+𝑤1+𝑤)1+𝑤². (39) By means of transformation 𝜉=𝑧𝑧⁴−1^−1/4,𝜂=𝑤𝑤⁴−1^−1/4, (40) system (39) is reduced to a linear system.
Thus, the origin of system (17) is a complex isochronous center.

(b) about (1,0) of system (25).

When condition (28) holds, substituting 𝜆=(1/2)(𝐴₃₂−4𝐴₅₀), 𝛽=4𝐴₅₀, into the recursive formulae in Theorem 2, we easily obtain the first two period constants 𝜏₁𝐴=−³²−2𝐴₅₀𝐴₃₂−4𝐴₅₀,𝜏₂=−2. (41) Thus (1,0) could not be an isochronous center when condition (28) holds.

Theorem 16. (1,0) of system (25) is a complex isochronous center if and only if condition (37) holds.

Proof. When condition (29) holds, substituting 𝐵₅₀ into the recursive formulae in Theorem 2, we easily obtain the first four period constants: 𝜏₁=−4𝜆+10𝐴₃₂−4𝐴₅₀+𝛽2,𝜏₂=80𝜆−13392𝜆²+80𝐴₅₀+17856𝜆𝐴₅₀−5952𝐴²₅₀−20𝛽−15624𝜆𝛽+10416𝐴₅₀𝛽−4557𝛽²(200)⁻¹,𝜏₃=18000−237312𝜆²+2939328𝜆³+208896𝜆𝐴₅₀−5878656𝜆²𝐴₅₀−46592𝐴²₅₀−870912𝐴³₅₀−299104𝜆𝛽+5143824𝜆²𝛽+162816𝐴₅₀𝛽−6858432𝐴₅₀𝜆𝛽+2286144𝐴²₅₀𝛽−103592𝛽²+3000564𝜆𝛽²−2000376𝐴₅₀𝛽²+583443𝛽³+3919104𝜆𝐴²₅₀,𝜏₄=140000−51200𝜆²[phone]𝜆³[phone]𝜆⁴+57600𝜆𝐴₅₀−29498368𝜆²𝐴₅₀−46282752𝜆³𝐴₅₀+6545472𝜆𝐴²₅₀[phone]𝜆²𝐴²₅₀+421504𝐴³₅₀−20570112𝜆𝐴³₅₀+3428352𝐴⁴₅₀−70400𝜆𝛽+40046432𝜆²𝛽+40497408𝜆³𝛽+39600𝐴₅₀𝛽−11200𝐴²₅₀−80994816𝜆²𝐴50𝛽+996112𝐴²₅₀𝛽+53996544𝜆𝐴²₅₀𝛽−11999232𝐴³₅₀𝛽−24200𝛽²[phone]𝜆𝛽²[phone]𝜆²𝛽²−3856888𝐴₅₀𝛽²−47246976𝜆𝐴₅₀𝛽²−24098656𝜆𝐴50𝛽+15748992𝐴²₅₀𝛽²+2329754𝛽³[phone]𝜆𝛽³−9186912𝐴₅₀𝛽³+2009637𝛽⁴. (42)
Denoting by 𝑔₁=𝑅𝜏₂,𝜏₃,𝜆,𝑔₂=𝑅𝜏₂,𝜏₄,𝜆, (43) where 𝑅(𝜏_𝑖,𝜏_𝑗,𝐴) is the resultant of 𝜏_𝑖,𝜏_𝑗 with respect to 𝐴, we have 𝑅(𝑔₁,𝑔₂,𝛽)=0. From 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑜𝑎𝑙𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟[𝑔₂,𝑔₁,𝐴₅₀]=0, we could get that 𝛽=2𝐴₅₀. When 𝛽=2𝐴₅₀, 𝜏₂=𝜏₃=𝜏₄=0 deduce 𝐴₃₂=𝐵₅₀=0, 𝐴₅₀=−2𝜆, 𝛽=−4𝜆.
Besides, when condition (37) holds, system (25) is changed into 𝑑𝑧=1𝑑𝑇4𝑧(𝑧+1)(𝑧+2)𝑧²+2𝑧+2,𝑑𝑤1𝑑𝑇=−4(1+𝑤)𝑤(2+𝑤)2+2𝑤+𝑤². (44) Letting that 𝑧(𝜉=2+𝑧)2+2𝑧+𝑧²(𝑧+1)⁴,𝑤()𝜂=2+𝑤2+2𝑤+𝑤²(𝑤+1)⁴, (45) system (44) is reduced to a linear system.

From Theorems 15 and 16, we can conclude the following theorem.

Theorem 17. The infinity and four elementary singular points (±1,0), (0,±1) are five isochronous centers of (1) if and only if condition (37) holds; namely 𝐴₃₂=𝐵₅₀=0, 𝐴₅₀=−2𝜆, 𝛽=−4𝜆.

Acknowledgments

This research is partially supported by the National Nature Science Foundation of China (11201211, 111101126, and 1371373) and Nature Science Foundation of Shandong Province (ZR2012AL04) and Applied Mathematics Enhancement Program of Linyi University.