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

Theorem 1 (see [11].)For system (3), we can derive successively the terms of the following formal series:

such that 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:

Theorem 2 (see [12].)For system (3), we can derive uniquely the following formal series:

where $$, k = 1,2, …, such that and when k − j − 1 ≠ 0, $$ and $$ are determined by the following recursive formulae: and for any positive integer j, $$ and $$ are determined by the following recursive formulae: In the above expression, we have let $$, $$, and if α < 0 or β < 0, let $$.

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

Then system (14) is a Z4-equivariant polynomial system about point (−1,0).

Theorem 4. The first two singular point quantities at the origin of system (17) are as follows:

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

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

Remark (21) is symmetric with v axis.

Theorem 7. The first five singular point quantities at the origin of system (25) are

where

Theorem 8. For system (25), the first five singular point quantities are zero if and only if one of the following conditions holds:

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

System (30) has an integrating factor Under condition (29), system (25) becomes where (32) is symmetric with x axis.

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.

Theorem 11. For system (14), the (1,0) is a fifth order weak focus if and only if

Proof. When u₁ = u₂ = u₃ = u₄ = 0, we could find suitable β and A₅₀ to satisfy

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, A₅₀

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.

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.

Theorem 15. Under condition (20), the origin of system (17) is a complex isochronous center if and only if the following conditions holds

Proof. If condition (20) holds,substituting B₅₀ = 0 into the recursive formulae in Theorem 2, we can easily get the first four period constants

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

By means of transformation system (39) is reduced to a linear system.

Thus, the origin of system (17) is a complex isochronous center.

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:

Denoting by

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 polynomioal remainder[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

Letting that system (44) is reduced to a linear system.

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.