3.1. The formal series method of computing degenerate singular point quantities on center manifold

Let us consider the real n-dimensional systems with two zero eigenvalues and zero linear part as follows

#### Mutiple Hopf Bifurcation on Center Manifold http://dx.doi.org/10.5772/65674 11

$$\begin{cases} \frac{d\mathbf{x}}{dt} = (\delta \mathbf{x} - \mathbf{y})(\mathbf{x}^2 + \mathbf{y}^2)^q + \sum\_{k+j+1=2q+2}^{\infty} A\_{kj1} \mathbf{x}^k \mathbf{y}^1 \mathbf{u}^1 = X(\mathbf{x}, \mathbf{y}, \mathbf{u}),\\ \frac{d\mathbf{y}}{dt} = (\mathbf{x} - \delta \mathbf{y})(\mathbf{x}^2 + \mathbf{y}^2)^q + \sum\_{k+j+1=2q+2}^{\infty} B\_{kj1} \mathbf{x}^k \mathbf{y}^1 \mathbf{u}^1 = Y(\mathbf{x}, \mathbf{y}, \mathbf{u}),\\ \frac{d\mathbf{u}\_i}{dt} = -d\_i \mathbf{u}\_i + \sum\_{k+j+1=2}^{\infty} d\_{kj} \mathbf{z}^k w^j \mathbf{u}^1 = \mathcal{U}\_i(\mathbf{x}, \mathbf{y}, \mathbf{u}), \ i = 1, 2, \dots, n-2 \end{cases} \tag{23}$$

where the subscript "kj1" denotes "kjl1⋯ln<sup>−</sup>2", <sup>u</sup><sup>1</sup> <sup>¼</sup> <sup>u</sup><sup>l</sup><sup>1</sup> <sup>1</sup> ul<sup>2</sup> <sup>2</sup> <sup>⋯</sup>uln<sup>−</sup><sup>2</sup> <sup>n</sup>−2, and l ¼ ∑ n−2 i¼1 li, all di > 0, x; y; ui; t; δ; Akjl; Bkjl; dkjl∈R, q; k; j; li∈N. Obviously, the origin of system (23) is a high-order degenerate singular point with two zero eigenvalues and n−2 negative ones.

In order to discuss the calculation method of the focal values on center manifold of the system (23), from the center manifold theorem [1], we take an approximation to the center manifold:

$$\mathbf{u} = \mathbf{u}(x, y) = \mathbf{u}\_2(x, y) + \mathbf{h}.\\ \mathbf{o}.\\ \text{t.} \tag{24}$$

where u ¼ ðx1;x2;⋯;xn<sup>−</sup>2Þ <sup>T</sup>, u<sup>2</sup> is a quadratic homogeneous polynomial vector in x and y, and h. o.t. denotes the terms with orders greater than or equal to 3. Substituting u ¼ uðx;yÞ into the equations of system (23), we obtain a real planar polynomial differential system as follows

$$\begin{cases} \frac{\text{dx}}{\text{dt}} = (\delta \text{x} - y)(\text{x}^2 + y^2)^q + \sum\_{k=2q+2}^{\infty} \text{X}\_k(\text{x}, y) = \tilde{\text{X}}(\text{x}, y),\\ \frac{\text{dy}}{\text{dt}} = (\text{x} - \delta y)(\text{x}^2 + y^2)^q + \sum\_{k=2q+2}^{\infty} \text{Y}\_k(\text{x}, y) = \tilde{\text{Y}}(\text{x}, y) \end{cases} \tag{25}$$

where Xkðx;yÞ, Ykðx;yÞ are homogeneous polynomials of degree k, and the origin is degenerate with a zero linear part.

For system (25), some significant works have been done in Refs. [26] and [27]. Let us recall the related notions and results.

By means of transformation (5)

<sup>f</sup> <sup>1</sup> <sup>¼</sup> <sup>8</sup>a<sup>3</sup>ce <sup>þ</sup> <sup>8</sup>a<sup>3</sup>c−8a<sup>3</sup>e−8a<sup>3</sup>−2a<sup>2</sup>bce <sup>þ</sup> <sup>2</sup>a<sup>2</sup>be <sup>þ</sup> <sup>8</sup>a<sup>2</sup>ce <sup>þ</sup> <sup>8</sup>a<sup>2</sup><sup>c</sup>

ð2ae þ 2a−bÞðe þ 1),

<sup>d</sup><sup>1</sup> <sup>¼</sup> <sup>8</sup>a<sup>3</sup>c−8a<sup>3</sup>−2a<sup>2</sup>bc <sup>þ</sup> <sup>2</sup>a<sup>2</sup><sup>b</sup> <sup>þ</sup> <sup>8</sup>a<sup>2</sup>c−8a<sup>2</sup> <sup>þ</sup> <sup>3</sup>ab<sup>2</sup> <sup>þ</sup> <sup>3</sup>b<sup>2</sup>

<sup>f</sup> <sup>4</sup> <sup>¼</sup> <sup>8</sup>a<sup>5</sup>c<sup>2</sup>−16a<sup>5</sup><sup>c</sup> <sup>þ</sup> <sup>8</sup>a<sup>5</sup>−2a<sup>4</sup>bc<sup>2</sup> <sup>þ</sup> <sup>2</sup>a<sup>4</sup>bc <sup>þ</sup> <sup>8</sup>a<sup>4</sup>c<sup>2</sup>−16a<sup>4</sup><sup>c</sup> <sup>þ</sup> <sup>8</sup>a<sup>4</sup> <sup>þ</sup> <sup>2</sup>a<sup>3</sup>b<sup>2</sup>

−b<sup>3</sup> ;

Theorem 6. For the flow on center manifold of the system (14), the first 2 focal values of the origin are

Remark 3. In contrast to the result and process in [36], one can easily see that our first quantity is basically consistent with its characteristic exponent of bifurcating periodic solutions, and our algorithm is easy to realize with computer algebra system due to the linear recursion formulas,

Theorem 7. At least two small limit cycles can be bifurcated from the origin of the 4D-hyoerchaotic

The rigorous proof of the above theorem is very similar to the previous ones in [14, 16], namely, by calculating the Jacobian determinant with respect to the functions v3; v<sup>5</sup> and its variables,

Up till now, study on bifurcation of limit cycles from the degenerate singularity of higher dimensional nonlinear systems (1) is hardly seen in published references. Here, we will investigate the Hopf bifurcation problem from the high-order critical point on the center manifold.

3.1. The formal series method of computing degenerate singular point quantities on center

Let us consider the real n-dimensional systems with two zero eigenvalues and zero linear part

and more convenient to investigate the multiple Hopf bifurcation on center manifold.

Considering its Hopf bifurcation form of Theorem 6, we have the following:

system (14), which lie in the neighborhood of the origin restricted to the center manifold.

<sup>e</sup> <sup>þ</sup> <sup>2</sup>ab<sup>2</sup> <sup>þ</sup> <sup>2</sup>abc−2ab <sup>þ</sup> <sup>3</sup>b<sup>2</sup>

<sup>c</sup> <sup>þ</sup> <sup>4</sup>a<sup>2</sup>b<sup>3</sup> <sup>þ</sup> <sup>2</sup>a<sup>2</sup>b<sup>2</sup>

Þðc−1Þ 3=2 ; c

;

v<sup>3</sup> ¼ iπμ1; v<sup>5</sup> ¼ iπμ<sup>2</sup> (22)

<sup>e</sup> <sup>þ</sup> <sup>3</sup>b<sup>2</sup> ;

c

<sup>c</sup> <sup>þ</sup> <sup>3</sup>ab<sup>2</sup>

<sup>−</sup>8a<sup>2</sup>e−8a<sup>2</sup> <sup>þ</sup> ab<sup>2</sup>

<sup>d</sup><sup>3</sup> <sup>¼</sup> <sup>9</sup>a<sup>2</sup>c−8a<sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>a</sup> <sup>þ</sup> <sup>1</sup>;

3

<sup>d</sup><sup>0</sup> ¼ ða<sup>2</sup><sup>c</sup> <sup>þ</sup> <sup>2</sup><sup>a</sup> <sup>þ</sup> <sup>1</sup>Þð4a<sup>2</sup>c−4a<sup>2</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup>

From Remark 1 and the singular point quantities (21), we have

where the expression of v<sup>5</sup> is obtained under the condition of v<sup>3</sup> ¼ 0.

<sup>f</sup> <sup>3</sup> <sup>¼</sup> <sup>4</sup>a<sup>2</sup><sup>e</sup> <sup>þ</sup> <sup>4</sup>a<sup>2</sup>−3abe−2ab <sup>þ</sup> <sup>4</sup>ae <sup>þ</sup> <sup>4</sup><sup>a</sup> <sup>þ</sup> <sup>b</sup>;

<sup>−</sup>4a<sup>3</sup>bc <sup>þ</sup> <sup>4</sup>a<sup>3</sup>b−5a<sup>2</sup>b<sup>3</sup>

<sup>−</sup>2a<sup>2</sup>bc <sup>þ</sup> <sup>2</sup>a<sup>2</sup>b−2ab<sup>3</sup>

<sup>d</sup><sup>2</sup> <sup>¼</sup> <sup>8</sup>a<sup>2</sup><sup>e</sup> <sup>þ</sup> <sup>8</sup>a<sup>2</sup>−2abe <sup>þ</sup> <sup>8</sup>ae <sup>þ</sup> <sup>8</sup><sup>a</sup> <sup>þ</sup> <sup>b</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>b;

and the above expression of μ<sup>2</sup> is obtained under the condition of μ<sup>1</sup> ¼ 0.

f <sup>2</sup> ¼ ð2a þ b þ 2Þ

10 Manifolds - Current Research Areas

−2a<sup>3</sup>b<sup>2</sup>

−2a<sup>2</sup>b<sup>2</sup>

as follow

which will not be given here.

manifold

as follows

3. Case of the degenerate singular point

$$z = x + y\mathbf{i},\ \ w = x - y\mathbf{i},\ \ u = \nu,\ \ T = \mathbf{i}\ t,\ \ \mathbf{i} = \sqrt{-1},$$

system (25) is transformed into following system:

$$\begin{cases} \frac{d\mathbf{z}}{dT} = (\mathbf{1} \cdot \mathbf{i}\delta)\mathbf{z}^{q+1}w^{q} + \sum\_{k+j=2q+2}^{\infty} a\_{kj}\mathbf{z}^{k}w^{j} = \mathbf{Z}(\mathbf{z}, \mathbf{w}),\\ \frac{d\mathbf{w}}{dT} = -(\mathbf{1} + \mathbf{i}\delta)\mathbf{z}^{q}w^{q+1} - \sum\_{k+j=2q+2}^{\infty} b\_{kj}\mathbf{z}^{k}w^{j} = -\mathbf{W}(\mathbf{z}, \mathbf{w}) \end{cases} \tag{26}$$

where z; w; T are complex variables and for any positive integer k; j, we have akj ¼ bkj, then systems (25) and (26) are called concomitant.

For any positive integer k, we denote

$$f\_k(z, w) = \sum\_{\alpha + \beta = k} c\_{\alpha\beta} z^{\alpha} w^{\beta}$$

a homogeneous polynomial of degree k with c<sup>00</sup> ¼ 1; ckk ¼ 0; k ¼ 1, 2,⋯.

Theorem 8 ([26, 27]). For system (26) with δ ¼ 0, we can derive successively the terms of the following formal series:

$$F(z, w) = zw \left[ 1 + \sum\_{m=1}^{\infty} \frac{f\_{m(2\eta+3)}(z, w)}{(z\varpi)^{m(\eta+1)}} \right] \tag{27}$$

such that

$$\frac{\partial F}{\partial T} = \frac{\partial F}{\partial z} Z - \frac{\partial F}{\partial w} W = (zw)^q \sum\_{m=1}^{\infty} \mu\_m (zw)^{m+1}.\tag{28}$$

Definition 3. If δ ¼ 0 holds, μ<sup>m</sup> in expression (28) is called the mth singular point quantity at the degenerate singular point for system (26) or (1.3.26) is also called the mth singular point quantity of the origin on the center manifold of system (23), where m ¼ 1; 2;⋯:

Similar to Theorem 2, there also exists a equivalence between the mth singular point quantity and the mth focal value v2mþ<sup>1</sup>ð2πÞ at the origin on center manifold of system (23).

Theorem 9. For system (23) with δ ¼ 0, and any positive integer m, the following assertion holds: <sup>v</sup>2mþ<sup>1</sup>ð2πÞ~iπμm, namely

$$w\_{2m+1}(2\pi) = i\pi \left(\mu\_m + \sum\_{k=1}^{m-1} \xi\_m^{(k)} \mu\_k \right),\tag{29}$$

where ξðk<sup>Þ</sup> <sup>m</sup> ðk ¼ 1; 2;⋯;m−1Þ are polynomial functions of coefficients of system (26). Then, the relation between v2mþ<sup>1</sup>ð2πÞ and μ<sup>m</sup> is called the algebraic equivalence.

Remark 4. In fact, from Theorem 2, for any positive integer m ¼ 2; 3;⋯, if μ<sup>1</sup> ¼ μ<sup>2</sup> ¼ ⋯ ¼ μm−<sup>1</sup> ¼ 0 and v1ð2πÞ ¼ v3ð2πÞ⋯ ¼ v2m−1ð2πÞ ¼ 0 hold, and vice versa. And more the stability and bifurcation of the origin of system (23) can be figured out by calculating the singular point quantities.

Corollary 2. The origin of system (23) is a center restricted to the center manifold if and only if μ<sup>m</sup> ¼ 0 for all m.

#### 3.2. An example of three-dimensional system

Now we consider an example for system (23) with n ¼ 3, it can be put in its concomitant form as follows

#### Mutiple Hopf Bifurcation on Center Manifold http://dx.doi.org/10.5772/65674 13

$$\begin{cases} \frac{d\mathbf{z}}{dT} = (\mathbf{1} \cdot \mathbf{i}\delta)\mathbf{z}^2 w + \imath\omega \left(a\_{20}\mathbf{z}^2 + a\_{11}\varpi w + a\_{02}w^2\right) = \mathbf{Z},\\ \frac{d\mathbf{w}}{dT} = -(\mathbf{1} + \mathbf{i}\delta)\varpi w^2 - \imath\omega w \left(b\_{20}w^2 + b\_{11}\imath\varpi + b\_{02}\mathbf{z}^2\right) = -\mathcal{W},\\ \frac{d\mathbf{u}}{dT} = \mathbf{i}\mathbf{u} + \mathbf{i}d\_1\varpi w = \mathcal{U}, \end{cases} \tag{30}$$

where d1≠0 and

f <sup>k</sup>ðz;wÞ ¼ ∑

a homogeneous polynomial of degree k with c<sup>00</sup> ¼ 1; ckk ¼ 0; k ¼ 1, 2,⋯.

dF <sup>d</sup><sup>T</sup> <sup>¼</sup> <sup>∂</sup><sup>F</sup> ∂z

between v2mþ<sup>1</sup>ð2πÞ and μ<sup>m</sup> is called the algebraic equivalence.

3.2. An example of three-dimensional system

Fðz;wÞ ¼ zw 1 þ ∑

<sup>Z</sup><sup>−</sup> <sup>∂</sup><sup>F</sup> ∂w

quantity of the origin on the center manifold of system (23), where m ¼ 1; 2;⋯:

and the mth focal value v2mþ<sup>1</sup>ð2πÞ at the origin on center manifold of system (23).

v2mþ<sup>1</sup>ð2πÞ ¼ iπ μ<sup>m</sup> þ ∑

following formal series:

12 Manifolds - Current Research Areas

<sup>v</sup>2mþ<sup>1</sup>ð2πÞ~iπμm, namely

where ξðk<sup>Þ</sup>

quantities.

for all m.

as follows

such that

αþβ¼k

Theorem 8 ([26, 27]). For system (26) with δ ¼ 0, we can derive successively the terms of the

∞ m¼1

W ¼ ðzwÞ

Definition 3. If δ ¼ 0 holds, μ<sup>m</sup> in expression (28) is called the mth singular point quantity at the degenerate singular point for system (26) or (1.3.26) is also called the mth singular point

Similar to Theorem 2, there also exists a equivalence between the mth singular point quantity

Theorem 9. For system (23) with δ ¼ 0, and any positive integer m, the following assertion holds:

cαβz<sup>α</sup>w<sup>β</sup>

<sup>f</sup> <sup>m</sup>ð2qþ<sup>3</sup>Þðz;w<sup>Þ</sup> ðzwÞ

" #

<sup>q</sup> ∑ ∞ m¼1

> m−1 k¼1 ξðk<sup>Þ</sup> <sup>m</sup> μ<sup>k</sup>

<sup>m</sup> ðk ¼ 1; 2;⋯;m−1Þ are polynomial functions of coefficients of system (26). Then, the relation

Remark 4. In fact, from Theorem 2, for any positive integer m ¼ 2; 3;⋯, if μ<sup>1</sup> ¼ μ<sup>2</sup> ¼ ⋯ ¼ μm−<sup>1</sup> ¼ 0 and v1ð2πÞ ¼ v3ð2πÞ⋯ ¼ v2m−1ð2πÞ ¼ 0 hold, and vice versa. And more the stability and bifurcation of the origin of system (23) can be figured out by calculating the singular point

Corollary 2. The origin of system (23) is a center restricted to the center manifold if and only if μ<sup>m</sup> ¼ 0

Now we consider an example for system (23) with n ¼ 3, it can be put in its concomitant form

� �

mðqþ1Þ

μmðzwÞ

mþ1

(27)

: (28)

, (29)

$$a\_{i\rangle} = A\_i + \mathbf{i}B\_i, \ b\_{i\rangle} = A\_i \mathbf{-i}B\_i, A\_i, B\_i \in \mathbb{R}, \, i, j = 0, 1, 2,\tag{31}$$

namely, aij ¼ bij. Then for the center manifold of system (30), from the transformation (5), we can determine the formal expression (24): u ¼ uðx;yÞ ¼ u~ðz;wÞ, thus obtain

$$\begin{cases} \frac{d\mathbf{z}}{dT} = (\mathbf{1} \cdot \mathbf{i} \delta) \mathbf{z}^2 w + \check{u} z \left( a\_{20} \mathbf{z}^2 + a\_{11} \mathbf{z} w + a\_{02} w^2 \right) = \check{\mathbf{Z}},\\ \frac{d\mathbf{w}}{dT} = -(\mathbf{1} + \mathbf{i} \delta) \mathbf{z} w^2 - \check{u} w \left( b\_{20} w^2 + b\_{11} w \mathbf{z} + b\_{02} \mathbf{z}^2 \right) = -\check{\mathbf{W}} \end{cases}.\tag{32}$$

Remark 5. For system (32), the corresponding n ¼ 1 in (27) and (28) of Theorem 8, we figure out that each μ<sup>m</sup> is related to only the coefficients of the first 2m þ 3 order terms of system (32), m ¼ 1; 2;⋯. Here, we determine the above u~ just to the sixth-order term as follows

$$
\tilde{\boldsymbol{\mu}}(\boldsymbol{z}, \boldsymbol{w}) = \sum\_{k=2}^{6} \tilde{\boldsymbol{\mu}}\_{k}(\boldsymbol{z}, \boldsymbol{w}) \tag{33}
$$

where u~<sup>k</sup> is a homogeneous polynomial in z;w of degree k and

$$\begin{aligned} \|\boldsymbol{\tilde{u}}\_{2} = -d\_{1}zw, \ \boldsymbol{\tilde{u}}\_{4} = 2\delta d\_{1}z^{2}w^{2}, \ \boldsymbol{\tilde{u}}\_{3} = \boldsymbol{\tilde{u}}\_{4} = \boldsymbol{\tilde{u}}\_{5} = \boldsymbol{0}, \\ \boldsymbol{\tilde{u}}\_{6} = -\mathbf{i}d\_{1}zwz( (a\_{02} - b\_{20})d\_{1}w^{3}z + (a\_{11}d\_{1} - b\_{11}d\_{1} - 8\mathbf{i}\delta^{2})w^{2}z^{2} \\ + (a\_{20} - b\_{02})d\_{1}wz^{3}). \end{aligned} \tag{34}$$

Hence, Z~ and W~ in system (32) are two polynomials with degree 9.

Theorem 10. For system (32) with δ ¼ 0, we can derive successively the terms of the formal series (27), such that (28) holds (cαβ, μ<sup>m</sup> in Appendix A).

Applying the powerful symbolic computation function of the Mathematica system and the recursive formulas in Theorem 10, and from Remark 5, we obtain the first three singular point quantities as follows

$$\begin{aligned} \mu\_1 &= -d\_1(a\_{11} - b\_{11}), \\ \mu\_2 &= d\_1^2(b\_{20}b\_{02} - a\_{20}a\_{02}), \\ \mu\_3 &= -2\text{id}\_1^2(a\_{02}a\_{20} + b\_{02}b\_{20} - a\_{02}b\_{02} - a\_{20}b\_{20}) \end{aligned} \tag{35}$$

In the above expression of each μk; k ¼ 2; 3, we have already let μ<sup>1</sup> ¼ ⋯ ¼ μk−<sup>1</sup> ¼ 0.

Thus, from Theorem 9 and Eqs. (35) and (31), we have

Theorem 11. For the flow on center manifold of system (30),<sup>δ</sup>¼<sup>0</sup>, the first three focal values v2iþ<sup>1</sup>ð2πÞ ði ¼ 1; 2; 3Þ of the origin are as follows

$$\begin{aligned} \upsilon\_3 &= 2\pi d\_1 B\_1, \\ \upsilon\_5 &= 2\pi d\_1^2 \left( A\_2 B\_0 + A\_0 B\_2 \right), \\ \upsilon\_7 &= 2\pi d\_1^2 \left[ \left( A\_0 - A\_2 \right)^2 + \left( B\_0 + B\_2 \right)^2 \right] \end{aligned} \tag{36}$$

Theorem 12. For the flow on center manifold of (30)<sup>δ</sup>¼<sup>0</sup>, the origin is a three-order weak focus, i.e., v<sup>3</sup> ¼ v<sup>5</sup> ¼ 0; v<sup>7</sup> ≠ 0 if and only if

$$B\_1 = 0, A\_2 B\_0 + A\_0 B\_2 = 0 \text{ and } \left(A\_0 \text{--} A\_2\right)^2 + \left(B\_0 + B\_2\right)^2 \neq 0 \tag{37}$$

Remark 6. For the coefficients of system (30)<sup>δ</sup>¼0, there exists necessarily a group of critical values: Ai ¼ A� <sup>i</sup> ; Bi ¼ B� <sup>i</sup> ði ¼ 0; 1; 2Þ such that the conditions (37) hold, for example:

$$A\_1^\* = B\_1^\* = 0, A\_0^\* = B\_0^\* = 1, B\_2^\* = -A\_2^\* = 13\tag{38}$$

Now we consider Hopf bifurcation of limit cycles from the origin for perturbed system (30).

Theorem 13. At least three limit cycles can be bifurcated from the origin of system (30) restricted to the center manifold, which lie in the neighborhood of the origin.

Proof. From Theorem 11, one can easily calculate the Jacobian determinant with respect to the functions v3;v5;v<sup>7</sup> and variables B1;B0;A0,

$$f = \frac{\partial(\upsilon\_3, \upsilon\_5, \upsilon\_7)}{\partial(B\_1, B\_0, A\_0)} = -2\pi^3 d\_1^5 [8(A\_0 A\_2 - A\_2^2 - B\_0 B\_2 - B\_2^2)]\tag{39}$$

Considering the conditions (37) of Theorem 12 and substituting the group of critical values of Eq. (38) into Eq. (39), we obtain <sup>J</sup> <sup>¼</sup> <sup>649</sup>π<sup>3</sup>d<sup>5</sup> <sup>1</sup> ≠ 0. Thus, we take some appropriate perturbations for the coefficients of system (32) to make the following two conditions:

$$(\upsilon\_1(2\pi) - 1)\upsilon\_3 < 0, \upsilon\_3\upsilon\_5 < 0, \upsilon\_5\upsilon\_7 < 0\tag{40}$$

and

$$|e^{2\pi b} - 1| \ll |\upsilon\_3| \ll |\upsilon\_5| \ll |\upsilon\_7| \tag{41}$$

hold, one must obtain that the succession function on the center manifold has three small real positive roots, just the system (30) has at least three limit cycles in the neighborhood of the origin. We can refer to references [16, 26, 27] for more details about the construction of limit cycles.

Remark 7. In general, in order to find more limit cycles in the neighborhood of the origin of system (30), we should add more higher order terms of u~ðz;wÞ determined in Eq. (33). Here we propose a conjecture that system (30) has at most three limit cycles through Hopf bifurcation restricted to a center manifold from the origin. However, the center conditions or integrability at the degenerate singularity will need further study.
