4. Conclusion and discussion

The two classes of methods for computing the nondegenerate and degenerate singular point quantities on center manifold of the three-, four-, and more higher dimensional polynomial systems are discussed here, and more as the applications of them, the multiple limit cycles or Hopf cyclicity of two typical nonlinear dynamic systems restricted to the corresponding center manifolds are investigated.

## Appendix A

Theorem 11. For the flow on center manifold of system (30),<sup>δ</sup>¼<sup>0</sup>, the first three focal values

<sup>1</sup> ðA2B<sup>0</sup> þ A0B2Þ,

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.,

Remark 6. For the coefficients of system (30)<sup>δ</sup>¼0, there exists necessarily a group of critical

<sup>0</sup> ¼ B�

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

Proof. From Theorem 11, one can easily calculate the Jacobian determinant with respect to the

Considering the conditions (37) of Theorem 12 and substituting the group of critical values of

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

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

<sup>¼</sup> <sup>−</sup>2π<sup>3</sup> d5

<sup>i</sup> ði ¼ 0; 1; 2Þ such that the conditions (37) hold, for example:

1½8ðA0A2−A<sup>2</sup>

<sup>2</sup> ¼ −A�

<sup>0</sup> ¼ 1; B�

<sup>2</sup> þ ðB<sup>0</sup> <sup>þ</sup> <sup>B</sup>2<sup>Þ</sup>

2 �

<sup>2</sup> þ ðB<sup>0</sup> <sup>þ</sup> <sup>B</sup>2<sup>Þ</sup>

2−B0B2−B<sup>2</sup>

ðv1ð2πÞ−1Þv<sup>3</sup> < 0; v3v<sup>5</sup> < 0; v5v<sup>7</sup> < 0 (40)

<sup>2</sup>πδ−1j≪jv3j≪jv5j≪jv7<sup>j</sup> (41)

<sup>1</sup> ≠ 0. Thus, we take some appropriate perturbations

(36)

<sup>2</sup> ≠ 0 (37)

<sup>2</sup> ¼ 13 (38)

<sup>2</sup>Þ� (39)

<sup>1</sup> ½ðA0−A2Þ

v<sup>3</sup> ¼ 2πd1B1; <sup>v</sup><sup>5</sup> <sup>¼</sup> <sup>2</sup>πd<sup>2</sup>

B<sup>1</sup> ¼ 0; A2B<sup>0</sup> þ A0B<sup>2</sup> ¼ 0 and ðA0−A2Þ

<sup>1</sup> ¼ 0; A�

<sup>v</sup><sup>7</sup> <sup>¼</sup> <sup>2</sup>πd<sup>2</sup>

v2iþ<sup>1</sup>ð2πÞ ði ¼ 1; 2; 3Þ of the origin are as follows

v<sup>3</sup> ¼ v<sup>5</sup> ¼ 0; v<sup>7</sup> ≠ 0 if and only if

14 Manifolds - Current Research Areas

<sup>i</sup> ; Bi ¼ B�

functions v3;v5;v<sup>7</sup> and variables B1;B0;A0,

Eq. (38) into Eq. (39), we obtain <sup>J</sup> <sup>¼</sup> <sup>649</sup>π<sup>3</sup>d<sup>5</sup>

A� <sup>1</sup> ¼ B�

center manifold, which lie in the neighborhood of the origin.

<sup>J</sup> <sup>¼</sup> <sup>∂</sup>ðv3;v5;v7<sup>Þ</sup> ∂ðB1;B0;A0Þ

for the coefficients of system (32) to make the following two conditions:

je

values: Ai ¼ A�

and

cycles.

c½α;β� ¼ 1 5ðα−βÞ d1{b<sup>2</sup> <sup>02</sup>ð3β−2αÞ þ <sup>a</sup>20b02ð20−β−αÞ−a<sup>2</sup> <sup>20</sup>ð20 þ 2β−3αÞÞ · d1c½α−17;β−13� þ ðða11b<sup>02</sup> þ a20b11Þð20−β−αÞ−2b02b11ð5−3β þ 2αÞ− 2a11a20ð15 þ 2β−3αÞÞd1c½α−16;β−14� þ ðða02b<sup>02</sup> þ a11b<sup>11</sup> þ a20b20Þð20− <sup>β</sup>−αÞ−ða<sup>2</sup> <sup>11</sup> <sup>þ</sup> <sup>2</sup>a02a20Þð<sup>10</sup> <sup>þ</sup> <sup>2</sup>β−3αÞ−ðb<sup>2</sup> <sup>11</sup> þ 2b02b20Þð10−3β þ 2αÞÞd1c½α− 15;β−15� þ ðða02b<sup>11</sup> þ a11b20Þð20−β−αÞ−2b11b20ð15−3β þ 2αÞ−2a02a11ð5þ <sup>2</sup>β−3αÞÞd1c½α−14;β−16�þða02b20ð20−β−αÞ−b<sup>2</sup> <sup>20</sup>ð20−3β þ 2αÞ− a2 <sup>02</sup>ð2β−3αÞÞd1c½α−13;β−17�−b02ð5 þ 3β−2αÞ þ a20ð5 þ 2β−3αÞÞic½α− 6;β−4�−ðb11ð3β−2αÞ þ a11ð2β−3αÞÞi c½α−5;β−5� þðb20ð5−3β þ 2αÞ þ a02ð5−2β þ 3αÞÞic½α−4;β−6� μ~ ½α� ¼ − d1 <sup>5</sup> {ða<sup>2</sup> <sup>20</sup>ðα−20Þ þ <sup>2</sup>a20b02ð10−αÞ þ <sup>b</sup><sup>2</sup> <sup>02</sup>αÞd1c½α−17;α−13�

$$\begin{cases} \begin{aligned} &+(2a\_{11}a\_{20}(\alpha-15)-2(a\_{11}b\_{02}+a\_{20}b\_{11})(\alpha-10)+2b\_{02}b\_{11}(\alpha-5))d\_{1}c[\alpha-\\ &16,\alpha-14]+((a\_{11}^{2}+2a\_{02}a\_{20}-2a\_{02}b\_{02}-2a\_{11}b\_{11}+b\_{11}^{2}-2a\_{20}b\_{20}+2b\_{02}b\_{20})(\alpha-10) \end{aligned} &\text{ $\alpha$ } \\ \begin{aligned} &10) &d\_{1}c[\alpha-15,\alpha-15]+2((a\_{02}b\_{11}+a\_{11}b\_{20})(10-\alpha)-b\_{11}b\_{20}(15-\alpha)-a\_{02}a\_{11}(5-\alpha)] \end{aligned} &\text{ $\alpha$ } \\ &(10) &d\_{1}c[\alpha-14,\alpha-16]+(b\_{20}^{2}(\alpha-20)-2(a\_{02}b\_{20})(\alpha-10)+a\_{02}^{2}\alpha)d\_{1}c[\alpha-13,\alpha-10] \end{aligned} &\text{ $\alpha-13$ } \\ &17] &+(a\_{20}(\alpha-5)-b\_{02}(5+\alpha))\text{i }c[\alpha-6,\alpha-4]+(a\_{11}-b\_{11})\alpha \text{ii }c[\alpha-5,\alpha-10] \\ &(10) &d\_{2}(\alpha-5)-a\_{02}(5+\alpha))\text{i }c[\alpha-4,\alpha-6]), \\ & \mu\_{m}=\ddot{\mu}\ [5m], \end{aligned}$$

where c½k;j� ¼ ckj.
