## 1. Introduction

This chapter is concerned with Hopf bifurcation restricted to the center manifold from the equilibrium for three‐, four‐, and more higher‐dimensional nonlinear dynamical systems.

Let us first consider the generic real systems which take the form

where

where

Suppose that

The first equation in Eq. (3) is called the restriction of system (2) to its center manifold at the origin. The local center manifold, which is tangent to the

If

If

## 2. Case of the nondegenerate singular point

In this section, we consider Hopf bifurcation from the nondegenerate origin of system (1) restricted to the center manifold, in which the Jacobian matrix

### 2.1. The formal series method of computing nondegenerate singular point quantities on center manifold

Considering the Jacobian matrix

(4) |

where

Here, we recall first the calculation method of the singular point quantities on center manifold for the above real three‐dimensional nonlinear dynamical systems. By means of transformation

system (4) is also transformed into the following complex system:

(6) |

where

**Theorem 1** (see [16]). *For system (6), using the program of term by term calculations, we can determine a formal power series:*

*such that*

*where*

**Definition 1.** The

**Theorem 2** (see [16, 34]). *For the* *th singular point quantity and the* *th focal value at the origin on center manifold of system (4), i.e.,* *and* *, we have the following relation:*

*where* *are polynomial functions of coefficients of system (6). Usually, it is called algebraic equivalence and written as* *.*

Based on the previous work in Ref. [16], we have developed the calculation method of the focal values on the center manifold for real four‐dimensional nonlinear dynamical systems in Ref. [35]. In fact, here Theorem 1 can be generalized in the

where

By means of transformation of Eq. (5), system (10) can be transformed into the following complex system

(11) |

where the subscript “

**Theorem 3.** *For system (11), using the program of term by term calculations, we can determine a formal power series:*

*such that*

*where the subscript* “*” denotes “**”,* *, and* *, and more setting* *with* *except for* *, and* *with* *.*

*Proof.* It is very similar to the proving course of Theorem 1.3.1 in [16], by computing carefully and comparing the above power series with the two sides of (13), we can obtain the expression of

**Definition 2.** The

*Remark* 1. Similar to Theorem 2, there exists a equivalence between

**Corollary 1.** *The origin of system (10) or (11) is a center restricted to the center manifold if and only if* *for all* *.*

*Remark* 2. From the relation given by Remark 1 and Corollary 1, the center‐focus problem and Hopf bifurcation of equilibrium point restricted to the center manifold can be figured out by the calculation of singular point quantities for system (10).

### 2.2. An example of four‐dimensional system

Recently, the study of chaos has become a hot research topic, and the attention of many researchers is turning to 4D systems from 3D dynamical systems, for example, the authors of Ref. [36] investigated Hopf bifurcation of a 4D‐hyoerchaotic system by applying the normal form theory in 2012, but its multiple Hopf bifurcation on the center manifold have not been considered. Here, we will investigate the system further by computing the singular point quantities of its equilibrium point, which takes the following form

where

with the characteristic equation:

To guarantee that

Thus, we obtain the critical condition of Hopf bifurcation at

where

such that

Namely, we can use the nondegenerate transformation and the time rescaling:

(18) |

where

where

According to Theorem 3, we obtain the recursive formulas of

**Theorem 5.** *For system (18), setting* *with* *except for* *, and* *with* *, we can derive successively and uniquely the terms of the following formal series (12) with* *, such that (13) with* *holds and if* *or* *,* *is determined by following recursive formula:*

(19) |

*and for any positive integer* *is determined by following recursive formula:*

*and when* *or* *or* *or* *or* *, we* *have let* *, and where each* *denotes* *.*

By applying the above formulas in the Mathematica symbolic computation system, we figure out easily the first two singular point quantities of the origin of system (18):

where

and the above expression of

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

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

*where the expression of* *is obtained under the condition of* *.*

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

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

**Theorem 7.** *At least two small limit cycles can be bifurcated from the origin of the 4D‐hyoerchaotic system (14), which lie in the neighborhood of the origin restricted to the center manifold.*

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

## 3. Case of the degenerate singular point

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 manifold

Let us consider the real

(23) |

where the subscript “

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:

where **h**.**o**.**t**. denotes the terms with orders greater than or equal to 3. Substituting

where

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)

system (25) is transformed into following system:

where

For any positive integer

a homogeneous polynomial of degree

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

such that

**Definition 3.** If

Similar to Theorem 2, there also exists a equivalence between the

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

*where* *are polynomial functions of coefficients of system (26). Then, the relation between* *and* *is called the algebraic equivalence.*

*Remark* 4. In fact, from Theorem 2, for any positive integer

**Corollary 2.** *The origin of system (23) is a center restricted to the center manifold if and only if* *for all* *.*

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

Now we consider an example for system (23) with

(30) |

where

namely,

*Remark* 5. For system (32), the corresponding **Theorem 8**, we figure out that each

where

(34) |

Hence,

**Theorem 10.** *For system (32) with* *, we can derive successively the terms of the formal series (27), such that (28) holds (**,* *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

In the above expression of each

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

**Theorem 11.** *For the flow on center manifold of system (30),**, the first three focal values* *of the origin are as follows*

**Theorem 12.** *For the flow on center manifold of (30),**, the origin is a three‐order weak focus, i.e.,* *if and only if*

*Remark* 6. For the coefficients of system (30)

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

Considering the conditions (37) of Theorem 12 and substituting the group of critical values of Eq. (38) into Eq. (39), we obtain

and

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

## 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

where