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

$$
\dot{\mathbf{x}} = A\mathbf{x} + \mathbf{f}(\mathbf{x}) \tag{1}
$$

where <sup>x</sup> ¼ ðx1; <sup>x</sup>2; <sup>⋯</sup>; xn<sup>Þ</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>, <sup>A</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> · <sup>n</sup>; <sup>n</sup> <sup>∈</sup> <sup>N</sup>, and <sup>f</sup>ðx<sup>Þ</sup> is sufficiently smooth with <sup>f</sup>ð0Þ ¼ <sup>0</sup>, Dfð0Þ ¼ 0. Then the origin is an equilibrium. For dynamical analysis of systems (1), it is very important to discuss the asymptotic behavior and the existence of periodic orbits at the origin. When the Jacobi matrix A has an eigenvalue with zero real part, the phase portraits in the vicinity of the origin is not easy to be determined. In particular, a system (1) has the following form

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#### 4 Manifolds - Current Research Areas

$$\begin{cases}
\dot{\mathbf{x}}\_1 = A\_1 \mathbf{x}\_1 + \mathbf{f}\_1(\mathbf{x}\_1, \mathbf{x}\_2) \\
\dot{\mathbf{x}}\_2 = A\_2 \mathbf{x}\_2 + \mathbf{f}\_2(\mathbf{x}\_1, \mathbf{x}\_2)
\end{cases} \tag{2}$$

where x1 ¼ ðx1; x2; …; xnc Þ <sup>T</sup> <sup>∈</sup> <sup>R</sup>nc ; x2 ¼ ðxncþ<sup>1</sup>;…; xn<sup>Þ</sup> <sup>T</sup><sup>∈</sup> <sup>R</sup>ns with nc <sup>þ</sup> ns <sup>¼</sup> <sup>n</sup>; <sup>A</sup><sup>1</sup> and <sup>A</sup><sup>2</sup> are constant matrices, and f1ðx1, x2Þ, f2ðx1, x2Þ are functions with

$$\mathbf{f\_1(0,0) = 0, f\_2(0,0) = 0, Df\_1(0,0) = 0, Df\_2(0,0) = 0}$$

Suppose that A<sup>1</sup> has nc critical eigenvalues (i.e., eigenvalues with Re λ = 0) and all ns eigenvalues of A<sup>2</sup> satisfy Re λ < 0. According to the Center Manifold Theorem (see, e.g., [1, 2]), there exists a (local) center manifold x2 ¼ hðx1Þ with hð0Þ ¼ 0; Dhð0Þ ¼ 0; and system (2) is topologically equivalent near ð0, 0Þ to the system

$$\begin{cases}
\dot{\mathbf{x}}\_1 = A\_1 \mathbf{x}\_1 + \mathbf{f}\_1(\mathbf{x}\_1, \mathbf{h}(\mathbf{x}\_1)) \\
\dot{\mathbf{x}}\_2 = A\_2 \mathbf{x}\_2.
\end{cases} \tag{3}$$

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 ðx1; x2;…; xnc Þ-plane (hyperplane) at the origin and which contains all the recurrent behavior of system (2) in a neighborhood of the origin, since the second equation in (3) is linear and has exponentially decaying solutions (see, e.g., [3]). Thus, the dynamics of Eq. (2) near a nonhyperbolic equilibrium are determined by this restriction. Generally, the local center manifold is not necessarily unique, but if the origin is a center restricted to a local center manifold for system (2), then the center manifold is unique and analytic, which is presented by the Lyapunov Center Theorem proved in Ref. [4].

If A has a simple pair of purely imaginary eigenvalues �ωi (ω > 0), system (1) undergoes a Hopf bifurcation or multiple Hopf bifurcation in a neighborhood of the origin on the local center manifold under proper perturbations of parameters. The computation of focal values (Lyapunov coefficients) plays an important role in the study of small-amplitude limit cycles appearing in these bifurcations (see [5–14] and references therein). The projection method was used for computing the first and the second focal values (see [2]), and a perturbation technique based on multiple time scales was used for computing focal values (see [15]). For a class of three-dimensional systems, the formal series method was presented with a recursive formula for computing singular point quantities (see [16]), here the theory and methodology described in Refs. [16, 17] can be applied to n-dimensional systems, where n ≥ 4.

If A has some zero eigenvalues for system (1), the Hopf bifurcation problem at the origin on the local center manifold becomes generally more difficult in comparison to the nondegenerate case. Take the degenerate singular point with a zero linear part in planar system, for example, the investigation of Hopf bifurcation from the equilibrium has to involve detecting the monodromy and distinguishing between a center and a focus [18, 19]. For that matter, several available approaches and corresponding results can be seen in [18–25], and one can easily find that the results on the bifurcation of limit cycles are very less. Remarkably, the author of reference [26] in 2001 gave the formal series method of calculating the singular point quantities of the degenerate critical point, which made it possible to investigate multiple Hopf bifurcation of higher degree polynomial systems [27, 28]. Here we extend its application to the local center manifold of more higher-dimensional system.
