The equivalence between singular point quantities and Liapunov constants on center manifold

Abstract

The algorithm of singular point quantities for an equilibrium of three-dimensional dynamics system is studied. The explicit algebraic equivalent relation between singular point quantities and Liapunov constants on center manifold is rigorously proved. As an example, the calculation of singular point quantities of the Lü system is applied to illustrate the advantage in investigating Hopf bifurcation of three-dimensional system.

MR (2000) Subject Classification: 34C23, 34C28, 37Gxx.

1 Introduction

We consider the real three-dimensional differential autonomous systems which take the form

$$\dot{\mathbf{x}}=A\mathbf{x}+\mathbf{f}\left(\mathbf{x}\right)$$

(1)

where x ∈ ℝ³, A ∈ ℝ^3×3, f(x) ∈ C² with f(0) = 0, Df(0) = 0, then the origin is an equilibrium. The systems (1) usually involve many important nonlinear dynamical models such as Lotka-Volterra system [1, 2] and Lorenz system [3, 4].

As far as Hopf bifurcation of the origin of systems (1) is concerned, the Jacobian matrix A at the origin should have a pair of purely imaginary eigenvalues and a negative one. In general, one can apply firstly the center manifold theorem to reduce the system (1) to a two-dimensional system [5], then compute Liapunov constants or the bifurcation formulae based on Liapunov functions-Poincaré theory [6, 7]. However, this traditionary way has quite complicated course of determining coefficients of the two-dimensional reduced equations, and for bifurcation formulae or Liapunov coefficient [6, 8], usually, only the first value is obtained, thus just one single limit cycle in the vicinity of the origin can be found.

Recently, the authors Wang et al. [9] introduced an algorithm of computing the singular point quantities on center manifold, which misses the above tedious course. In contrast to the usual ones, this algorithm is more convenient to investigate the multiple Hopf bifurcation at equilibrium of a three-dimensional system. However, it is possibly difficult to be approbated. For this reason, this paper will give the explicit relation between the singular point quantities and Liapunov constants of the origin of system (1). And more we hope that the results presented here will stimulate the analysis of topological structure and dynamical behavior for a higher-dimensional system.

This paper is organized as follows. In Section 2, we give some preliminaries about Liapunov constants, the focal values and singular point quantities on center manifold for a three-dimensional system (1). In Section 3, the relation between the singular point quantities and Liapunov constants is investigated, and their algebraic equivalence is proved rigorously. In Section 4, the singular point quantities of the Lü system as an example are computed, then the existence of four limit cycles for this system is judged.

2 The related definitions

2.1 Liapunov constants on center manifold

We give firstly the definition of Liapunov constants for a three-dimensional system. Considering the Jacobian matrix A at the origin of system (1) has a pair of purely imaginary eigenvalues and a negative one, then the system (1) can be put in the following form:

$$\left\{\begin{array}{c}{ẋ}_1={\ell }_1{x}_1+{\ell }_2{x}_2+\sum _{k=2}^{\mathrm{\infty }}{X}_k\left(x_1,x_2,x_3\right),\hfill \\ {ẋ}_2=\ell x_1-{\ell }_1{x}_2+\sum _{k=2}^{\mathrm{\infty }}{Y}_k\left(x_1,x_2,x_3\right),\hfill \\ {ẋ}_3=-d{x}_3+\sum _{k=2}^{\mathrm{\infty }}{U}_k\left(x_1,x_2,x_3\right)\hfill \end{array}\right.$$

(2)

Where ${ẋ}_i=\frac{\mathsf{\text{d}}{x}_i}{\mathsf{\text{d}}\tau }\left(i=1,2,3\right)$, and d, λ, λ₁, ${\ell }_2\in ℝ\left(d>0,{\ell }_1^2+\ell {\ell }_2<0\right)$, and X _k , Y _k , U _k are homogeneous polynomials in x₁, x₂, x₃ of degree k.

According to the center manifold theorem [5], the three-dimensional system (2) has an approximation to the center manifold taking the form

$$u=u\left(x_1,x_2\right)=u_2\left(x_1,x_2\right)+\mathsf{\text{h}}\mathsf{\text{.o}}\mathsf{\text{.t}}.$$

(3)

where u₂ is a quadratic homogeneous polynomial in x₁ and x₂, and h.o.t. denotes the terms with orders greater than or equal to 3. Substituting u = u(x₁, x₂) into the equations of system (2), we can obtain a generic two-dimensional differential system with center-focus type linear part as follows

$$\left\{\begin{array}{c}{ẋ}_1={\ell }_1{x}_1+{\ell }_2{x}_2+\sum _{k=2}^{\mathrm{\infty }}{\tilde{X}}_k\left(x_1,x_2\right),\hfill \\ {ẋ}_2=\ell x_1-{\ell }_1{x}_2+\sum _{k=2}^{\mathrm{\infty }}{Ỹ}_k\left(x_1,x_2\right)\hfill \end{array}\right.$$

(4)

where ${\tilde{X}}_k,{Ỹ}_k$ are homogeneous polynomials in x₁, x₂ of degree k, and their coefficients are polynomial functions of coefficients of the system (2). System (4) is often called the reduction system of (2). Correspondingly, one can take a generic Liapunov function

$$H\left(x_1,x_2\right)=\kappa \left(\ell x_1^2-2{\ell }_1{x}_1{x}_2-{\ell }_2{x}_2^2\right)+\sum _{k=3}^{\mathrm{\infty }}{H}_k\left(x_1,x_2\right)$$

(5)

where κ can be an arbitrary non-zero real number, H _k is a homogeneous polynomial in x₁, x₂ of degree k and the coefficients of H should satisfy

$${\left.\frac{\mathsf{\text{d}}H}{\mathsf{\text{d}}\tau }\right|}_{\left(4\right)}=V_2{\mathrm{\Phi }}^2+V_4{\mathrm{\Phi }}^3+\cdots +V_{2m}{\mathrm{\Phi }}^{m+1}+\cdots$$

(6)

where $\mathrm{\Phi }\left(x_1,x_2\right)=\left(x_1^2+x_2^2\right)$ or $x_1^2$ or $x_2^2$ or (x₁ + x₂)² or other suitable forms [7, 10, 11].

Definition 2.1. V_2min (6) is called the m th Liapunov constant of the origin for system (2) or (4), m = 1, 2, ....

2.2 The focal values on center manifold

In this part, we give the definition of the focal values for a three-dimensional system. One transformation matrix P can be found such that the coefficient matrix A of linear part of system (2) becomes the matrix B as follows

$$P^{-1}AP=P^{-1}\left(\begin{array}{ccc}\hfill {\ell }_1\hfill & \hfill {\ell }_2\hfill & \hfill 0\hfill \\ \hfill \ell \hfill & \hfill -{\ell }_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -d\hfill \end{array}\right)P=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill -\omega \hfill & \hfill 0\hfill \\ \hfill \omega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -d\hfill \end{array}\right)\triangleq B$$

(7)

where |P| ≠ 0 and $\omega ={\left(-{\ell }_1^2-\ell {\ell }_2\right)}^{1/2}>0$. Thus by a nondegenerate transformation (x₁, x₂, x₃)' = P (x, y, u)', and after a time scaling: t = ωτ, the system (2) can be changed into the following system

$$\left\{\begin{array}{c}\frac{\mathsf{\text{d}}x}{\mathsf{\text{d}}t}=-y+\sum _{k+j+l=2}^{\mathrm{\infty }}{A}_{kjl}{x}^k{y}^j{u}^l=X\left(x,y,u\right),\hfill \\ \frac{\mathsf{\text{d}}y}{\mathsf{\text{d}}t}=x+\sum _{k+j+l=2}^{\mathrm{\infty }}{B}_{kjl}{x}^k{y}^j{u}^l=Y\left(x,y,u\right),\hfill \\ \frac{\mathsf{\text{d}}u}{\mathsf{\text{d}}t}=-d_{0}u+\sum _{k+j+l=2}^{\mathrm{\infty }}{d}_{kjl}{x}^k{y}^j{u}^l=U\left(x,y,u\right)\hfill \end{array}\right.$$

(8)

where x, y, u, A _kjl , B _kjl , d _kjl ∈ ℝ (k, j, l ∈ ℕ) and d₀ = d/ω. Similarly, according to the center manifold theorem, putting certain approximation with the same form as (3) into (8), we get the following real planar polynomial differential system

$$\left\{\begin{array}{c}\frac{\mathsf{\text{d}}x}{\mathsf{\text{d}}t}=-y+\sum _{k+j=2}^{\mathrm{\infty }}{A}_{kj}{x}^k{y}^j=X\left(x,y\right),\hfill \\ \frac{\mathsf{\text{d}}y}{\mathsf{\text{d}}t}=x+\sum _{k+j=2}^{\mathrm{\infty }}{B}_{kj}{x}^k{y}^j=Y\left(x,y\right)\hfill \end{array}\right.$$

(9)

where A _kj , B _kj ∈ ℝ(k, j ∈ ℕ) and all A _kj , B _kj are polynomial functions of coefficients of the system (8) or (2). System (9) is also called the equations on the center manifold or reduction system of (8). It is well-known, the origin of system (9) is center-focus type, and some significant work about it has been done in [12–14].

In order to define the focal values, we transform system (9) into the following form under the polar coordinates: x = r cos θ, y = r sin θ,

$$\frac{\mathsf{\text{d}}r}{\mathsf{\text{d}}t}=r\sum _{k=1}^{\mathrm{\infty }}{\phi }_k\left(\theta \right)r^k,\frac{\mathsf{\text{d}}\theta }{\mathsf{\text{d}}t}=1+\sum _{k=1}^{\mathrm{\infty }}{\psi }_k\left(\theta \right)r^k$$

(10)

where φ _k (θ) and ψ _k (θ), k = 1, 2, 3, ... are analytic. System (10) is again transformed into

$$\frac{\mathsf{\text{d}}r}{\mathsf{\text{d}}\theta }=r\sum _{k=0}^{\mathrm{\infty }}{R}_k\left(\theta \right)r^k,$$

(11)

where the function on the right-hand side of Equation (11) is convergent in the range θ ∈ [− 4π, 4π], |r| < r₀ (r₀ is certain positive constant) and

$$R_k\left(\theta +\pi \right)={\left(-1\right)}^k{R}_k\left(\theta \right),k=0,1,2,\dots .$$

(12)

For sufficiently small h, let

$$\mathrm{\Delta }\left(h\right)=r\left(2\pi ,h\right)-h,r=r\left(\theta ,h\right)=\sum _{m=1}^{\mathrm{\infty }}{v}_m\left(\theta \right)h^m$$

(13)

be the Poincaré succession function and the solution of Equation (11) satisfying the initial-value condition r|_θ=0= h. Moreover, for (13) we have

$$v_1\left(\theta \right)=1,v_m\left(0\right)=0,m=2,3,\dots .$$

(14)

Definition 2.2. For the succession function in (13), if v₂(2π) = v₃(2π) = · · · = v_2k(2π) = 0 and v_2k+1(2π) ≠ 0, then the origin is called the fine focus or weak focus of order k, and the quantity of v_2k+1(2π) is called the k th focal value at the origin on center manifold of system (8) or (2), k = 1, 2, ....

Remark 1. For the coefficients of the form solution in (13), we have the following property [13, 15]: for every positive integer m = 1, 2, ..., there exists expression of the relation

$$v_{2m}\left(2\pi \right)=\frac{1}{1+v_1\left(\pi \right)}\sum _{k=1}^{m-1}{\xi }_m^{\left(k\right)}{v}_{2k+1}\left(2\pi \right)$$

(15)

where every ${\xi }_m^{\left(k\right)}$ is a polynomial of v₁(π), v₂(π), ..., v_2m(π) and v₁(2π), v₂(2π), ..., v_2m(2π) with rational coefficients. Particularly, if for each 1 ≤ k ≤ m − 1, v_2k+1(2π) = 0 holds, we can get v₂(2π) = v₄(2π) = ··· = v_2m(2π) = 0.

2.3 The singular point quantities on center manifold

Here, we recall the definition of the singular point quantities on center manifold. By means of transformation

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

(16)

system (8) is also transformed into the following complex system

$$\left\{\begin{array}{c}\frac{\mathsf{\text{d}}z}{\mathsf{\text{d}}T}=z+\sum _{k+j+l=2}^{\mathrm{\infty }}{a}_{kjl}{z}^k{w}^j{u}^l=Z\left(z,w,u\right),\hfill \\ \frac{\mathsf{\text{d}}w}{\mathsf{\text{d}}T}=-w-\sum _{k+j+l=2}^{\mathrm{\infty }}{b}_{kjl}{w}^k{z}^j{u}^l=-W\left(z,w,u\right),\hfill \\ \frac{\mathsf{\text{d}}u}{\mathsf{\text{d}}T}=\mathbf{i}{d}_{0}u+\sum _{k+j+l=2}^{\mathrm{\infty }}{\tilde{d}}_{kjl}{z}^k{w}^j{u}^l=Ũ\left(z,w,u\right)\hfill \end{array}\right.$$

(17)

where z, w, T, a _kjl , b _kjl , ${\tilde{d}}_{kjl}$ ∈ ℂ (k, j, l ∈ ℕ), the systems (8) and (17) are called concomitant. If no confusion arises, ${\tilde{d}}_{kjl}$, Ũ are still written as d _kjl and U.

Lemma 2.1 (see [9]). For system (17), using the program of term by term calculations, we can determine a formal power series:

$$F\left(z,w,u\right)=zw+\sum _{\alpha +\beta +\gamma =3}^{\mathrm{\infty }}{c}_{\alpha \beta \gamma }{z}^{\alpha }{w}^{\beta }{u}^{\gamma }$$

(18)

such that

$$\frac{\mathsf{\text{d}}F}{\mathsf{\text{d}}T}=\frac{\partial F}{\partial z}Z-\frac{\partial F}{\partial y}W+\frac{\partial F}{\partial u}U=\sum _{m=1}^{\mathrm{\infty }}{\mu }_m{\left(zw\right)}^{m+1}$$

(19)

where c₁₁₀ = 1, c₁₀₁ = c₀₁₁ = c₂₀₀ = c₀₂₀ = 0, c_{kk 0}= 0, k = 2, 3, .... And μ _m is called the mth singular point quantity at the origin on center manifold of system (17) or (8) or (2).

Lemma 2.2 (see [9]). For the mth singular point quantity and the mth focal value at the origin on center manifold of system (8), i.e. µ _m and v_2m+1, m = 1, 2, ..., we have the following relation:

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

(20)

where${\xi }_m^{\left(k\right)}\left(k=1,2,\dots ,m-1\right)$are polynomial functions of coefficients of system (17). Usually it is called algebraic equivalence and written as v_2m+1~ i πµ _m .

3 The conclusions and proofs

3.1 The equivalence for singular point quantities on center manifold

In this subsection, we give firstly the results about the equivalence. Then the key theorem, i.e. the following Theorem 3.1 will be proved in next subsection.

Theorem 3.1. For the mth Liapunov constant of the origin for system (2) and the mth focal value at the origin for system (8), i.e. V_2mand v_2m+1, for every positive integer m, we have the following relation:

$$V_{2m}={\sigma }_m{v}_{2m+1}\left(2\pi \right)+\sum _{k=1}^{m-1}{\xi }_m^{\left(k\right)}{v}_{2k+1}\left(2\pi \right)$$

(21)

where${\xi }_m^{\left(k\right)}\left(k=1,2,\dots ,m-1\right)$are polynomial functions of coefficients of system (8) and

$${\sigma }_m=2\kappa \ell {\omega }^3\left({\nu }^2+{\upsilon }^2\right){\left(\underset{0}{\overset{2\pi }{\int }}{\left[\varphi \left(\theta \right)\right]}^{m+1}\mathsf{\text{d}}\theta \right)}^{-1},$$

(22)

with

$$\varphi \left(\theta \right)=\left\{\begin{array}{cc}{x}_{1\theta }^2,\hfill & if\mathrm{\Phi }=x_1^2\hfill \\ x_{2\theta }^2,\hfill & if\mathrm{\Phi }=x_2^2\hfill \\ x_{1\theta }^2+x_{2\theta }^2,\hfill & if\mathrm{\Phi }=x_1^2+x_2^2\hfill \\ {\left(x_{1\theta }+x_{2\theta }\right)}^2,\hfill & if\mathrm{\Phi }={\left(x_1+x_2\right)}^2\hfill \end{array}\right.$$

(23)

where

$$x_{1\theta }=\left[\left({\ell }_1\upsilon +\omega \nu \right)\mathsf{\text{cos}}\left(\theta \right)+\left({\ell }_1\nu -\omega \upsilon \right)\mathsf{\text{sin}}\left(\theta \right)\right]/\ell ,x_{2\theta }=\upsilon \mathsf{\text{cos}}\left(\theta \right)+\nu \mathsf{\text{sin}}\left(\theta \right)$$

(24)

and ν , υ are the two constants given by the real transformation matrix P in (7) with ν² + υ² ≠ 0. Then, we also call the relation algebraic equivalence and write as V_2m~ σ _m v_2m+1(2π), m = 1, 2, ....

How to determine the above v, υ is shown in the next elementary lemma.

Lemma 3.2. The nondegenerate real transformation matrix in (7) possesses necessarily the following generic form:

$$P=\left(\begin{array}{ccc}\hfill \frac{\upsilon {\ell }_1+\nu \omega }{\ell }\hfill & \hfill \frac{\nu {\ell }_1-\upsilon \omega }{\ell }\hfill & \hfill 0\hfill \\ \hfill \upsilon \hfill & \hfill \nu \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \zeta \hfill \end{array}\right)$$

(25)

where ς , ν , υ are three real numbers such that ς(υ² + ν²) ≠ 0 holds.

Proof. Considering that the coefficient matrix A in (7) has a pair of purely imaginary eigenvalues i ω, − i ω and one real eigenvalue −d, one can select freely their eigenvectors respectively, for example

$${\eta }_1={\left(\frac{{\ell }_1+\mathbf{i}\omega }{\ell },1,0\right)}^{\prime },{\eta }_2={\left(\frac{{\ell }_1-\mathbf{i}\omega }{\ell },1,0\right)}^{\prime },{\eta }_3={\left(0,0,1\right)}^{\prime }.$$

Thus, there must exist only a generic transformation matrix T₁ = k₁η₁+k₂η₂+k₃η₃ with the following form

$$T_1=\left(\begin{array}{ccc}\hfill \frac{k_1\left({\ell }_1+i\omega \right)}{\ell }\hfill & \hfill \frac{k_2\left({\ell }_1-i\omega \right)}{\ell }\hfill & \hfill 0\hfill \\ \hfill k_1\hfill & \hfill k_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill k_3\hfill \end{array}\right)$$

such that

$$T_1^{-1}A{T}_1=\left(\begin{array}{ccc}\hfill \mathbf{i}\omega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\mathbf{i}\omega \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -d\hfill \end{array}\right)\triangleq J$$

(26)

where k₁, k₂ and k₃ are three arbitrary non-zero constants.

Similarly, we obtain also that for the matrix B in (7), there must exist only a generic transformation matrix T₂ with the following form

$$T_2=\left(\begin{array}{ccc}\hfill \mathbf{i}{j}_1\hfill & \hfill -\mathbf{i}{j}_2\hfill & \hfill 0\hfill \\ \hfill j_1\hfill & \hfill j_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill j_3\hfill \end{array}\right)$$

such that

$$T_2^{-1}B{T}_2=\left(\begin{array}{ccc}\hfill \mathbf{i}\omega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\mathbf{i}\omega \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -d\hfill \end{array}\right)=J$$

(27)

where j₁, j₂ and j₃ are also three arbitrary non-zero constants.

Then from (26) and (27), there must exist only a generic transformation matrix P as follows

$$P=T_1{T}_2^{-1}=\left(\begin{array}{ccc}\hfill \frac{{\ell }_1\left(K_2-K_1\right)\mathbf{i}+\left(K_2+K_1\right)\omega }{2\ell }\hfill & \hfill \frac{\left(K_2+K_1\right){\ell }_1-\left(K_2-K_1\right)\omega \mathbf{i}}{2\ell }\hfill & \hfill 0\hfill \\ \hfill \frac{\left(K_2-K_1\right)\mathbf{i}}{2}\hfill & \hfill \frac{K_2+K_1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \zeta \hfill \end{array}\right)$$

(28)

such that P^{− 1}AP = B in (7) holds, and where K₁ = k₁/j₁, K₂ = k₂/j₂ and ς= k₃/j₃ are also arbitrary non-zero numbers because of the property of k₁, k₂, k₃, j₁, j₂ and j₃. Furthermore in order to guarantee that the transformation in (7) is real and nondegenerate, form (28), there is no other choice, only we can let K₁ and K₂ conjugate, i.e.

$$K_1=\nu +\mathbf{i}\upsilon ,K_2=\nu -\mathbf{i}\upsilon ,\nu ,\upsilon ,\in ℝ,$$

at the same time, ς, υ and ν are three real numbers such that |P| ≠ 0 holds, thus we obtain the generic form of the transformation matrix P.      □

Remark 2. From the expression σ _m , m = 1, 2, ... in (22), one should notice that once the coefficient matrix A and transformation matrix P are given explicitly, then σ _m is never an arbitrary and undefined value. In particular, when λ = − λ₂ = ω, λ₁ = 0 in system (2), i.e. A = B in (7), we can get the identity matrix E as the simplest transformation matrix P, namely ν= 1, υ= 0, if we choose

$$\kappa =\frac{1}{2}{\omega }^{-4}\mathsf{\text{and}}\mathrm{\Phi }=x_1^2+x_2^2$$

in (6), then at this time every ${\sigma }_m=\frac{1}{2\pi }$.

Furthermore, from Lemma 2.2 and Theorem 3.1, we have

Theorem 3.3. For the mth Liapunov constant of the origin for system (2) and the mth singular point quantity of the origin for system (8) or (17), i.e. V_2mand µ _m , m = 1, 2, ..., there exists the following relation:

$$V_{2m}=\mathbf{i}\pi {\sigma }_m{\mu }_m+\mathbf{i}\pi {\sigma }_m\sum _{k=1}^{m-1}{\xi }_m^{\left(k\right)}{\mu }_k$$

(29)

where σ _m has been given in (22) of Theorem 3.1 and${\xi }_m^{\left(k\right)}\left(k=1,2,\dots ,m-1\right)$are polynomial functions of coefficients of system (17). Similarly we also call it algebraic equivalence and write as V_2m~ i π σ _m µ _m .

Remark 3. From (20) in Lemma 2.2 and (29) in Theorem 3.3, we get that:

$$V_2={\sigma }_1{v}_3\left(2\pi \right)=\mathbf{i}\pi {\sigma }_1{\mu }_1$$

(30)

and if for all k = 1, 2, ..., m − 1, V_2k= 0 or v_2k+1= 0 or µ _k = 0 holds, then we have

$$V_{2m}={\sigma }_m{v}_{2m+1}=\mathbf{i}\pi {\sigma }_m{\mu }_m,m=2,3,\dots .$$

(31)

Thus the stability of the origin for the systems (1) or (2) can be figured out directly by calculating the singular point quantities of the origin for system (17).

3.2 Proof of theorem 3.1

Firstly, from Lemma 3.2 and the nondegenerate transformation: ${\left(x_1,x_2,x_3\right)}^{\prime }=P{\left(x,y,u\right)}^{\prime }$ in (7), we obtain

$$\left(x_1,x_2,x_3\right)=\left(\frac{\upsilon {\ell }_1+\nu \omega }{\ell }x+\frac{\nu {\ell }_1-\upsilon \omega }{\ell }y,\upsilon x+\nu y,\zeta u\right),$$

(32)

and more under the polar coordinates x = r cos θ, y = r sin θ, we have

$$\begin{array}{ll}\hfill x_1& =\frac{r}{\ell }\left[\left(\upsilon {\ell }_1+\nu \omega \right)\mathsf{\text{cos}}\left(\theta \right)+\left(\nu {\ell }_1-\upsilon \omega \right)\mathsf{\text{sin}}\left(\theta \right)\right]=r{x}_{1\theta },\\ \hfill x_2& =r\left[\upsilon \mathsf{\text{cos}}\left(\theta \right)+\nu \mathsf{\text{sin}}\left(\theta \right)\right]=r{x}_{2\theta }.\end{array}$$

(33)

Substituting (32) into the Liapunov function (5), then we denote

$$H\left(x_1,x_2\right)=H\left[x,y\right]=\sum _{k=2}^{\mathrm{\infty }}{H}_k\left[x,y\right]$$

(34)

where H _k [x, y] is a homogeneous polynomial in x, y of degree k and

$$H_2\left[x,y\right]=-\frac{\kappa }{\ell }\left({\nu }^2+{\upsilon }^2\right)\left({\ell }_1^2+\ell {\ell }_2\right)\left(x^2+y^2\right)$$

(35)

with $H_2\left[1,0\right]=\frac{\kappa }{\ell }\left({\nu }^2+{\upsilon }^2\right){\omega }^2$. And more from the expressions (13) and (34), we have

$$\begin{array}{ll}\hfill H\left[x,y\right]& =H\left[r\left(\theta ,h\right)\mathsf{\text{cos}}\left(\theta \right),r\left(\theta ,h\right)\mathsf{\text{sin}}\left(\theta \right)\right]=\sum _{k=2}^{\mathrm{\infty }}{H}_k\left[\mathsf{\text{cos}}\left(\theta \right),\mathsf{\text{sin}}\left(\theta \right)\right]r^k\left(\theta ,h\right)\\ =Ĥ\left(\theta ,h\right).\end{array}$$

(36)

Next, we investigate $\mathrm{\Delta }Ĥ=Ĥ\left(2\pi ,h\right)-Ĥ\left(0,h\right)$ based on the Equation 6.

On the one hand, from (14), we denote

$$\begin{array}{ll}\hfill Ĥ\left(2\pi ,h\right)-Ĥ\left(0,h\right)& =\sum _{k=2}^{\mathrm{\infty }}{H}_k\left[1,0\right]\left[r^k\left(2\pi ,h\right)-r^k\left(0,h\right)\right]\\ =\sum _{k=2}^{\mathrm{\infty }}{H}_k\left[1,0\right]\left[r^k\left(2\pi ,h\right)-h^k\right]\\ =\mathrm{\Delta }{Ĥ}_1\end{array}$$

(37)

where

$$\begin{array}{ll}\hfill r^k\left(2\pi ,h\right)-h^k& =\left(r\left(2\pi ,h\right)-h\right)\left(r^{k-1}\left(2\pi ,h\right)+r^{k-2}\left(2\pi ,h\right)h+\cdots +h^{k-1}\right)\\ =\left(r\left(2\pi ,h\right)-h\right)\left(k{h}^{k-1}+o\left(h^{k-1}\right)\right),\end{array}$$

(38)

and from the expression (13), one can get

$$\begin{array}{ll}\hfill \mathrm{\Delta }{Ĥ}_1& =\left(r\left(2\pi ,h\right)-h\right)\sum _{k=2}^{\mathrm{\infty }}{H}_k\left[1,0\right]\left(k{h}^{k-1}+o\left(h^{k-1}\right)\right)\\ =\left(r\left(2\pi ,h\right)-h\right)\left[\frac{2}{\ell }\kappa \left({\nu }^2+{\upsilon }^2\right){\omega }^{2}h+o\left(h\right)\right]\\ =\frac{2}{\ell }\kappa \left({\nu }^2+{\upsilon }^2\right){\omega }^2\sum _{m=2}^{\mathrm{\infty }}{v}_m\left(2\pi \right)h^{m+1}\left(1+o\left(1\right)\right).\end{array}$$

(39)

On the other hand, we denote

$$Ĥ\left(2\pi ,h\right)-Ĥ\left(0,h\right)=\underset{0}{\overset{2\pi }{\int }}\frac{\mathsf{\text{d}}Ĥ}{\mathsf{\text{d}}\theta }\mathsf{\text{d}}\theta =\underset{0}{\overset{2\pi }{\int }}\frac{\mathsf{\text{d}}Ĥ}{\mathsf{\text{d}}\tau }\frac{\mathsf{\text{d}}\tau }{\mathsf{\text{d}}\theta }\mathsf{\text{d}}\theta =\frac{1}{\omega }\underset{0}{\overset{2\pi }{\int }}\frac{\mathsf{\text{d}}Ĥ}{\mathsf{\text{d}}\tau }\frac{\mathsf{\text{d}}t}{\mathsf{\text{d}}\theta }\mathsf{\text{d}}\theta =\mathrm{\Delta }{Ĥ}_2.$$

(40)

From system (10) and the expression (13), we can get

$$\frac{\mathsf{\text{d}}t}{\mathsf{\text{d}}\theta }={\left[1+\sum _{k=2}^{\mathrm{\infty }}{r}^{k-1}{\psi }_{k+1}\left(\theta \right)\right]}^{-1}=1+c_1\left(\theta \right)h+c_2\left(\theta \right)h^2+\cdots =1+o\left(1\right)$$

(41)

where ψ_k+1(θ), c_{k− 1}(θ) (k = 2, 3, ...) are analytic. At the same time, putting (33) in the right side of (6), we have

$$\frac{\mathsf{\text{d}}Ĥ}{\mathsf{\text{d}}\tau }=\sum _{m=1}^{\mathrm{\infty }}{V}_{2m}{\left[{\varphi }^{\frac{1}{2}}\left(\theta \right)r\left(\theta ,h\right)\right]}^{2m+2}=\sum _{m=1}^{\mathrm{\infty }}{V}_{2m}{h}^{2m+2}{\left[\varphi \left(\theta \right)\right]}^{m+1}\left(1+o\left(1\right)\right),$$

(42)

thus

$$\mathrm{\Delta }{Ĥ}_2=\frac{1}{\omega }\sum _{m=1}^{\mathrm{\infty }}\left[{V_2}_m{h}^{2m+2}\left(\underset{0}{\overset{2\pi }{\int }}{\left[\varphi \left(\theta \right)\right]}^{m+1}\mathsf{\text{d}}\theta \mathsf{\text{ + }}o\mathsf{\text{(1)}}\right)\right].$$

(43)

According to (39), (43) and applying (15) in Remark 1 and mathematical induction to m, we complete the proof.

4 Singular point quantities for the Lü system

In this section, we investigate the singular point quantities of the equilibrium point of the Lü system which bridges the gap between the Lorenz and Chen attractors [7], and takes the following form

$$\left\{\begin{array}{c}{ẋ}_1=a\left(x_2-x_1\right)\hfill \\ {ẋ}_2=c{x}_2-x_1{x}_3\hfill \\ {ẋ}_3=x_1{x}_2-b{x}_3\hfill \end{array}\right.$$

(44)

where a > 0, b > 0, c > 0. Obviously, system (44) has three equilibrium points: O(0, 0, 0), $O_1\left(\sqrt{bc},\sqrt{bc},c\right)$ and $O_2\left(-\sqrt{bc},-\sqrt{bc},c\right)$. Because the Jacobian matrix at the origin O has no purely imaginary eigenvalues, it is unnecessary to consider its singular point quantities. And more the equations in (44) are invariant under the transformation:

$$\left(x_1,x_2,x_3\right)\mapsto \left(-x_1,-x_2,x_3\right),$$

(45)

which means that system (44) is symmetrical. Therefore, we only need to consider O₁.

The Jacobian matrix of system (44) at O₁ is

$$A_o=\left(\begin{array}{ccc}\hfill -a\hfill & \hfill a\hfill & \hfill 0\hfill \\ \hfill -c\hfill & \hfill c\hfill & \hfill -\sqrt{bc}\hfill \\ \hfill \sqrt{bc}\hfill & \hfill \sqrt{bc}\hfill & \hfill -b\hfill \end{array}\right)$$

with the characteristic equation: λ³ + (a + b − c)λ + abλ + 2abc = 0. To guarantee that A _o has a pair of purely imaginary eigenvalues ± i ω(ω > 0) and one negative real eigenvalue λ₀, we let its characteristic equation take the form

$$\left({\lambda }^2+{\omega }^2\right)\left(\lambda -{\lambda }_0\right)=0.$$

Thus we obtain the critical condition of Hopf bifurcation at O₁: c = (a + b)/ 3, then

$$\omega =\sqrt{bc},{\lambda }_0=-2c$$

(46)

namely b = ω²/a, c = (a² + ω²)/(3a). By the translation: (x₁, x₂, x₃) → (x₁ + ω, x₂ + ω, x₃ + c), we make the equilibrium O₁ become the origin and change system (44) into

$$\dot{\mathbf{x}}=A_o\left(\begin{array}{c}\hfill x_1+\omega \hfill \\ \hfill x_2+\omega \hfill \\ \hfill x_3+c\hfill \end{array}\right)+\left(\begin{array}{c}\hfill 0\hfill \\ \hfill -\left(x_1+\omega \right)\left(x_3+c\right)\hfill \\ \hfill \left(x_1+\omega \right)\left(x_2+\omega \right)\hfill \end{array}\right).$$

(47)

Under the conditions (46), one can find a nondegenerate matrix

$$P_o=\left(\begin{array}{ccc}\hfill \frac{\sqrt{3a}\left(2a^2-{\omega }^2\right)}{d_1\sqrt{d_0}}\hfill & \hfill \frac{3\sqrt{3}{a}^2\omega }{d_1\sqrt{d_0}}\hfill & \hfill -\frac{\sqrt{3}{a}^2}{2\omega \sqrt{d_0}}\hfill \\ \hfill \frac{2a\sqrt{3d_0}}{d_1}\hfill & \hfill \frac{\omega \sqrt{3d_0}}{d_1}\hfill & \hfill \frac{2{\omega }^2-a^2}{2\omega \sqrt{3d_0}}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right)$$

such that

$$P_o^{-1}{A}_o{P}_o=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill -\omega \hfill & \hfill 0\hfill \\ \hfill \omega \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -2d_0/\left(3a\right)\hfill \end{array}\right)$$

(48)

where d₀ = a² + ω², d₁ = 4a²+ ω².

Then we can use the nondegenerate transformation x = P _o y and the time re-scaling: t → t/ω to make the system (47) become the following form:

$$\begin{array}{ll}\hfill \mathbf{ẏ}& =\frac{1}{\omega }\left[P_o^{-1}*\mathsf{\text{diag}}\left(P_{o}y+E\right)A_o{P}_{o}y\right]\\ =\frac{1}{\omega }\left[P_o^{-1}{A}_o{P}_{o}y+P_o^{-1}*\mathsf{\text{diag}}\left(P_{o}y\right)A_o{P}_{o}y\right]\end{array}$$

(49)

where y = (y₁, y₂, y₃), and E is the 3 × 3 identity matrix.

Now we put y₁ = (z + w)/ 2, y₂ = (z − w)i / 2, y₃ = u, t = −T i in system (49) and obtain the following same form as the complex system (17):

$$\left\{\begin{array}{c}\frac{\mathsf{\text{d}}z}{\mathsf{\text{d}}T}=z+a_{101}uz+a_{011}uw+a_{110}zw+a_{200}{z}^2+a_{020}{w}^2+a_{002}{u}^2=Z\hfill \\ \frac{\mathsf{\text{d}}w}{\mathsf{\text{d}}T}=-\left(w+b_{011}uz+b_{101}uw+b_{110}zw+b_{020}{z}^2+b_{200}{w}^2+b_{002}{u}^2\right)=-W\hfill \\ \frac{\mathsf{\text{d}}u}{\mathsf{\text{d}}T}=d_{001}u+d_{101}uz+d_{011}uw+d_{110}zw+d_{200}{z}^2+d_{020}{w}^2+d_{002}{u}^2=U\hfill \end{array}\right.$$

(50)

where u ∈ ℝ, z, w, T ∈ ℂ, and

$$\begin{array}{ll}\hfill a_{200}& =\frac{3a\left(7a^2+4{\omega }^2\right)}{2d_1{d}_2}+\frac{9a^4}{2d_1{d}_2\omega }\mathbf{i},\\ \hfill a_{020}& =-\frac{3a\left(16a^6-93a^2{\omega }^4+4{\omega }^6\right)}{2d_1^3{d}_2}+\frac{3a^2\left(32{\omega }^6-16a^6-72a^4{\omega }^2-105a^2{\omega }^4\right)}{2d_1^3{d}_2\omega }\mathbf{i},\\ \hfill a_{002}& =-\frac{3a^3\left(8a^4+19a^2{\omega }^2+20{\omega }^4\right)}{8d_0{d}_1{d}_2{\omega }^2}-\frac{a^2\left(4a^6-21a^4{\omega }^2-36a^2{\omega }^4+16{\omega }^6\right)}{8d_0{d}_1{d}_2{\omega }^3}\mathbf{i},\\ \hfill a_{101}& =-\frac{d_{0}a\left(5a^2-4{\omega }^2\right)}{d_1{d}_2{\omega }^2}+\frac{3a^2\left(5a^2-4{\omega }^2\right)}{2d_1{d}_2\omega }\mathbf{i},\\ \hfill a_{011}& =-\frac{a\left(a^2-2{\omega }^2\right)\left(8a^6+30a^4{\omega }^2+45a^2{\omega }^4-4{\omega }^6\right)}{2d_0{d}_1^2{d}_2{\omega }^2}+\frac{3a^2\left(a^2-2{\omega }^2\right)\left(4a^4+5a^2{\omega }^2-8{\omega }^4\right)}{2d_0{d}_1^2{d}_2\omega }\mathbf{i},\\ \hfill a_{110}& =\frac{18a^3\left(2a^2+5{\omega }^2\right)}{d_1^2{d}_2}+\frac{3a^2\left(8{\omega }^4-4a^4-23a^2{\omega }^2\right)}{d_1^2{d}_2\omega }\mathbf{i},\\ \hfill b_{kjl}& ={ā}_{kjl}\left(kjl=200,020,002,101,011,110\right),\\ \hfill d_{200}& =-\frac{6a\left(16a^6-36a^4{\omega }^2-21a^2{\omega }^4+4{\omega }^6\right)}{d_1^3{d}_2}+\frac{54a^2\omega \left(4a^4-a^2{\omega }^2-2{\omega }^4\right)}{d_1^3{d}_2}\mathbf{i},\\ \hfill d_{020}& =\frac{6a\left(16a^6-36a^4{\omega }^2-21a^2{\omega }^4+4{\omega }^6\right)}{d_1^3{d}_2}+\frac{54a^2\omega \left(4a^4-a^2{\omega }^2-2{\omega }^4\right)}{d_1^3{d}_2}\mathbf{i},\\ \hfill d_{002}& =-\frac{3a^2\left(5a^2-4{\omega }^2\right)}{2d_1{d}_2\omega }\mathbf{i},\\ \hfill d_{101}& =-\frac{3a\left(16a^4-7a^2{\omega }^2+4{\omega }^4\right)}{d_1^2{d}_2}-\frac{12a^2\left(2a^2-{\omega }^2\right)\left(a^2-2{\omega }^2\right)}{d_1^2{d}_2\omega }\mathbf{i},\\ \hfill d_{011}& =\frac{3a\left(16a^4-7a^2{\omega }^2+4{\omega }^4\right)}{d_1^2{d}_2}-\frac{12a^2\left(2a^2-{\omega }^2\right)\left(a^2-2{\omega }^2\right)}{d_1^2{d}_2\omega }\mathbf{i},\\ \hfill d_{110}& =\frac{36a^2\omega \left(a^2-2{\omega }^2\right)}{d_1^2{d}_2}\mathbf{i},\\ \hfill d_{001}& =\frac{2d_1}{3a\omega }\mathbf{i}\end{array}$$

where d₂ = a² + 4ω², and ${ā}_{kjl}$ denotes the conjugate complex number of a _kjl .

According to Lemma 2.1, we obtain the recursive formulas c _αβγ and µ _m in Appendix. By applying the formulas in the Mathematica symbolic computation system, we figure out easily the first twenty singular point quantities of the origin of system (50):

$$\begin{array}{ll}\hfill {\mu }_1& =-243\mathbf{i}{a}^5\left(2a^2-{\omega }^2\right)\left(a^2-5{\omega }^2\right)\omega /\left(d_0{d}_1^2{d}_2{d}_3\right),\\ \hfill {\mu }_2& =-891849714087780\mathbf{i}{a}^7\left(2a^2-{\omega }^2\right){\omega }^{21}/\left(d_0^3{d}_1^5{d}_2^3{d}_3^2{d}_4\right),\\ \hfill {\mu }_3& ={\mu }_4=\cdots ={\mu }_{20}=0\end{array}$$

(51)

where d₃ = a⁴ + 11a²ω² + ω⁴, d₄ = 4a⁴ + 89a²ω² + 4ω⁴, and in the above expression of each µ _k , k = 2, 3, ..., we have already let µ₁ = ··· = µ_{k− 1}= 0.

From the remark 3 and the singular point quantities (51), if we let $P_o^{-1}{A}_o{P}_o$ in (48) become A in (7), and considering the particular case in the Remark 2, then we have

Theorem 4.1. For the flow on center manifold of the system (44), the first two focal values and Liapunov constants of the origin are as follow

$$\left(i\right)v_3=\mathbf{i}\pi {\mu }_1,v_5=\mathbf{i}\pi {\mu }_2$$

(52)

$$\left(ii\right)V_2=\frac{1}{2}\mathbf{i}{\mu }_1,V_4=\frac{1}{2}\mathbf{i}{\mu }_2$$

(53)

where the expression of v₅is obtained under the condition of v₃ = 0, and V₄is obtained under the condition of V₂ = 0.

Remark 4. Considering Hopf bifurcation at the two symmetrical equilibria O₁ and O₂, form the Theorem 4.1, we conclude that the Lü system (44) at least 4 small limit cycles, which will be proved rigorously in a following paper.

Appendix

$$\begin{array}{ll}\hfill c_{\alpha \beta \gamma }& =\frac{1}{\beta -\alpha -\gamma d_0}[d_{200}\left(1+\gamma \right)c_{\alpha -2,\beta ,\gamma +1}-b_{020}\left(1+\beta \right)c_{\alpha -2,\beta +1,\gamma }+d_{110}\left(1+\gamma \right)c_{\alpha -1,\beta -1,\gamma +1}-\\ \left(a_{200}+b_{110}\beta -a_{200}\alpha -d_{101}\gamma \right)c_{\alpha -1,\beta ,\gamma }-b_{011}\left(1+\beta \right)c_{\alpha -1,\beta +1,\gamma -1}+\\ d_{020}\left(\gamma +1\right)c_{\alpha ,\beta -2,\gamma +1}+\left(b_{200}-b_{200}\beta +a_{110}\alpha +d_{011}\gamma \right)c_{\alpha ,\beta -1,\gamma }-(d_{002}+b_{101}\beta -\\ a_{101}\alpha -d_{002}\gamma )c_{\alpha ,\beta ,\gamma -1}-b_{002}(1+\beta )c_{\alpha ,\beta +1,\gamma -2}+a_{020}(1+\alpha )c_{\alpha +1,\beta -2,\gamma }+\\ a_{011}(1+\alpha )c_{\alpha +1,\beta -1,\gamma -1}+a_{002}(1+\alpha )c_{\alpha +1,\beta ,\gamma -2}],\\ \hfill {\mu }_m& =d_{200}{c}_{m-2,m,1}-b_{020}\left(1+m\right)c_{m-2,m+1,0}+d_{110}{c}_{m-1,m-1,1}-\left(a_{200}-a_{200}m+b_{110}m\right)\\ c_{m-1,m,0}+d_{020}{c}_{m,m-2,1}+\left(b_{200}+a_{110}m-b_{200}m\right)c_{m,m-1,0}+a_{020}\left(1+m\right)c_{m+1,m-2,0}.\end{array}$$