Advertisement

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

  • Qinlong WangEmail author
  • Wentao Huang
Open Access
Research

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.

Keywords

three-dimensional system multiple Hopf bifurcation singular point quantities center manifold 

1 Introduction

We consider the real three-dimensional differential autonomous systems which take the form
x ˙ = A x + f ( x ) Open image in new window
(1)

where x ∈ ℝ3, A ∈ ℝ3×3, 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:
1 = 1 x 1 + 2 x 2 + k = 2 X k ( x 1 , x 2 , x 3 ) , 2 = x 1 - 1 x 2 + k = 2 Y k ( x 1 , x 2 , x 3 ) , 3 = - d x 3 + k = 2 U k ( x 1 , x 2 , x 3 ) Open image in new window
(2)

Where i = d x i d τ ( i = 1 , 2 , 3 ) Open image in new window, and d, λ, λ1, 2 ( d > 0 , 1 2 + 2 < 0 ) Open image in new window, 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 ( x 1 , x 2 ) = u 2 ( x 1 , x 2 ) + h .o .t . Open image in new window
(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
1 = 1 x 1 + 2 x 2 + k = 2 X ̃ k ( x 1 , x 2 ) , 2 = x 1 - 1 x 2 + k = 2 k ( x 1 , x 2 ) Open image in new window
(4)
where X ̃ k , k Open image in new window 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 ( x 1 , x 2 ) = κ ( x 1 2 - 2 1 x 1 x 2 - 2 x 2 2 ) + k = 3 H k ( x 1 , x 2 ) Open image in new window
(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
d H d τ ( 4 ) = V 2 Φ 2 + V 4 Φ 3 + + V 2 m Φ m + 1 + Open image in new window
(6)

where Φ ( x 1 , x 2 ) = ( x 1 2 + x 2 2 ) Open image in new window or x 1 2 Open image in new window or x 2 2 Open image in new window or (x1 + x2)2 or other suitable forms [7, 10, 11].

Definition 2.1. V2min (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 A P = P - 1 1 2 0 - 1 0 0 0 - d P = 0 - ω 0 ω 0 0 0 0 - d B Open image in new window
(7)
where |P| ≠ 0 and ω = ( - 1 2 - 2 ) 1 / 2 > 0 Open image in new window. 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
d x d t = - y + k + j + l = 2 A k j l x k y j u l = X ( x , y , u ) , d y d t = x + k + j + l = 2 B k j l x k y j u l = Y ( x , y , u ) , d u d t = - d 0 u + k + j + l = 2 d k j l x k y j u l = U ( x , y , u ) Open image in new window
(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
d x d t = - y + k + j = 2 A k j x k y j = X ( x , y ) , d y d t = x + k + j = 2 B k j x k y j = Y ( x , y ) Open image in new window
(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, 13, 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,
d r d t = r k = 1 φ k ( θ ) r k , d θ d t = 1 + k = 1 ψ k ( θ ) r k Open image in new window
(10)
where φ k (θ) and ψ k (θ), k = 1, 2, 3, ... are analytic. System (10) is again transformed into
d r d θ = r k = 0 R k ( θ ) r k , Open image in new window
(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 ( θ + π ) = ( - 1 ) k R k ( θ ) , k = 0 , 1 , 2 , . Open image in new window
(12)
For sufficiently small h, let
Δ ( h ) = r ( 2 π , h ) - h , r = r ( θ , h ) = m = 1 v m ( θ ) h m Open image in new window
(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 ( θ ) = 1 , v m ( 0 ) = 0 , m = 2 , 3 , . Open image in new window
(14)

Definition 2.2. For the succession function in (13), if v2(2π) = v3(2π) = · · · = v2k(2π) = 0 and v2k+1(2π) ≠ 0, then the origin is called the fine focus or weak focus of order k, and the quantity of v2k+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 2 m ( 2 π ) = 1 1 + v 1 ( π ) k = 1 m - 1 ξ m ( k ) v 2 k + 1 ( 2 π ) Open image in new window
(15)

where every ξ m ( k ) Open image in new window is a polynomial of v1(π), v2(π), ..., v2m(π) and v1(2π), v2(2π), ..., v2m(2π) with rational coefficients. Particularly, if for each 1 ≤ k ≤ m − 1, v2k+1(2π) = 0 holds, we can get v2(2π) = v4(2π) = ··· = v2m(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 i , w = x - y i , u = u , T = i t , i = - 1 , Open image in new window
(16)
system (8) is also transformed into the following complex system
d z d T = z + k + j + l = 2 a k j l z k w j u l = Z ( z , w , u ) , d w d T = - w - k + j + l = 2 b k j l w k z j u l = - W ( z , w , u ) , d u d T = i d 0 u + k + j + l = 2 d ̃ k j l z k w j u l = Ũ ( z , w , u ) Open image in new window
(17)

where z, w, T, a kjl , b kjl , d ̃ k j l Open image in new window ∈ ℂ (k, j, l ∈ ℕ), the systems (8) and (17) are called concomitant. If no confusion arises, d ̃ k j l Open image in new window, 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 ( z , w , u ) = z w + α + β + γ = 3 c α β γ z α w β u γ Open image in new window
(18)
such that
d F d T = F z Z - F y W + F u U = m = 1 μ m ( z w ) m + 1 Open image in new window
(19)

where c110 = 1, c101 = c011 = c200 = c020 = 0, ckk 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 v2m+1, m = 1, 2, ..., we have the following relation:
v 2 m + 1 ( 2 π ) = i π μ m + i π k = 1 m - 1 ξ m ( k ) μ k Open image in new window
(20)

where ξ m ( k ) ( k = 1 , 2 , , m - 1 ) Open image in new windoware polynomial functions of coefficients of system (17). Usually it is called algebraic equivalence and written as v2m+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. V2mand v2m+1, for every positive integer m, we have the following relation:
V 2 m = σ m v 2 m + 1 ( 2 π ) + k = 1 m - 1 ξ m ( k ) v 2 k + 1 ( 2 π ) Open image in new window
(21)
where ξ m ( k ) ( k = 1 , 2 , , m - 1 ) Open image in new windoware polynomial functions of coefficients of system (8) and
σ m = 2 κ ω 3 ( ν 2 + υ 2 ) 0 2 π [ ϕ ( θ ) ] m + 1 d θ - 1 , Open image in new window
(22)
with
ϕ ( θ ) = x 1 θ 2 , i f Φ = x 1 2 x 2 θ 2 , i f Φ = x 2 2 x 1 θ 2 + x 2 θ 2 , i f Φ = x 1 2 + x 2 2 ( x 1 θ + x 2 θ ) 2 , i f Φ = ( x 1 + x 2 ) 2 Open image in new window
(23)
where
x 1 θ = [ ( 1 υ + ω ν ) cos ( θ ) + ( 1 ν - ω υ ) sin ( θ ) ] / , x 2 θ = υ cos ( θ ) + ν sin ( θ ) Open image in new window
(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 V2m~ σ m v2m+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 = υ 1 + ν ω ν 1 - υ ω 0 υ ν 0 0 0 ζ Open image in new window
(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
η 1 = 1 + i ω , 1 , 0 , η 2 = 1 - i ω , 1 , 0 , η 3 = ( 0 , 0 , 1 ) . Open image in new window
Thus, there must exist only a generic transformation matrix T1 = k1η1+k2η2+k3η3 with the following form
T 1 = k 1 ( 1 + i ω ) k 2 ( 1 - i ω ) 0 k 1 k 2 0 0 0 k 3 Open image in new window
such that
T 1 - 1 A T 1 = i ω 0 0 0 - i ω 0 0 0 - d J Open image in new window
(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 = i j 1 - i j 2 0 j 1 j 2 0 0 0 j 3 Open image in new window
such that
T 2 - 1 B T 2 = i ω 0 0 0 - i ω 0 0 0 - d = J Open image in new window
(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 = 1 ( K 2 - K 1 ) i + ( K 2 + K 1 ) ω 2 ( K 2 + K 1 ) 1 - ( K 2 - K 1 ) ω i 2 0 ( K 2 - K 1 ) i 2 K 2 + K 1 2 0 0 0 ζ Open image in new window
(28)
such that P 1AP = 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 = ν + i υ , K 2 = ν - i υ , ν , υ , , Open image in new window

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
κ = 1 2 ω - 4 and Φ = x 1 2 + x 2 2 Open image in new window

in (6), then at this time every σ m = 1 2 π Open image in new window.

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. V2mand µ m , m = 1, 2, ..., there exists the following relation:
V 2 m = i π σ m μ m + i π σ m k = 1 m - 1 ξ m ( k ) μ k Open image in new window
(29)

where σ m has been given in (22) of Theorem 3.1 and ξ m ( k ) ( k = 1 , 2 , , m - 1 ) Open image in new windoware polynomial functions of coefficients of system (17). Similarly we also call it algebraic equivalence and write as V2m~ 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 = σ 1 v 3 ( 2 π ) = i π σ 1 μ 1 Open image in new window
(30)
and if for all k = 1, 2, ..., m 1, V2k= 0 or v2k+1= 0 or µ k = 0 holds, then we have
V 2 m = σ m v 2 m + 1 = i π σ m μ m , m = 2 , 3 , . Open image in new window
(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: ( x 1 , x 2 , x 3 ) = P ( x , y , u ) Open image in new window in (7), we obtain
( x 1 , x 2 , x 3 ) = ( υ 1 + ν ω x + ν 1 - υ ω y , υ x + ν y , ζ u ) , Open image in new window
(32)
and more under the polar coordinates x = r cos θ, y = r sin θ, we have
x 1 = r [ ( υ 1 + ν ω ) cos ( θ ) + ( ν 1 - υ ω ) sin ( θ ) ] = r x 1 θ , x 2 = r [ υ cos ( θ ) + ν sin ( θ ) ] = r x 2 θ . Open image in new window
(33)
Substituting (32) into the Liapunov function (5), then we denote
H ( x 1 , x 2 ) = H [ x , y ] = k = 2 H k [ x , y ] Open image in new window
(34)
where H k [x, y] is a homogeneous polynomial in x, y of degree k and
H 2 [ x , y ] = - κ ( ν 2 + υ 2 ) ( 1 2 + 2 ) ( x 2 + y 2 ) Open image in new window
(35)
with H 2 [ 1 , 0 ] = κ ( ν 2 + υ 2 ) ω 2 Open image in new window. And more from the expressions (13) and (34), we have
H [ x , y ] = H [ r ( θ , h ) cos ( θ ) , r ( θ , h ) sin ( θ ) ] = k = 2 H k [ cos ( θ ) , sin ( θ ) ] r k ( θ , h ) = Ĥ ( θ , h ) . Open image in new window
(36)

Next, we investigate Δ Ĥ = Ĥ ( 2 π , h ) - Ĥ ( 0 , h ) Open image in new window based on the Equation 6.

On the one hand, from (14), we denote
Ĥ ( 2 π , h ) - Ĥ ( 0 , h ) = k = 2 H k [ 1 , 0 ] [ r k ( 2 π , h ) - r k ( 0 , h ) ] = k = 2 H k [ 1 , 0 ] [ r k ( 2 π , h ) - h k ] = Δ Ĥ 1 Open image in new window
(37)
where
r k ( 2 π , h ) - h k = ( r ( 2 π , h ) - h ) ( r k - 1 ( 2 π , h ) + r k - 2 ( 2 π , h ) h + + h k - 1 ) = ( r ( 2 π , h ) - h ) ( k h k - 1 + o ( h k - 1 ) ) , Open image in new window
(38)
and from the expression (13), one can get
Δ Ĥ 1 = ( r ( 2 π , h ) - h ) k = 2 H k [ 1 , 0 ] ( k h k - 1 + o ( h k - 1 ) ) = ( r ( 2 π , h ) - h ) 2 κ ( ν 2 + υ 2 ) ω 2 h + o ( h ) = 2 κ ( ν 2 + υ 2 ) ω 2 m = 2 v m ( 2 π ) h m + 1 ( 1 + o ( 1 ) ) . Open image in new window
(39)
On the other hand, we denote
Ĥ ( 2 π , h ) - Ĥ ( 0 , h ) = 0 2 π d Ĥ d θ d θ = 0 2 π d Ĥ d τ d τ d θ d θ = 1 ω 0 2 π d Ĥ d τ d t d θ d θ = Δ Ĥ 2 . Open image in new window
(40)
From system (10) and the expression (13), we can get
d t d θ = 1 + k = 2 r k - 1 ψ k + 1 ( θ ) - 1 = 1 + c 1 ( θ ) h + c 2 ( θ ) h 2 + = 1 + o ( 1 ) Open image in new window
(41)
where ψk+1(θ), ck− 1(θ) (k = 2, 3, ...) are analytic. At the same time, putting (33) in the right side of (6), we have
d Ĥ d τ = m = 1 V 2 m [ ϕ 1 2 ( θ ) r ( θ , h ) ] 2 m + 2 = m = 1 V 2 m h 2 m + 2 [ ϕ ( θ ) ] m + 1 ( 1 + o ( 1 ) ) , Open image in new window
(42)
thus
Δ Ĥ 2 = 1 ω m = 1 V 2 m h 2 m + 2 0 2 π [ ϕ ( θ ) ] m + 1 d θ  +  o (1) . Open image in new window
(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
1 = a ( x 2 - x 1 ) 2 = c x 2 - x 1 x 3 3 = x 1 x 2 - b x 3 Open image in new window
(44)
where a > 0, b > 0, c > 0. Obviously, system (44) has three equilibrium points: O(0, 0, 0), O 1 ( b c , b c , c ) Open image in new window and O 2 ( - b c , - b c , c ) Open image in new window. 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:
( x 1 , x 2 , x 3 ) ( - x 1 , - x 2 , x 3 ) , Open image in new window
(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 = - a a 0 - c c - b c b c b c - b Open image in new window
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
( λ 2 + ω 2 ) ( λ - λ 0 ) = 0 . Open image in new window
Thus we obtain the critical condition of Hopf bifurcation at O1: c = (a + b)/ 3, then
ω = b c , λ 0 = - 2 c Open image in new window
(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
x ˙ = A o x 1 + ω x 2 + ω x 3 + c + 0 - ( x 1 + ω ) ( x 3 + c ) ( x 1 + ω ) ( x 2 + ω ) . Open image in new window
(47)
Under the conditions (46), one can find a nondegenerate matrix
P o = 3 a ( 2 a 2 - ω 2 ) d 1 d 0 3 3 a 2 ω d 1 d 0 - 3 a 2 2 ω d 0 2 a 3 d 0 d 1 ω 3 d 0 d 1 2 ω 2 - a 2 2 ω 3 d 0 0 1 1 Open image in new window
such that
P o - 1 A o P o = 0 - ω 0 ω 0 0 0 0 - 2 d 0 ( 3 a ) Open image in new window
(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:
= 1 ω [ P o - 1 * diag ( P o y + E ) A o P o y ] = 1 ω [ P o - 1 A o P o y + P o - 1 * diag ( P o y ) A o P o y ] Open image in new window
(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):
d z d T = z + a 101 u z + a 011 u w + a 110 z w + a 200 z 2 + a 020 w 2 + a 002 u 2 = Z d w d T = - ( w + b 011 u z + b 101 u w + b 110 z w + b 020 z 2 + b 200 w 2 + b 002 u 2 ) = - W d u d T = d 001 u + d 101 u z + d 011 u w + d 110 z w + d 200 z 2 + d 020 w 2 + d 002 u 2 = U Open image in new window
(50)
where u ∈ ℝ, z, w, T ∈ ℂ, and
a 200 = 3 a ( 7 a 2 + 4 ω 2 ) 2 d 1 d 2 + 9 a 4 2 d 1 d 2 ω i , a 020 = - 3 a ( 16 a 6 - 93 a 2 ω 4 + 4 ω 6 ) 2 d 1 3 d 2 + 3 a 2 ( 32 ω 6 - 16 a 6 - 72 a 4 ω 2 - 105 a 2 ω 4 ) 2 d 1 3 d 2 ω i , a 002 = - 3 a 3 ( 8 a 4 + 19 a 2 ω 2 + 20 ω 4 ) 8 d 0 d 1 d 2 ω 2 - a 2 ( 4 a 6 - 21 a 4 ω 2 - 36 a 2 ω 4 + 16 ω 6 ) 8 d 0 d 1 d 2 ω 3 i , a 101 = - d 0 a ( 5 a 2 - 4 ω 2 ) d 1 d 2 ω 2 + 3 a 2 ( 5 a 2 - 4 ω 2 ) 2 d 1 d 2 ω i , a 011 = - a ( a 2 - 2 ω 2 ) ( 8 a 6 + 30 a 4 ω 2 + 45 a 2 ω 4 - 4 ω 6 ) 2 d 0 d 1 2 d 2 ω 2 + 3 a 2 ( a 2 - 2 ω 2 ) ( 4 a 4 + 5 a 2 ω 2 - 8 ω 4 ) 2 d 0 d 1 2 d 2 ω i , a 110 = 18 a 3 ( 2 a 2 + 5 ω 2 ) d 1 2 d 2 + 3 a 2 ( 8 ω 4 - 4 a 4 - 23 a 2 ω 2 ) d 1 2 d 2 ω i , b k j l = ā k j l ( k j l = 200 , 020 , 002 , 101 , 011 , 110 ) , d 200 = - 6 a ( 16 a 6 - 36 a 4 ω 2 - 21 a 2 ω 4 + 4 ω 6 ) d 1 3 d 2 + 54 a 2 ω ( 4 a 4 - a 2 ω 2 - 2 ω 4 ) d 1 3 d 2 i , d 020 = 6 a ( 16 a 6 - 36 a 4 ω 2 - 21 a 2 ω 4 + 4 ω 6 ) d 1 3 d 2 + 54 a 2 ω ( 4 a 4 - a 2 ω 2 - 2 ω 4 ) d 1 3 d 2 i , d 002 = - 3 a 2 ( 5 a 2 - 4 ω 2 ) 2 d 1 d 2 ω i , d 101 = - 3 a ( 16 a 4 - 7 a 2 ω 2 + 4 ω 4 ) d 1 2 d 2 - 12 a 2 ( 2 a 2 - ω 2 ) ( a 2 - 2 ω 2 ) d 1 2 d 2 ω i , d 011 = 3 a ( 16 a 4 - 7 a 2 ω 2 + 4 ω 4 ) d 1 2 d 2 - 12 a 2 ( 2 a 2 - ω 2 ) ( a 2 - 2 ω 2 ) d 1 2 d 2 ω i , d 110 = 36 a 2 ω ( a 2 - 2 ω 2 ) d 1 2 d 2 i , d 001 = 2 d 1 3 a ω i Open image in new window

where d2 = a2 + 4ω2, and ā k j l Open image in new window 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):
μ 1 = - 243 i a 5 ( 2 a 2 - ω 2 ) ( a 2 - 5 ω 2 ) ω / ( d 0 d 1 2 d 2 d 3 ) , μ 2 = - 891849714087780 i a 7 ( 2 a 2 - ω 2 ) ω 21 / ( d 0 3 d 1 5 d 2 3 d 3 2 d 4 ) , μ 3 = μ 4 = = μ 20 = 0 Open image in new window
(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 = ··· = µk− 1= 0.

From the remark 3 and the singular point quantities (51), if we let P o - 1 A o P o Open image in new window 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
( i ) v 3 = i π μ 1 , v 5 = i π μ 2 Open image in new window
(52)
( i i ) V 2 = 1 2 i μ 1 , V 4 = 1 2 i μ 2 Open image in new window
(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

c α β γ = 1 β - α - γ d 0 [ d 200 ( 1 + γ ) c α - 2 , β , γ + 1 - b 020 ( 1 + β ) c α - 2 , β + 1 , γ + d 110 ( 1 + γ ) c α - 1 , β - 1 , γ + 1 - ( a 200 + b 110 β - a 200 α - d 101 γ ) c α - 1 , β , γ - b 011 ( 1 + β ) c α - 1 , β + 1 , γ - 1 + d 020 ( γ + 1 ) c α , β - 2 , γ + 1 + ( b 200 - b 200 β + a 110 α + d 011 γ ) c α , β - 1 , γ - ( d 002 + b 101 β - a 101 α - d 002 γ ) c α , β , γ - 1 - b 002 ( 1 + β ) c α , β + 1 , γ - 2 + a 020 ( 1 + α ) c α + 1 , β - 2 , γ + a 011 ( 1 + α ) c α + 1 , β - 1 , γ - 1 + a 002 ( 1 + α ) c α + 1 , β , γ - 2 ] , μ m = d 200 c m - 2 , m , 1 - b 020 ( 1 + m ) c m - 2 , m + 1 , 0 + d 110 c m - 1 , m - 1 , 1 - ( a 200 - a 200 m + b 110 m ) c m - 1 , m , 0 + d 020 c m , m - 2 , 1 + ( b 200 + a 110 m - b 200 m ) c m , m - 1 , 0 + a 020 ( 1 + m ) c m + 1 , m - 2 , 0 . Open image in new window

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Project No. 10961011) and the Science Fund of Hubei Province Education Department in China (Project No. Q20091209).

References

  1. 1.
    Hofbauer J, Sigmund K: Evolutionary Games and Population Dynamics. Cambridge University Press, Cambridge; 1998.zbMATHCrossRefGoogle Scholar
  2. 2.
    Zeeman ML: Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems. Dyn Stab Syst 1993, 8: 189–217. 10.1080/02681119308806158zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Sparrow C: The Lorenz Equations: Bifcations, Chaos, and Strange Attractors. Springer, New York; 1982.CrossRefGoogle Scholar
  4. 4.
    Ćelikovský S, Chen G: On the generalized Lorenz canonical form. Chaos, Solitons Fractals 2005, 26: 1271–1276. 10.1016/j.chaos.2005.02.040zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Carr J: Applications of Center Manifold Theory. Springer, New York; 1981.CrossRefGoogle Scholar
  6. 6.
    Hassard B, Kazarinoff N, Wan Y: Theory and Application of Hopf Bifurcation. Cambridge University Press, Cambridge; 1981.Google Scholar
  7. 7.
    Lu Z, Luo Y: Two limit cycles in three-dimensional Lotka-Volterra systems. Comput Math Appl 2002, 44: 51–66. 10.1016/S0898-1221(02)00129-3zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kuznestov YA: Elements of applied bifurcation theory. Springer, New York; 1998.Google Scholar
  9. 9.
    Wang Q, Liu Y, Chen H: Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems. Bull des Sci Math 2010, 134: 786–798. 10.1016/j.bulsci.2009.12.001zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Andronov AA, Leontovich EA, Gordon II, Maier AG: Theory of Bifurcation of Dynamic Systems on a Plane. Wiley, New York; 1973.Google Scholar
  11. 11.
    Wang D: Mechanical manipulation for a class of differential systems. J Symb Comput 1991, 12: 233–254. 10.1016/S0747-7171(08)80127-7zbMATHCrossRefGoogle Scholar
  12. 12.
    Liu Y, Li J: Theory of values of singular point in complex autonomous differential system. Sci China Ser A 1990, 33: 10–24.zbMATHMathSciNetGoogle Scholar
  13. 13.
    Liu Y: Theory of center-focus for a class of higher-degree critical points and infinite points. Sci China Ser A 2001, 44: 37–48.Google Scholar
  14. 14.
    Chen H, Liu Y: Linear recursion formulas of quantities of singular point and applications. Appl Math Comput 2004, 148: 163–171. 10.1016/S0096-3003(02)00835-4zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Liu Y, Huang W: A cubic system with twelve small amplitude limit cycles. Bull des Sci Math 2005, 129: 83–98. 10.1016/j.bulsci.2004.05.004zbMATHCrossRefGoogle Scholar

Copyright information

© Wang and Huang; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.School of Information and MathematicsYangtze UniversityJingzhouPeople's Republic of China
  2. 2.School of Mathematics and Computing ScienceGuilin University of Electronic TechnologyGuilinPeople's Republic of China

Personalised recommendations