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The equivalence between singular point quantities and Liapunov constants on center manifold
Advances in Difference Equations volume 2012, Article number: 78 (2012)
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.
We consider the real three-dimensional differential autonomous systems which take the form
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 , 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.  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:
Where , and d, λ, λ1, , and X k , Y k , U k are homogeneous polynomials in x1, x2, x3 of degree k.
According to the center manifold theorem , the three-dimensional system (2) has an approximation to the center manifold taking the form
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
where 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
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
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
where |P| ≠ 0 and . 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
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
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,
where φ k (θ) and ψ k (θ), k = 1, 2, 3, ... are analytic. System (10) is again transformed into
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
For sufficiently small h, let
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
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, ....
where every 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
system (8) is also transformed into the following complex system
where z, w, T, a kjl , b kjl , ∈ ℂ (k, j, l ∈ ℕ), the systems (8) and (17) are called concomitant. If no confusion arises, , Ũ are still written as d kjl and U.
Lemma 2.1 (see ). For system (17), using the program of term by term calculations, we can determine a formal power series:
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 ). 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:
whereare 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:
whereare polynomial functions of coefficients of system (8) and
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:
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
Thus, there must exist only a generic transformation matrix T1 = k1η1+k2η2+k3η3 with the following form
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
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
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.
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
in (6), then at this time every .
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:
where σ m has been given in (22) of Theorem 3.1 andare 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:
and if for all k = 1, 2, ..., m − 1, V2k= 0 or v2k+1= 0 or µ k = 0 holds, then we have
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: in (7), we obtain
and more under the polar coordinates x = r cos θ, y = r sin θ, we have
Substituting (32) into the Liapunov function (5), then we denote
where H k [x, y] is a homogeneous polynomial in x, y of degree k and
with . And more from the expressions (13) and (34), we have
Next, we investigate based on the Equation 6.
On the one hand, from (14), we denote
and from the expression (13), one can get
On the other hand, we denote
From system (10) and the expression (13), we can get
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
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 , and takes the following form
where a > 0, b > 0, c > 0. Obviously, system (44) has three equilibrium points: O(0, 0, 0), and . 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:
which means that system (44) is symmetrical. Therefore, we only need to consider O1.
The Jacobian matrix of system (44) at O1 is
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
Thus we obtain the critical condition of Hopf bifurcation at O1: c = (a + b)/ 3, then
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
Under the conditions (46), one can find a nondegenerate matrix
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:
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):
where u ∈ ℝ, z, w, T ∈ ℂ, and
where d2 = a2 + 4ω2, and 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):
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 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
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.
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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).
The authors declare that they have no competing interests.
QW completed the main study, carried out the results of this article and drafted the manuscript. WH checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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Wang, Q., Huang, W. The equivalence between singular point quantities and Liapunov constants on center manifold. Adv Differ Equ 2012, 78 (2012). https://doi.org/10.1186/1687-1847-2012-78
- three-dimensional system
- multiple Hopf bifurcation
- singular point quantities
- center manifold