The equivalence between singular point quantities and Liapunov constants on center manifold
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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.
Keywordsthree-dimensional system multiple Hopf bifurcation singular point quantities center manifold
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
Where , and d, λ, λ1, , and X k , Y k , U k are homogeneous polynomials in x1, x2, x3 of degree k.
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
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].
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
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.
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).
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.
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.
where ς , ν , υ are three real numbers such that ς(υ2 + ν2) ≠ 0 holds.
where k1, k2 and k3 are three arbitrary non-zero constants.
where j1, j2 and j3 are also three arbitrary non-zero constants.
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. □
in (6), then at this time every .
Furthermore, from Lemma 2.2 and Theorem 3.1, we have
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 .
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
Next, we investigate based on the Equation 6.
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
which means that system (44) is symmetrical. Therefore, we only need to consider O1.
where d0 = a2 + ω2, d1 = 4a2+ ω2.
where y = (y1, y2, y3), and E is the 3 × 3 identity matrix.
where d2 = a2 + 4ω2, and denotes the conjugate complex number of a kjl .
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
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.
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).
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