No periodic orbits in the Bianchi models B

Abstract In this paper we prove that the Bianchi models B do not have periodic solutions.


Introduction and statement of the main results
Bianchi [1,2] was the first to classify three dimensional Lie algebras in terms of the reduction to the canonical forms for their structure constants. There are nine types of models based on the dimension n of the algebra. a) n = 0 Type I; b) n = 1 Type II, III; c) n = 2 Type IV , V , V I, V II; d) n = 3 Type V III, IX.
To be of cosmological interest, this classification is called Bianchi universe models, which are spatially homogeneous cosmological models that in general are anisotropic. Taub [13] introduced this work of Bianchi into relativistic cosmology and derived the dynamic equations for the generic vacuum Bianchi geometries.
Ellis and MacCallum [8] gave this classification, introduced the type A/B of Bianchi model notation that is now in common use [Type A: a = 0, Type B: a ̸ = 0], see also [3]. The different types correspond to all nonequivalent sets of their structure constants. Let {X 1 , X 2 , X 3 } be an appropriate basis of the three dimensional Lie Algebra. Following Bogoyavlensky's approach, see [3], we can obtain the classification conditions on the property of the structure constants, and write in the form of commutation relations.
where a ∈ R, [ , ] is the Lie bracket and (n 1 , n 2 , n 3 ) with n i ∈ {+1, −1, 0}. In particular, for a = 0 we obtain models of type A and for a ̸ = 0 we obtain models of type B. we arrive at the following list of types A and B of homogeneous spaces. In the the table the roman numeral is labeled the type of the Bianchi classification  According to Bogoyavlensky [3], for the homogeneous cosmological models of Class B Einstein's system of equations reduces to the following dynamical system in the phase space p i , q i , p φ , φ, i = 1, 2, 3, where the function H is System (1) in an explicit form writes as with 0 ≤ k ≤ 1 andH = T + V G . Using dynamical systems methods, Collins [7] first introduce phase planes with compactified boundaries, to characterize the evolution of particular Bianchi classes of universe models, and Bogoyavlensky systematically introduce its study and application in [3]. After that many dynamical properties of the Bianchi models have been studied, for example, the integrability and the existence of periodic orbits, see [4,5,6,9,10,12] and the references quoted there.
In this paper we study the periodic solutions of the type B of Bianchi model (2). It is known that all the Bianchi class A models do not have periodic orbits. Using evolutions equations associated to these models, and showing that such equations always have some monotone function evaluated on the orbits, Wainwright and Ellis [14] prove that these models cannot exhibit periodic motion. Buzzi and Llibre [4] provide a new, direct and easier proof on the non-existence of periodic orbits for the 6 models of Bianchi class A. On the other hand for the Einstein's field equation, Llibre and Yu [11] studied the periodic orbits of the static, spherically symmetric Einstein-Yang-Mills equations. We shall mainly follow the idea of the qualitative analysis of dynamical systems shown in [4] and [11] in this paper.
Our main result is the following.
Theorem 1. The Bianchi models B, i.e. III, IV , V, VI and VII in Table  2 have no periodic solutions.
We shall provide the proof of Theorem 1 for the type V of the Bianchi models B in section 2, the proof of Theorem 1 for the type IV of the Bianchi models B in section 3, and the proof of Theorem 1 for the type III, VI and VII of the Bianchi models B in section 4.

The Bianchi V model
Consider Bianchi V model. Noticing that for Bianchi V we have p φ = 0 andφ = 0. After the change of coordinates and time we obtain a simplest model of six equations, and write it as the homogeneous polynomial differential system of degree 2. (3)ẋ where 0 ≤ k ≤ 1 and Λ = x 2 4 + x 2 5 + x 2 6 − 2(x 4 x 5 + x 4 x 6 + x 5 x 6 ). Proposition 2. The Bianchi V system (3) has no periodic solutions.
Proof. Suppose that system (3) has a periodic solution Γ(t) = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 )(t) with period T > 0. We have from (3) thatẋ 4 =ẋ 5 =ẋ 6 , then where a and b are any constants in R. Substituting into the first three equations of (3) we haveẋ From this differential system it follows that Hence we get x 1 x 3 = C 1 e 2bt and x 2 x 3 = C 2 e 2at . Since the exponential function is not periodic, so the Bianchi V system (3) has no periodic solutions.

The Bianchi IV model
Consider Bianchi IV model, after the change of coordinates and time system (2) can be written as the six-dimensional homogeneous polynomial differential system of degree 3. (4)ẋ where 0 ≤ k ≤ 1 and Proof. Obviously the hyperplane {x 1 = 0} is invariant manifold for system (4).
Suppose that system (4) has a periodic solution Γ(t) = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 )(t) with period T > 0 in x 1 = 0. We have that x 1 = 0, x 2 = a and x 3 = b where a, b are constants, and x 4 , x 5 , x 6 satisfy The set of singular points of (5) is the plane x 4 − x 5 + x 6 = 0. All the orbits of (5) shall be attracted or repelled by this plane. Moreover x 4 , x 5 and x 6 are monotone. Hence if (x 4 , x 5 , x 6 ) is a periodic solution of (4), then (x 4 , x 5 , x 6 ) has to be a singular point in the phase portrait of (5), i.e. a constant vector. So Γ(t) is a singular point instead of a periodic solution.

The Bianchi III, VI and VII model
Consider the rest of the cases, i.e. the type III, VI and VIII of Bianchi models. After the change of coordinates and time where N = x 1 − n 2 x 2 and n 2 = ±1, system (2) can be written as the six-dimensional homogeneous polynomial differential system of degree 5. (6)ẋ where 0 ≤ k ≤ 1 and Proof. Let X the vector field associated to the differential system (6). Hence The hyperplane N = 0 is invariant by the flows of system (6).
Suppose that system (6) has a periodic solution Γ(t) = (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 )(t) with period T > 0 in N = 0. Then we have that x 1 = c 1 , x 2 = c 2 and x 3 = c 3 where c i are constants, and x 4 , x 5 , x 6 satisfy Hence if (x 4 , x 5 , x 6 ) is a periodic solution of (7), using the same kind of arguments as in the proof of Lemma 3, (x 4 , x 5 , x 6 ) has to be a constant vector, i.e. a singular point in the phase portrait of (7). So Γ(t) is a singular point instead of a periodic solution.
Proposition 6. The type III, VI and VII of Bianchi system (6) has no periodic solutions.