Dynamics of two Einstein–Friedmann cosmological models

We describe completely the dynamics of the two Einstein–Friedmann cosmological models, which can be characterized by the Hamiltonians H=12(py2-px2)+e2xV(y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H = \frac{1}{2} (p_{y}^2 - p_{x}^2) + e^{2 x} V(y), \end{aligned}$$\end{document}with the cosmological potentials V(y)=eλy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(y)=e^{\lambda y}$$\end{document}, or V(y)=(a+by)ey\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(y)=(a+by)e^{y}$$\end{document} with λab≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda a b\ne 0$$\end{document}.


Introduction and statement of the main results
The present work is devoted to the Einstein-Friedmann cosmological models, which can be characterized by the Hamiltonian where V (y) = e λy or V (y) = (a + by)e y with λab = 0 are cosmological potentials. For more details on these two special models see subsections 2.2 and 3.1 [10], and for more details on the general Einstein-Friedmann cosmological models see [4,7]. The Hamiltonian system with two degrees of freedom associated with the Hamiltonian a e-mail: innabasak@gmail.com b e-mail: jllibre@mat.uab.cat (corresponding author) The above Hamiltonian H has the additional first integral as it is easy to check. The rank of the 2 × 4 matrix is two except at a zero Lebesgue measure set in the phase space E = R 4 with coordinates (x, y, p x , p y ). Therefore the first integrals H and F are independent. One can verify that the Poisson bracket {H, F} = 0, so the first integrals H and F are in involution. Therefore, the Hamiltonian system (3) is completely integrable in the sense of the Liouville-Arnold Theorem, see, for instance, [1,2]. Clearly, the above Hamiltonian system has no equilibrium points. Therefore, all values (h, f ) for the map H × F are regular; in particular, the level (H, is not empty, it is a two-dimensional manifold; for more details, see, for instance, [5].
Since H and F are first integrals, the sets are invariant by the Hamiltonian flow, i.e., if an orbit solution of the Hamiltonian system has a point in one of the previous three sets, the whole orbit is contained in that set. From physical reasons we are only interested in the dynamics of the Hamiltonian system (3) on the energy level H = 0, see [10]. Following the Liouville-Arnold Theorem, since the values (0, f ) are regular for all f ∈ R, every connected component of the invariant 2-manifold I 0 f is diffeomorphic either to a torus, to a cylinder or to a plane, see Theorem 3, and the dynamics on them are conjugated to a linear flow when the flow is complete, i.e., when the orbits are defined for all time t ∈ R.
that can be obtain from Hamiltonian (1)  Following [10] we consider the potential V (y) = (a + by)e y with ab = 0.
Then the Hamiltonian system with Hamiltonian (6) has the first integral The condition {H, F} = 0 proves that F is a first integral of the following Hamiltonian systemẋ Therefore this Hamiltonian system is completely integrable in the sense of Liouville-Arnold Theorem, because the rank of the 2 × 4 matrix is two except in a zero Lebesgue measure set of the phase space E = R 4 again in the coordinates (x, y, p x , p y ). Hence the first integrals H and F are independent.
As in the previous section the above Hamiltonian system has no equilibrium points and the levels (H, F) −1 (0, f ) are regular.

Theorem 2
The following statements hold for the Hamiltonian system (9): (a) The set I 0 f is empty if b < 0 and f ≤ 0; otherwise, I 0 f is an invariant two-dimensional manifold diffeomorphic to two copies of R 2 . (b) All the orbits of the Hamiltonian system (3) restricted to I 0 f come from the infinity and go to infinity.

Proof of Theorem 1
In order to describe the dynamics of a complete integrable Hamiltonian system of two degrees of freedom we shall use the Liouville-Arnold Theorem, which can be stated as follows; for more details, see [1,2,5]. Unfortunately, we cannot apply to our two Hamiltonian systems (3) and (9) the results of Theorem 3 because we do not know whether the flows of these Hamiltonian systems are complete, because we cannot obtain explicitly the solutions of these systems in function of the time. But as we shall see later on we can apply to them the Markus-Neumann-Peixoto theorem.
From F(x, y, p x , p y ) = f we obtain that Assume that |λ| = 2. Then, from H x, y, p x , we get that Substituting p x from (11) in (10) we obtain In order to prove Theorem 1 we must characterize the topology of the two-dimensional manifolds I 0 f given by Note that when in the expression of p y there is a plus (respectively, minus) in the expression of p x there is a plus (respectively, minus). In short, when |λ| = 2 from (13) it follows that I 0 f is empty if f = 0 and |λ| > 2; otherwise, I 0 f is an invariant two-dimensional manifold diffeomorphic to two copies of R 2 . This completes the statements (a) and (b) of Theorem 1.
Assume λ = 2. Then, from F(x, y, p x , p y ) = f we obtain again that p y = ( f + 2 p x )/2, and from H (x, y, p x , ( f + 2 p x )/2) = 0, we have that p x = −2e 2(x+y)) / f − f /4. Substituting this expression of p x into the previous expression of p y we obtain that p y = −2e 2(x+y)) / f + f /4. In order to prove statement (c) of Theorem 1 we must characterize the topology of the two-dimensional manifolds I 0 f given by (2021) 136:8 Page 5 of 7 8 Therefore, when λ = 2 from (14) it follows that I 0 f is empty if f = 0; otherwise, I 0 f is an invariant two-dimensional manifold diffeomorphic to R 2 . This completes the statement (c) of Theorem 1. Assume now λ = −2. Then, from F(x, y, p x , p y ) = f we obtain again that p y = ( f −2 p x )/2, and from H (x, y, p x , ( f −2 p x )/2) = 0, we have that p x = 2e 2(x−y)) / f + f /4. Substituting this expression of p x into the previous expression of p y we obtain that p y = −2e 2(x−y)) / f + f /4. Again in order to prove statement (d) of Theorem 1 we must characterize the topology of the two-dimensional manifolds I 0 f given by Therefore, when λ = −2 from (15) it follows that I 0 f is empty if f = 0; otherwise, I 0 f is an invariant two-dimensional manifold diffeomorphic to R 2 . This completes the statement (d) of Theorem 1.
In order to prove statement (e) we need some preliminary results. A phase portrait of the Hamiltonian system (3) restricted to the two-dimensional manifold I 0 f is the decomposition of I 0 f as union of the orbits of this differential system.
Two phase portraits on I 0 f 1 and on I 0 f 2 are topologically equivalent if there is a homeomorphism h : I 0 f 1 −→ I 0 f 2 which send orbits in I 0 f 1 into orbits of I 0 f 2 , preserving or reversing the sense of all the orbits.
A separatrix of the Hamiltonian system (3) restricted to the two-dimensional manifold I 0 f is one of following orbits: the equilibrium points, the limit cycles, and the two orbits at the boundary of every hyperbolic sector of an equilibrium point, see for more details on the separatrices [6,8]. Recall that a limit cycle of a differential system is a periodic orbit isolated in the set of all periodic orbits of the differential system. For a definition of a hyperbolic sector see page 18 of [3].
The set of all separatrices of the Hamiltonian system (3) restricted to the two-dimensional manifold I 0 f , denoted by 0 f , is a closed set (see [8]).
A canonical region of I 0 f is an open connected component of I 0 f \ 0 f . The union of the set 0 f with an orbit of each canonical region form the separatrix configuration of the Hamiltonian system (3) restricted to the two-dimensional manifold I 0 f and is denoted by 0 f . We say that the flow of the Hamiltonian system (3) restricted to a two-dimensional manifold I 0 f is parallel if it is topologically equivalent to one of the following flows: (i) The flow defined on R 2 by the differential systemẋ = 1,ẏ = 0, which it is called the strip flow. (ii) The flow defined on R 2 \ {0} by the differential system given in polar coordinates by r = 0, θ = 1, which it is called the annulus flow. (iii) The flow defined on R 2 \ {0} by the differential system given in polar coordinates by r = r , θ = 0, which it is called the spiral or nodal flow. A main result is the following: The flow at every canonical region of a flow on a twodimensional manifold is parallel, given by either a strip, an annular, or a spiral flow, see [8].
According to the following theorem, which was proved by Markus [6], Neumann [8], and Peixoto [9], it is sufficient to investigate the separatrix configuration of a differential system on a two-dimensional manifold, for determining its phase portrait. From (19) it follows that I 0 f is empty if b < 0 and f ≤ 0; otherwise, I 0 f is an invariant two-dimensional manifold diffeomorphic to two copies of R 2 . So statement (a) of Theorem 2 is proved.
The proof of statement (b) of Theorem 2 is exactly the same than the proof of statement (e) of Theorem 1. This completes the proof of Theorem 2.