This is a preprint of: “Dynamics of the Szekeres system”, Jaume Llibre, Clàudia Valls, J. Math. Phys., vol. 62, 082502 (7 pages), 2021. DOI: [10.1063/5.0054051] DYNAMICS OF THE SZEKERES SYSTEM JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. The Szekeres model is a four-dimensional system which are the exact solutions of the Einstein field equations when there exists irrotational dust. It is completely integrable with two rational and one analytic first inte- gral. We describe the dynamics of the Szekeres system for any of the values of these two rational first integrals. 1. Introduction and statement of the results The exact solutions of the Einstein field equations with irrational dust are mod- elized by the Szekeres model (a four-dimensional system introduced in [15]). We note that the Lemaı̂tre-Tolman models (see [2] for details) can be obtained by the limiting cases of the Szekeres models. The equations of motion of the Szekeres system are ρ̇ = −θρ, 1 1 θ̇ = − θ2 − 6σ 2 − ρ, 3 2 (1) 2 2 σ̇ = σ − θσ − E, 3 1 Ė = −3Eσ − θE − ρσ, 2 where ρ is the energy density, θ is the expansion scalar, σ is the shear and E is the Weyl tensor and the dot means derivative with respect to t. As it is shown in [3], the Silent Universe system with σ1 = σ2 = σ and E1 = E2 = E yields the Szekeres system (here σ1 and σ2 are the independent eigenvalues of the traceless shear tensor and E1 and E2 are the traceless components of the Weyl tensor). In order to describe the propagation of the light in nonhomogeneous universe models, or to analyze the evolution and formation of the structure of the Universe some authors used the Szekeres system, see for instance [1, 8, 9, 14, 16] and the references therein. This clearly justifies the necessity of knowing the dynamics of such system near infinity. As it was shown in [5] the Szekeres system is completely integrable because it has three functional independent first integrals (two of them being rational and the 2010 Mathematics Subject Classification. 34C05, 34C07, 34C08. Key words and phrases. Szekeres system, phase portrait, Poincaré compactification. 1 2 J. LLIBRE AND C. VALLS third one being analytic). The two rational first integrals are (−18E − 3ρ + (θ + 3σ)2 )3 F = , (6E + ρ)2 (3θσ(2E + ρ) − E(18E + 2θ2 + 3ρ) + 9σ 2 (4E + ρ))3 H= . ρ3 (ρ + 6E)2 Setting F = f and H = h we will describe the dynamics of the Szekeres system (1) restricted to the invariant set defined by F = f and H = h. The next equality follows from system (1) ρ̇ 2(−ρĖ + E ρ̇) θ=− , σ= . ρ ρ(ρ + 6E) Introducing θ and σ in the equations of θ̇ and σ̇ in (1), solving them with respect to ρ̈ and Ë we get a differential system that has the form ρ̈ = f (ρ, E), Ë = g(ρ, E). Applying to that system the change of variables 6 x ρ= , E=− , (1 − x)y 3 (1 − x)y 3 we obtain the differential system (1) rewritten as 2ẋẏ 3x 1 (2) ẍ + − 3 = 0, ÿ + = 0. y y y2 Such a reduction (going from system (1) to system (2)) was done in [11]. We rewrite system (2) as ẋ = z, ẏ = w, 2zw 3x (3) ż = − + 3, y y 1 ẇ = − 2 . y In these new variables the first integrals F and H of system (3) become 2 x F = w2 − and H = xw2 + + wyz. y y Setting F = f and H = h we obtain 2 2(f h + 2wz − hw2 ) y= and x = − . w2 − f (f − w2 )(f − 3w2 ) Note that here the expressions for y and x are not well defined when (f − w2 )(f − 3w2 ) = 0. So, if f = 0 they are not wellpdefined for w = 0, and if f > 0 they are not √ well-defined for w = ± f and w = ± f /3. So in these cases the variables x and y are not defined and so they correspond to singularities of the Skezeres system. DYNAMICS OF THE SZEKERES SYSTEM 3 The differential system (3) restricted to F = f and H = h can be written as (f − w2 ) z0 = (3f 2 h + 10f zw − 6f hw2 − 18zw3 + 3hw4 ), (4) 4(f − 3w2 ) 1 w0 = − (f − w2 )2 . 4 We do a reparametrization of time in the form dt = (f − 3w2 )ds/(f − w2 ), and we rewrite system (4) as ż = 3f 2 h + 10f zw − 6f hw2 − 18zw3 + 3hw4 , (5) 1 ẇ = − (f − w2 )(f − 3w2 ), 4 where for this new system the dot means derivative with respect to s. The objective now is to know the dynamics of this differential system for all the values of f and h. To do so and taking into account that system (5) is a polynomial differential system, we shall use the Poincaré compactification (see [4, Chapter 5] for details and section 3). Roughly speaking the α-limit of an orbit is the place where it borns and the ω-limit of an orbit is the place where it dies. A more precise definition will be given at the end of section 3. We will investigate the α-limit and ω-limit of all the solutions of system (1). The dynamics of the Szekeres system (1) when f < 0 was completely determined in [6], there it is proved the following result. Theorem 1. All solutions of the Szekeres system (1) with f < 0 have α-limit and ω-limit at infinity, more precisely at the endpoints of the z-axis and w-axis. The main objective of this paper is to study the dynamics of Szekeres system (1) in the remaining cases, that is when either f = 0, or f > 0. Our main results are the following two results. Theorem 2. The following holds for the Szekeres system (1) with f > 0. √ (a) In the half-space w > f all solutions have α-limit √ the infinity at the endpoints of the z-axis, and ω-limit the point (0, f ) without reaching it (because on that point the dynamics is not defined), except the ones which have α-limit an infinite singular √ point between the endpoints of the z-axis and ω-limit the point (0, f ). √ p (b) The orbits between the hyperplanes w = f q and w q= f /3 have α-limit √  the point (0, f ) and ω-limit the point − f3 h, f3 without reaching them (on these points the dynamics is not defined). p p (c) The orbits betweenqthe hyperplanes q  w = f /3 and w q= − q f /3 have α-  f f f f limit the point − 3 h, 3 and ω-limit the point 3 h, − 3 without reaching them (on these points the dynamics is not defined). 4 J. LLIBRE AND C. VALLS p √ (d) The orbits between q the q hyperplanes w = − f /3 and w = − f have α-limit f f  √  the point 3 h, − 3 and ω-limit the point 0, − f without reaching them (on these points the dynamics is not defined). √ (e) In the half-space w < − f all solutions have ω-limit √ the infinity at the endpoints of the z-axis and α-limit the point (0, − f ) without reaching it (because on that point the dynamics is not defined), except the ones which have ω-limit an infinite singular √ point between the endpoints of the z-axis and α-limit the point (0, − f ). Theorem 3. All solutions of the Szekeres system (1) with f = 0 have α-limit the infinity at the endpoints of the z-axis, and ω-limit the origin, without crossing the hyperplane w = 0 where the dynamics is not defined, except the ones which are in w > 0 and have α-limit an infinite singular point between the endpoints of the z-axis and ω-limit the origin, and the ones which are in w < 0 and have ω-limit an infinite singular point between the endpoints of the z-axis and α-limit the origin. Theorems 3 and 2 are proved in section 3. 2. Preliminaries 2.1. Poincaré compactification. In this section we summarize some basic results about the Poincaré compactification, which was done by Poincaré in [13]. He pro- vided a tool for studying the behaviour of a planar polynomial differential system near the infinity. For more details on the Poincaré compactification, see [4, Chapter 5]. ∂ ∂ Let X = P +Q be a polynomial vector field of degree d. We consider the ∂x1 ∂x2 Poincaré sphere S = {y = (y1 , y2 , y3 ) ∈ R3 : y12 + y22 + y32 = 1}, its tangent plane 2 to the point (0, 0, 1) is identified with R2 . Now we consider the central projection f : R2 → S2 of the vector field X , which sends every point x ∈ R2 to the two intersection points of the straight line passing through the point x and the origin of coordinates with the sphere S2 . We note that the equator S1 = {y ∈ S2 : y3 = 0} of the sphere is in bijection with the infinity of R2 . The differential Df sends the vector field X on R2 into a vector field X 0 defined on S2 \ S1 , which is formed by two symmetric copies of X with respect to the origin of coordinates. We can extend the vector field X 0 analytically to a vector field on S2 multiplying X 0 by y3d . This new vector field is denoted by p(X ) and it is called the Poincaré compactification of the polynomial vector field X on R2 . The dynamics of p(X ) near S1 corresponds with the dynamics of X in the neighborhood of the infinity. Since S2 is a curved surface, for working with the vector field p(X ) on S2 , we need the expressions of this vector field in the local charts (Ui , φi ) and (Vi , ψi ), where Ui = {y ∈ S2 : yi > 0}, Vi = {y ∈ S2 : yi < 0}, φi : Ui −→ R2 and ψi : Vi −→ R2 for i = 1, 2, 3, with φi (y) = −ψi (y) = (ym /yi , yn /yi ) for m < n and m, n 6= i. In the local chart (U1 , φ1 ) the expression of p(X ) is        d 1 u 1 u d+1 1 u u̇ = v −uP , +Q , , v̇ = −v P , . v v v v v v DYNAMICS OF THE SZEKERES SYSTEM 5 In (U2 , φ2 ) the expression of p(X ) is        d u 1 u 1 d+1 u 1 u̇ = v P , − uQ , , v̇ = −v Q , , v v v v v v and for (U3 , φ3 ) is u̇ = P (u, v), v̇ = Q(u, v). The expressions for p(X ) in the local chart (Vi , ψi ) is the same as in the local chart (Ui , φi ) multiplied by (−1)d−1 for i = 1, 2, 3. The points of S1 in any local chart have its v coordinate equal to zero. We note that the equator S1 is invariant by the vector field p(X ). The infinite singular points of X are the singular points of p(X ) which lie in S1 . Note that if y ∈ S1 is an infinite singular point, then −y is also an infinite singular point and these two points have the same stability if the degree of vector field is odd. Such stability change to the opposite if the degree of the vector field is even. The image of the northern hemisphere of S2 onto the plane y3 = 0 under the projection π(y1 , y2 , y3 ) = (y1 , y2 ) is called the Poincaré disc which is denoted by D. The integral curves of S2 are symmetric with respect to the origin, therefore it is sufficient to investigate the flow of p(X ) only in the closed northern hemisphere. In order to draw the phase portrait on the Poincaré disc it is needed to project by π the phase portrait of p(X ) on the northern hemisphere of S2 . We note that the points (u, 0) are the points at infinity in the local charts Ui and Vi with i = 1, 2. Moreover, we remark that for studying the infinite singularities it is sufficient to study them on the local chart U1 , and to check if the origin of the local chart U2 is or not a singularity. 2.2. Topological equivalence of two polynomial vector fields. Let X1 and X2 be two polynomial vector fields on R2 . We say that they are topologically equivalent if there exists a homeomorphism on the Poincaré disc D which preserves the infinity S1 and sends the orbits of π(p(X1 )) to orbits of π(p(X2 )), preserving or reversing the orientation of all the orbits. A separatrix of the Poincaré compactification π(p(X )) is one of following orbits: all the orbits at the infinity S1 , the finite singular points, the limit cycles, and the two orbits at the boundary of a hyperbolic sector at a finite or an infinite singular point, see for more details on the separatrices [7, 10]. The set of all separatrices of π(p(X )), which we denote by ΣX , is a closed set (see [10]). A canonical region of π(p(X )) is an open connected component of D \ ΣX . The union of the set ΣX with an orbit of each canonical region form the separatrix con- figuration of π(p(X )) and is denoted by Σ0X . We denote the number of separatrices of a phase portrait in the Poincaré disc by S, and its number of canonical regions by R. Two separatrix configurations Σ0X1 and Σ0X2 are topologically equivalent if there is a homeomorphism h : D −→ D such that h(Σ0X1 ) = Σ0X2 . 6 J. LLIBRE AND C. VALLS According to the following theorem which was proved by Markus [7], Neumann [10] and Peixoto [12], it is sufficient to investigate the separatrix configuration of a polynomial differential system, for determining its global phase portrait. Theorem 4. Two Poincaré compactified polynomial vector fields π(p(X1 )) and π(p(X2 )) with finitely many separatrices are topologically equivalent if and only if their separatrix configurations Σ0X1 and Σ0X2 are topologically equivalent. 3. Proofs Figure 1. The phase portrait of system (5) in the Poincaré disc√ f > 0. On the four parallel straight lines w equal to ± f when p and ± f /3 of this phase portrait of the Szekeres system restricted to F = f > 0 and H = h is not defined. Figure 2. The phase portrait of system (5) in the Poincaré disc when f = 0. On the straight line w = 0 this phase portrait of the Szekeres system restricted to F = 0 and H = h is not defined. Theorem 5. The global phase portraits of system (5) with f > 0 are topologically equivalent to the one described in Figure 1, and the global phase portraits of system (5) with f = 0 are topologically equivalent to the one described in Figure 2. Proof. There are four finite singular points of system (5) when f > 0 which are p  p p  p p  p p1 = (0, f ), p2 = − f /3 h, f /3 , p3 = f /3 h, − f /3 , p4 = (0, − f ). DYNAMICS OF THE SZEKERES SYSTEM 7 Computing the eigenvalues of the Jacobian matrix evaluated at these points we get 3/2 3/2 3/2 3/2 √ are −8f , −f ), p2 is an unstable node that p1 is a stable node (the√eigenvalues (the eigenvalues √ are√4f / 3, f / 3), p3 is a stable node (the eigenvalues are −4f 3/2 / 3, −f 3/2 / 3) and p4 is an unstable node (the eigenvalues are 8f 3/2 , f 3/2 ). On the other hand if f = 0 system (5) can be written as ż = w3 (−18z + 3hw), (6) 3 ẇ = w4 , 4 and introducing the reparameterization of time ds = w3 dr we rewrite system (6) as ż = −18z + 3hw, (7) 3 ẇ = − w, 4 where now the dot means derivative with respect the new independent variable r. Note that the unique finite singular point of system (7) is the origin is a stable node because the eigenvalues of the Jacobian matrix at the origin are −18 and −3/4. Now we study the infinite singular points when f > 0. We can work in both cases with system (5). On the local chart U1 system (5) writes 1 u̇ = (69u4 − 36f u2 v 2 − f 2 v 4 − 12hu5 + 24f hu3 v 2 − 12f 2 huv 4 ), (8) 4 v̇ = −v(−18u3 + 10f uv 2 + 3hu4 − 6f hu2 v 2 + 3f 2 hv 4 ). The singular points at infinity in the local chart U1 are (u, v) = (0, 0) and (u, v) = (23/(4h), 0). So, if h 6= 0 we have two singular points on U1 , and if h = 0 we have only the origin as a singular point on U1 . If h 6= 0, computing the eigenvalues of the Jacobian matrix at (23/(4h), 0) we get that they are −839523/(256h3 ) and 36501/(256h3 ). So this point is a saddle. On the other hand computing the Jacobian matrix at the origin (0, 0) we get that it is identically zero. We need to do blow ups. We introduce the new variable w = v/u. In the new variables (u, w) we can rewrite system (10) as u4 u̇ = − (−69 + 12hu + 36f w2 − 24f huw2 + f 2 w4 + 12f 2 huw4 ), (9) 4 u3 w ẇ = (f w2 − 3)(f w2 − 1). 4 Now doing a rescaling of the independent variable we eliminate the term u3 in system (9) and we obtain u u̇ = − (−69 + 12hu + 36f w2 − 24f huw2 + f 2 w4 + 12f 2 huw4 ), 4 w ẇ = (f w2 − 3)(f w2 − 1). 4 √ √ The solutions p on u = p 0 of ẇ = 0 are precisely w = 0, w = −1/ f , w = 1/ f , w = − 3/f and w = 3/f . Computing the eigenvalues of the Jacobian matrix at these points we se that (0, 0) is an unstable node (the eigenvalues are 69/4, 3/4), 8 J. LLIBRE AND C. VALLS √ p (0, ±1/ f ) are saddles (the eigenvalues are 8, −1 for both of them), (0, ± 3/f ) are saddles (the eigenvalues are −12, 3 for both of them). Now going back through the changes of variables to system (10) we get that the origin of U1 is formed by six hyperbolic sectors and two parabolic sectors. Three hyperbolic sectors are separated from the other three by the line at infinity and adjacent to the line at infinity in both sides there is a parabolic sector. On the local chart U2 system (5) becomes 1 u̇ = (12h − 69u − 24f hv 2 + 36f uv 2 + 12f 2 hv 4 + f 2 uv 4 ), 4 1 v̇ = v(f v 2 − 3)(f v 2 − 1). 4 The origin of U2 is a singular point if and only if h = 0. In this last case computing the eigenvalues of the linear part of the differential system at the origin we get that they are −69/4 and 3/4. So the origin of U2 is a saddle. Gluing all these information together we get that the global phase portrait of system (5) when f > 0 in the Poincaré disc D is topologically equivalent to the one of Figure 1. Now we study the infinite singular points when f = 0. On the local chart U1 system (5) writes 3 4 (10) u̇ = u (23 − 24hu), v̇ = 3u3 v(6 − hu). 4 Doing a rescaling of the independent variable we eliminate the term u3 in (10) and we obtain 3 u̇ = u(23 − 24hu), v̇ = 3v(6 − hu). 4 The singular points at infinity in the local chart U1 are ((0, 0) and ((23/(4h), 0). So if h 6= 0 we have two infinite singular points on U1 , and if h = 0 we have only the origin as an infinite singular point on U1 . If h 6= 0, computing the eigenvalues of the Jacobian matrix at (23/(4h), 0) we get that they are −69/4 and 3/4. So this point is a saddle. On the other hand, computing the eigenvalues of the Jacobian matrix at (0, 0) we get that they are 69/4 and 18. So this point is an unstable node. On the local chart U2 system (5) becomes 3 3 u̇ = (4h − 23u), v̇ = v. 4 4 The origin of U2 is a singular point if and only if h = 0. In this last case computing the eigenvalues of the linear part of the differential system at the origin we get that they are −69/4 and 3/4. So the origin of U2 is a saddle. Gluing all these information together we get that the global phase portrait of system (5) when f = 0 in the Poincaré disc D is topologically equivalent to the one of Figure 2.  Let q ∈ D and denote by φt (q) the solution of the extended flow in D of system (5) that at time t = 0 pass through the point q. We recall that a point p ∈ D is an DYNAMICS OF THE SZEKERES SYSTEM 9 ω-limit (resp. α-limit) of a point q ∈ D if there are points φt1 (q), φt2 (q), . . . in the orbit of q such that tk → ∞ (resp. tk → −∞) and φtk (q) → p as k → ∞. Proof of Theorems 2 and 3. The proof of Theorem 2 comes interpreting the results provided in Theorem 5 on the gravitational Szekeres system (1) when f > 0, and the proof of Theorem 3 comes interpreting the results provided in Theorem 5 on the gravitational Szekeres system (1) when f = 0. More precisely, we prove statement (a) of Theorem 2, the other statements as well as the proof of Theorem√ 3 are proved in a similar way. Consider the orbits over the hyperplane w = f for fixed values of f > 0 and h ∈ R. From Figure 1 for the two fixed values of f and h there is a unique orbit γ with α-limit the infinite singular point (23/(4h),√ 0) ∈ U1 if h 6= 0, or (0, 0) ∈ U2 if h = 0, and ω-limit the finite stable node (0, f ). This orbit γ for the two fixed values of f and h separates two kind of orbits. The ones with α-limit √ the infinite singular point (0, 0) ∈ V1 and ω-limit the finite stable node (0, f ), and the others with α-limit the infinite singular point (0, 0) ∈ U1 and ω-limit the finite stable singular point. This completes the proof of statement (a) of Theorem 2.  4. Conclusion The gravitational Szekeres differential system (1) is completely integrable with two rational first integrals and an additional analytical first integral. One of these two rational first integrals is F = (−18E − 3ρ + (θ + 3σ)2 )3 /(6E + ρ)2 . The dynamics of the Szekeres system when the first integral F takes negative values was described in [6], showing that all the orbits come from the infinity of R4 in the variables (ρ, θ, σ, E) and go to infinity. In other words all orbits of the gravitational Szekeres differential system born at infinity and end at infinity when the first integral F < 0. In this paper we have described the dynamics when the first integral F takes positive and zero values. Thus for F > 0 there are essentially three different types of orbits, ones that born at infinity and tend to a finite point of the phase space, others that born at a finite point of the phase space and end at infinity, and finally orbits that born and end in two distinct finite points of the phase space. For more precise information about these kind of orbits see Theorem 2. If F = 0 then there are two types of orbits, the ones that born at infinity and tend to a finite point of the phase space, and the others that born at a finite point of the phase space and end at infinity. For more details on these orbits see Teorem 3. Acknowledgements The first author is supported by the Ministerio de Economı́a, Industria y Com- petitividad, Agencia Estatal de Investigación grant PID2019-104658GB-I00, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2019. 10 J. LLIBRE AND C. VALLS References [1] K. Bolejko, Structure fromation in the quasispherical Szekeres model, Phys. Rev. D 73 (2016), 123508. [2] K. Bolejko, A. Krasiński, C. Hellaby and M.N. Célérier, Structures in the Universe by exact methods–formation, evolution, interactions, Cambridge University Press, 2010. [3] M. Bruni, S. 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Quantum Grav. 14 (1997), 1151–1162. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt