Lyapunov function for cosmological dynamical system

Transcription

1 Demonstr. Mth. 207; 50: 5 55 Demonstrtio Mthemtic Open Access Reserch Article Mrek Szydłowski* nd Adm Krwiec Lypunov function for cosmologicl dynmicl system DOI 0.55/dem Received My 8, 206; ccepted November, 206. Abstrct: We prove the symptotic globl stbility of the de Sitter solution in the Friedmnn-Robertson-Wlker conservtive nd dissiptive cosmology. In the proof we construct Lypunov function in n exct form nd estblish its reltionship with the first integrl of dynmicl system determining evolution of the flt Universe. Our result is tht de-sitter solution is symptoticlly stble solution for generl form of eution of stte p D p.; H /, where dependence on the Hubble function H mens tht the effect of dissiption re included. Keywords: Lypunov function, First integrl, Dynmicl system, Mthemticl cosmology MSC: 4D2, 8F05 Introduction We ssume cosmologicl model with topology R M where M is mximlly symmetric -spce; the metric of spcetime is Lorenzin. ; C; C; C/ nd ssumes the following form! ds 2 D dt 2 D.t/ 2 dr 2 4 kr C r 2 d 2 C r 2 sin 2 d 2 () 2 where.t; r; ; / re the pseudo-sphericl coordintes [],.t/ > 0 is the function of the time coordinte which is clled the scle fctor, nd k D ; 0; is the spcil curvture. The -spce of constnt curvture is sptilly open for k D, sptilly closed for k D nd sptilly flt for k D 0. The dynmics of metric tensor g W ds 2 D g dx dx is determined from the Einstein field eution R 2 Rg C g D T (2) where R is the Ricci tensor, R D g R, (x ; x 2 ; x ; x 4 / D.t; r; ; ), is the Ricci sclr; we use nturl units such tht 8G D c D. Mtter is ssumed to be in the form of perfect fluid T D pg C. C p/u u () where u D. ; 0; 0; 0/ denotes the four-velocity of n observer comoving with the fluid. The functions p.t/,.t/ re pressure nd energy density of the mtter fluid, respectively. *Corresponding Author: Mrek Szydłowski: Astronomicl Observtory, Jgiellonin University, Orl 7, Krków, Polnd, E-mil: mrek.szydlowski@uj.edu.pl Adm Krwiec: Institute of Economics, Finnce nd Mngement, Jgiellonin University, Łojsiewicz 4, 0-48 Krków, Polnd, E-mil: dm.krwiec@uj.edu.pl 207 Szydłowski nd Krwiec, published by De Gruyter Open. This work is licensed under the Cretive Commons Attribution-NonCommercil-NoDerivs.0 License. Downlod Dte /24/8 6:7 AM

2 52 M. Szydłowski, A. Krwiec Tensor energy-momentum cn be generlised for non-perfect, viscous fluid [2], which for metric () ssumes the form ( p P T 0 0 D ; T i k D for i D k (4) 0 for i k where i; k D f; 2; g, is the viscosity coefficient nd H P is the Hubble prmeter. Formlly inclusion of viscous fluid is euivlent to replce pressure p by p P in the energy-momentum tensor. Such cosmologicl model with the imperfect fluid ( D const) hs been considered since Heller et l. [] nd Belinskii et l. [4]. The Einstein field eution (2) for this model reduces to [5] D C k C P2 (5) 2 2 p D 2 R P 2 k (6) 2 2 where is the cosmologicl constnt prmeter. The dependence of pressure p on H D P mens tht we considered viscous effects with the viscosity Eutions (5)-(6) cn be rewritten to the form of three-dimensionl utonomous dynmicl system HP D H 2 6. C p/ C (7) P D H. C p/ (8) P D H (9) where p D p.h; / is the generl form of ssumed eution of stte. The dynmicl system (7)-(9) is nlyzed in the present pper. Let us study the symptotic stbility of the solution (H 0 D ; 0 D 0) (future de Sitter solution). Definition.. A criticl point x 0 of the system Px D f.x/, is (Lypunov) stble point if for ll neighbourhoods U of x 0 there exists neighbourhood U of x 0 such tht if x 0 2 U t t D t 0 then t.x 0 / 2 U for ll t > t 0, where t is the flow of dynmicl system. If the criticl point x 0 is stble for ll x 2 U, lim t! jj t.x x 0 jj D 0. To determine the symptotic stbility of solution of the dynmicl system considered we construct the Lypunov function [6]. 2 Lypunov function Let us consider shortly some bsic definitions of first integrls [7]. Let M K n be n open subset in K n where the field K is R or C. We denote by X.M / nd F.M / the lgebr of vector fields nd functions on M, respectively. For simplicity, we ssume tht ll objects re of the clss C. Let us consider system of ordinry differentil eutions on M dx dt where the vector field X F D X.M / is given by D X F.x/ D F.x/; x D.x ; : : : ; x n / 2 M K n (0) X F D nx id i D n X id f i.x/@ i D f i.x/@ i () where.f.x/; : : : ; f n.x// re components of the vector field X F. We re looking for solution or clss of solutions of system (0). This is the motivtion of the following definition. Downlod Dte /24/8 6:7 AM

3 Lypunov function for cosmologicl dynmicl system 5 Definition 2.. We sy tht subset W M is invrint with respect to system (0) if W consists only of the system s phse curves. It seems tht it is extremely difficult to check if given set W is invrint with respect to (0) becuse, in generl, we do not know its solutions. However, for checking the invrince, it is enough to know if, for ll x 2 W, the vector of phse velocity in this point is tngent to M, i.e. if F.x/ 2 T x W for ll x 2 W. The most importnt invrint sets re those llowing to reduce the dimension of the system. For this purpose one invrint set is not enough, we need one prmeter fmily W c of.n /-dimensionl invrint sets tht gives folition of M. Such folition rises nturlly when we know first integrl. Definition 2.2. Function G 2 F.M / is clled the first integrl of system (0) if it is constnt on ll solutions of the system. It is euivlent to the condition X F.G/.x/ i G.x/f i.x/ D 0; for x 2 M: (2) It is well known tht level W c D fx 2 M jg.x/ D cg of first integrl is invrint, nd c! W c gives the folition mentioned erlier. When we cnnot find first integrl of the system, it is sometimes possible to find function whose one level is invrint. Definition 2. (Lypunov function). Let Px D f.x/ with x 2 X R n be smooth utonomous system of eutions with fixed point x 0. Let V W R n! R be continuous function in neighbourhood U of x 0, the V is clled Lypunov function for the point x 0 if. V is differentible in U n fx 0 g 2. V.x/ > V.x 0 /. PV 0 8x 2 U n fx 0 g. Theorem 2.4 (Lypunov stbility). Let x 0 be criticl point of the system Px D f.x/, nd let U be domin contining x 0. If there exists Lypunov function V.x/ for which PV 0, then x 0 is stble fixed point. If there exists Lypunov function V.x/ for which PV < 0, then x 0 is symptoticlly stble fixed point. Furthermore, if jjxjj! nd V.x/! for ll x, then x 0 is sid to be globlly stble or globlly symptoticlly stble, respectively. Let us return to our system (7)-(9). Theorem 2.5. The system (7)-(9) hs first integrl in the form H 2 C D k 2 : () Proof. After differentition of both sides of./ over time t nd substitution of right-hnd sides of (7)-(9) we obtin the form () of the first integrl. In the three-dimensionl phse spce the first integrl () defines surfces for different vlues of the prmeter. The dimension of the dynmicl system (7)-(9) cn be lowered over one due to this first integrl HP D H 2 6. C p/ C (4) P D H. C p/ (5) where p D p.h; / in generlly. For the de Sitter fixed point of (4)-(5), we hve p D, from eution (5). Then from eution (4) nd using the first integrl () we obtin tht k D 0. It mens tht fixed point is n intersection of the trjectory of the flt model nd the line C p.h; / D 0 in the phse spce.h; /. Downlod Dte /24/8 6:7 AM

4 54 M. Szydłowski, A. Krwiec Theorem 2.6. The de Sitter solution H 0 D is symptoticlly stble for H 0 > 0 nd symptoticlly unstble for H 0 < 0. Proof. Let us propose the following Lypunov function V.H; / ( H 2 C for k D 0; H 2 C for k D (6) which cn be obtined from () by putting k D 0. The surfce f.h; /W H 2 C D 0g divides the phse spce into two domins occupied by the trjectories with k D nd k D, respectively. Let us consider the first cse of non-negtive Lypunov function V.t/ for k D 0; in (6). PV.t/ D P 6H HP D H. C p/ 6H H 2 6. C p/ C D H C p C 2 H 2 6. C p/ C (7) D 2H H 2 C D 2H k 0; if H > 0: 2 Anlogously, we choose the second cse of Lypunov function V.t/ for k D in (6) to hve the function V.t/ to be non-negtive. Finlly, we obtin tht t both criticl points (H D ; 0 D 0) the Lypunov function (6) vnishes. So, the conditions of Lypunov stbility Theorem 2.4 re stisfied. We conclude tht while the stble de Sitter solution is symptoticlly stble, the unstble de Sitter solution is unstble. This result ws obtined by using globl methods of dynmics investigtions insted of the stndrd locl stbility nlysis. The choice of the Lypunov function in the form of first integrl is suitble for proving the symptotic stbility of the stble de Sitter solution of the model. This methodologicl result hs lso the cler cosmologicl interprettion: the stble de Sitter universe hs no hir like blck hole. Dynmics of stndrd cosmologicl model Let us consider homogeneous nd isotropic Friedmnn-Robertson-Wlker model with metric () nd mtter with energy density nd the cosmologicl constnt. We choose two phse spce vribles: the Hubble prmeter H D x nd energy density D y nd define the dynmicl system Px D x 2 6 y C (8) Py D xy (9) where the dot denotes derivtive with respect to time t nd > 0 is the cosmologicl constnt. It is specil cse of system (4)-(5). Let us pply the locl stbility nlysis for system (8)-(9). Remrk.. System (8)-(9) hs three criticl points: stble node (x D ; y D 0), nd sddle (x D 0; y D 2). ; y D 0), unstble node (x D From the chrcteristic eution det.a I / D 0, where the lineriztion mtrix of system (8)-(9) is " # 2x 0 A D 6 y 0 x 0 (20) Downlod Dte /24/8 6:7 AM

5 Lypunov function for cosmologicl dynmicl system 55 we hve tht the determinnt nd the trce of the lineriztion mtrix A nd the discriminnt of the chrcteristic eution re det A D 6x0 2 2 y 0, tr A D 5x 0 nd D x0 2 C 2y 0, respectively, where.x 0 ; y 0 / is criticl point. Therefore,. the criticl point (x 0 D ; y p 0 D 0) is n unstble node s det A D 2 > 0, tr A D 5 > 0, nd D > 0; 2. the criticl point (x 0 D ; y p 0 D 0) is stble node s det A D < 0, tr A D 5 < 0, nd D > 0;. the criticl point (x 0 D 0; y 0 D 2) is sddle s det A D 2 > 0, tr A D 0, nd D 4 > 0. The phse portrit of system (8)-(9) is presented in Figure. Fig.. The phse portrit of system (8)-(9). There three criticl points: point A represents the unstble de Sitter universe, point B represents the stble de Sitter universe, nd point C represents the Einstein-de Sitter universe. The red nd blue trjectories lie on unstble nd stble invrint mnifolds, respectively. It is ssumed is positive (for illustrtion D ). y C A B x Acknowledgement: The work ws supported by grnt NCN DEC-20/09/B/ST2/0455. References [] Wld R. M., Generl reltivity, The University of Chicgo Press, Chicgo, 984 [2] Weinberg S., Entropy genertion nd the survivl of protoglxies in n expnding universe, Astrophysicl Journl, 97, 68, [] Heller M., Klimek Z., Suszycki L., Imperfect fluid Friedmnnin cosmology, Astrophysics nd Spce Science, 97, 20, [4] Belinskii V. A., Khltnikov I. M., Influence of viscosity on the chrcter of cosmologicl evolution, Sov. Phys. JETP, 975, 42, [5] Szydłowski M., Heller M., Gold Z., Structurl stbility properties of Friedmn cosmology Generl Reltivity nd Grvittion, 984, 6, [6] Perko L., Differentil eutions nd dynmicl systems, rd ed., Springer, New York, 200 [7] Goriely A., Integrbility nd nonintegrbility of dynmicl systems, World Scientific, Singpore, 200 Downlod Dte /24/8 6:7 AM