Riemann geometry in theory of the first order systems of equations

1 RIEMANN GEOMETR Y IN THEOR Y OF THE FIRST ORDER S Y STEMS OF EQUA TIONS V alery Dryuma Institute of Mathematics and Informatics AS M oldov a, Kishinev E-mail: v alery@dryuma.com; cainar@mail.md Abstract Theory of Riemann extensions of the spaces with constant affine connection is prop osed to study of the pro per ties of nonlinea r the fir s t or der systems of d ifferential equations. As exa mple quadra tic system of differ en tial equations dx ds = P ( x, y ) , dy ds = Q ( x, y ) , (1) where P ( x, y ) = a 0 + a 1 x + a 2 y + a 11 x 2 + a 12 xy + a 22 y 2 , Q ( x, y ) = b 0 + b 1 x + b 2 y + b 11 x 2 + b 12 xy + b 22 y 2 , and a , b are the parameters, is pr e s ent ed in homogeneo us form a nd is co nsidered as geo desic of three- dimensional space with co nstant affine connection dep ending on the par ameters a, b . After the Riemann extension o ne get a six-dimensional space a nd its prop erties in r e la tion to the pa rameters ar e inv esti- gated. The Lo renz system of equations dx ds = σ ( y − x ) , dy ds = rx − y − xz , dz ds = xy − bz after pr e s ent atio n in homogeneous form is co ns idered as ge o desic equations of four dimensio nal space with consta n t affine connection. Based on the eight -dimens io nal Riemann extensio n of a giv en type space the pr op erties of the Lorenz system are studied. 1 In tro duction The sub ject o f cons ider ation is the first or der p olynomial systems of different ial equations dx i ds = c i + a i j x j + b i j k x j x k (2) depe nding on the par a meters a, b, c . They play an impor tant ro le in v ario us branches of mo dern mathema tics and its applica tions. In par ticular case of the s y stem of t wo equations dx ds = a 0 + a 1 x + a 2 y + a 1 1 x 2 + a 12 xy + a 22 y 2 , dy ds = b 0 + b 1 x + b 2 y + b 11 x 2 + b 12 xy + b 22 y 2 2 FR OM THE FIRST ORDE R SYSTEM OF EQUA TIONS TO THE SECOND ORDER SYSTEMS OF ODE 2 there ar e many uns o lved problems. The spa tial first o rder system of differential equatio ns dx ds = P ( x, y , z ) , dy ds = Q ( x, y , z ) , dz ds = R ( x, y , z ) (3) with the functions P , Q, R polynomia l on v ariables x, y , z are still mor e complicated ob ject for the studying of their pro per ties. As exa mple the s tudying of the Lor enz system o f equations dx ds = σ ( y − x ) , dy ds = rx − y − xz , dz ds = xy − bz (4) and the R¨ ossler sys tem dx ds = − y − z , dy ds = x + ay , dz ds = bx − cz + xz (5) which ar e the simplest exa mples of the spatia l systems have chaotic b ehaviour a t some v alues of pa - rameters repr esent the difficult tas k. 2 F rom the first ord er system of equations to the second order systems of O DE The systems o f the first order differential equations are not suitable ob ject of consideratio n fro m the usually p oint of Riemann geometr y . The sy stems of the second o rder differential equations in for m d 2 x i ds 2 + Π i kj ( x ) dx k ds dx j ds = 0 (6) are b est s uited to do that. They c an b e co nsidered as geo de s ics of the affinely connected space M k in lo cal co ordinates x k . The v alues Π i j k = Π i kj are the co efficients of affine connections o n M k . With the help of such co efficients can b e c o nstructed curv ature tens or and others geo metrical ob jects defined on v ariet y M k . There ar e many po ssibilities to present a given system of the fir st order of equatio ns in the form o f (6). One of them is a fo llowing naive presentation. F or the system dx ds = P ( x, y ) , dy ds = Q ( x, y ) (7) after differ e ntiation with resp ect the parameter ( s ) w e get the second or der system of differential equa - tions of the for m (6) d 2 x ds 2 = 1 P ( P x dx ds + P y dy ds ) dx ds , d 2 y ds 2 = 1 Q ( Q x dx ds + Q y dy ds ) dy ds . Such t yp e of the system contains the integral curves of the sy stem (7 ) as pa rt of its solutions and can be considered as the equations of geo desics of tw o dimensiona l spa ce M 2 ( x, y ) equipment b y affine connections with co efficients Π 1 11 = − P x P , Π 1 12 = − P y 2 P , Π 2 12 = − Q x 2 Q , Π 2 22 = − Q y Q . (8) 3 FR OM THE AFFINEL Y CONNECTED SP A CE TO THE RIEMANN SP ACE 3 It is appar en t that the properties of the system (7 ) hav e an influence on geometr y of the v a riety M 2 ( x, y ). Remark that a g iven sys tem is e quiv alen t the second o rder differential equation d 2 y dx 2 = ln( Q/P ) y  dy dx  2 + ln( Q/P ) x dy dx which has the so lution in fo rm dy dx = Q P . By analo gy the spa tial system of the first or der differential equations c an b e wr itten. As exa mple the L o renz sys tem of equations is e quiv alen t the second o rder ODE d 2 y dx 2 − 3 y  dy dx  2 +  αy − 1 x  dy dx + ǫxy 4 − γ y 3 − β x 3 y 4 − β x 2 y 3 + δ y 2 x = 0 , where α = b + σ + 1 σ , β = 1 σ 2 , γ = b ( σ + 1) σ 2 , δ = σ + 1 σ , ǫ = b ( r − 1) σ 2 , which can be obtained by the elimination of v ariable z from the system dy dx = rx − y − xz σ ( y − x ) , dz dx = xy − bz σ ( y − x ) . The se c ond order O DE’s d 2 y dx 2 + a 1 ( x, y )  dy dx  3 + 3 a 2 ( x, y )  dy dx  2 + 3 a 3 ( x, y ) dy dx + a 4 ( x, y ) = 0 , with ar bitrary co efficients a i ( x, y ) ar e form-inv arian t under the change of the co ordinates x = f ( u, v ) , y = h ( u , v ) and are equiv alent to the system d 2 x ds 2 − a 3 ( x, y )  dx ds  2 − 2 a 2 ( x, y )  dx ds   dy ds  − a 1 ( x, y )  dy ds  2 = 0 , d 2 y ds 2 + a 4 ( x, y )  dx ds  2 + 2 a 3 ( x, y )  dx ds   dy ds  + a 2 ( x, y )  dy ds  2 = 0 , having the form of geo desics of tw o dimensiona l affinely co nnected space ( with this aim the formulae d 2 y dx 2 = ˙ x ¨ y − ˙ y ¨ x ( ˙ x ) 3 ) was used. 3 F rom the affinely connected space to the Riemann sp ace Now we shall construct the Riemann space star ting from a g iven affinely connected s pace defined by the seco nd order O DE’s. With this aim we use the notion of the Riemann extensio n of nonrie ma nnian space which was used earlier in the a rticles of author. Remind the basic pr o pe r ties of this constructio n. 4 RIGOROUS APP R OA CH TO GEOMETR Y OF PLANAR SYSTEMS 4 With help of the co efficients o f affine connection o f a given n-dimensional space can be in tro duced 2n-dimensional Riemann space D 2 n in lo cal co o rdinates ( x i , Ψ i ) having the metric of form 2 n ds 2 = − 2 Π k ij ( x l )Ψ k dx i dx j + 2 d Ψ k dx k (9) where Ψ k are an additional co o rdinates. The imp ortant prop erty o f such type metric is that the geo desic equations o f metric (9) decomp oses int o t wo parts ¨ x k + Π k ij ˙ x i ˙ x j = 0 , (10) and δ 2 Ψ k ds 2 + R l kj i ˙ x j ˙ x i Ψ l = 0 , (11) where δ Ψ k ds = d Ψ k ds − Π l j k Ψ l dx j ds and R l kj i are the curv ature tensor of n-dimensio nal space with a given affine connection. The firs t part (10) of the full s ystem is the system of eq ua tions for geo desic of basic space with lo cal co ordinates x i and it do not contains the supplementary co ordinates Ψ k . The seco nd par t (11) of the system ha s the form of linear N × N matrix system of second order ODE’s for supplementary co o rdinates Ψ k d 2 ~ Ψ ds 2 + A ( s ) d ~ Ψ ds + B ( s ) ~ Ψ = 0 . (12) Remark that the full system o f geo des ics has the first in tegr a l − 2Π k ij ( x l )Ψ k dx i ds dx j ds + 2 d Ψ k ds dx k ds = ν (13) which is equiv alent to the rela tion 2Ψ k dx k ds = ν s + µ (14) where µ, ν are parameters . The geometry of extended space connects w ith geometry of basic space. F o r e xample the prop erty of the space to be Ricci-flat R ij = 0 or symmetrical R ij kl ; m = 0 keeps also for the extended space. It is imp ortant to note that for e xtended space having the metric (9) all s calar curv ature inv aria nts are v anished. As consequence the prop erties of linea r system of equatio n (11-12) dep ending fro m the the inv a riants of N × N matrix-function E = B − 1 2 dA ds − 1 4 A 2 under change of the co ordinates Ψ k can be of used for tha t. First applica tions the notion of extended s paces for the studying of no nlinea r second order ODE ’s connected with nonlinear dy namical systems hav e b e en cons idered by author (V.Dryuma 200 0-2008 ). 4 Rigorous approac h to geometry of planar sys tems The system of paths of tw o-dimensio nal space S 2 in ge ne r al form lo ok s as ¨ x + Π 1 11 ( ˙ x ) 2 + 2Π 1 12 ˙ x ˙ y + Π 1 22 ( ˙ y ) 2 = 0 , ¨ y + Π 2 11 ( ˙ x ) 2 + 2Π 2 12 ˙ x ˙ y + Π 2 22 ( ˙ y ) 2 = 0 , where the co efficients Π k ij = Π k j i . 4 RIGOROUS APP R OA CH TO GEOMETR Y OF PLANAR SYSTEMS 5 The Riema nn extension of the s pace S 2 is determined by the metric 4 ds 2 = − 2 z Π 1 11 dx 2 − 2 t Π 2 11 dx 2 − 4 z Π 1 12 dxdy − 4 t Π 2 12 dxdy − − 2 z Π 1 22 dy 2 − 2 t Π 2 22 dy 2 + 2 dxdz + 2 dy dt. W e shall apply such s ystem of e quations for the studying of the first order planar system of ODE’s. With this aim we use the facts ab out the geometrie s of paths fo r which the equations o f the pa ths admits indep endent linear first integrals. L.P .E i senhar t, 1 925 A necess ary condition to geo des ic admit the linear fir s t integral a i ( x, y ) dx i ds = const is a i ; j + a j ; i = 0 , where a i ; j = ∂ a i ∂ x j − a k Γ k ij , and Γ k ij are the Christoffel s ymbols of the metric. W e a pply this conditions and their co ns equence a i ; j ; k + R m kij a m = 0 where R i j kl is the curv ature tensor of the s pace to determination o f the co efficients o f equations Π k ij using the vector a i in for m a i = [ Q ( x, y ) , − P ( x, y ) , 0 , 0] . By this means that the fir st or der of equatio n dy dx = Q ( x, y ) P ( x, y ) or Q ( x, y ) dx − P ( x, y ) dy = 0 is an integral o f the paths equatio ns . With the help of these co nditions it is p ossible to state only three co e fficien ts of affine connectio ns Π k ij . F or deter mination of others co e fficien ts w e us e yet another the first order equation dy dx = − y ( y − 1) x ( x − 1) , with a first integral y ( x ) = C ( x − 1) x − C . In this case a second v ecto r b i is in form b i = [ y ( y − 1 ) , x ( x − 1) , 0 , 0] . The eq ua tion plays a n imp ortant role in theor y of of planar the first order system of equa tions and was used in a famous a rticle of Petrovsky-Landis (1956) ([ ? ]). 5 THE SE COND ORDER ODE ’S CUBIC O N THE FIRST DERIV A TIVE 6 So, from the co nditions on the metrics to admits tw o linear first integrals the co efficients Π k ij of the paths equatio n are uniquely determined and have the fo r m Π 1 11 =  ∂ ∂ x Q ( x, y )  x ( x − 1) y 2 P ( x, y ) − y P ( x, y ) + Q ( x, y ) x 2 − Q ( x, y ) x Π 1 22 = −  ∂ ∂ y P ( x, y )  x ( x − 1) y 2 P ( x, y ) − y P ( x, y ) + Q ( x, y ) x 2 − Q ( x, y ) x and cor r esp onding expressio ns for the co efficients Π 2 11 , Π 2 12 , Π 2 22 Remark that last tw o equations a re reduced at the independent equations d 2 z ds 2 + M ( s ) dz ds + N ( s ) z ( s ) + F ( s ) = 0 and d 2 t ds 2 + U ( s ) dt ds + V ( s ) t ( s ) + H ( s ) = 0 with the help o f the firs t integral of geo desic s z ( s ) dx ds + t ( s ) dy ds − α s 2 − β = 0 of the metric (18 ). It is in tere sted to note that such t yp e of non homogeneous linear second or der ODE’s a re connected with theory of fir st or der systems of O DE’s a s the eq uations on the p erio ds of c orresp onding Abel int egr als. 5 The second order ODE’s cub ic on the fi rst deriv ativ e The first tw o equations of geode s ic of the metric are equiv alent to the one seco nd order differen tial equation d 2 dx 2 y ( x ) +  ∂ ∂ y P ( x, y )  x 2 −  ∂ ∂ y P ( x, y )  x   d dx y ( x )  3 y 2 P ( x, y ) − y P ( x, y ) + Q ( x, y ) x 2 − Q ( x, y ) x + +  ( P x − Q y ) x 2 + ( Q y − P x − 2 P ) x + ( P y ) y 2 + ( − 2 P − P y ) y + 2 P   d dx y  2 y 2 P − y P + Qx 2 − Qx + +  − ( Q x ) x 2 + ( Q x + 2 Q ) x + ( P x − Q y ) y 2 + (2 Q − P x + Q y ) y − 2 Q  d dx y y 2 P − y P + Qx 2 − Qx + + −  ∂ ∂ x Q ( x, y )  y 2 +  ∂ ∂ x Q ( x, y )  y y 2 P ( x, y ) − y P ( x, y ) + Q ( x, y ) x 2 − Q ( x, y ) x = 0 . (14) The eq ua tion (14) is of the form d 2 y dx 2 + a 1 ( x, y )  dy dx  3 + 3 a 2 ( x, y )  dy dx  2 + 3 a 3 ( x, y ) dy dx + a 4 ( x, y ) = 0 , and it has the in v ariants dep ending from the co efficients a i ( x, y ) (R.Liouville, 1880, T.T ress e , 1886) under transfo rmations of v ariables ( x, y ). As it was shown by (E.Cartan, 1924) these inv ariant s in gener al case are sa me with the in v ari- ants of tw o-dimensio nal surface V 2 ( x, y ) in a four-dimens io nal pro jective space RP 4 ( ξ i ) (under the repara metr ization a nd the change of co ordinates ξ i ). 5 THE SE COND ORDER ODE ’S CUBIC O N THE FIRST DERIV A TIVE 7 F or example the in v ariants of e q uations with condition o n co efficients ν 5 = 0 same with the inv ar iants of developing surfaces in a thre e -dimensional pr o jective space R P 3 . The inv a riants o f R.Liouville hav e b een used succes sful in theory of ODE’s a nd their applications in works author (V.Dryuma, 19 84-). In particula r the second or der ODE with a Painleve pro per t y have the conditio n ν 5 = 0. In our case the equation has the particular integral dy dx = Q ( x, y ) P ( x, y ) and the function y ( x ) = C ( x − 1) x − C . as the first integral. F rom g eometrical p oint o f view the equation (14) co rresp onds tw o-dimensio na l surfa c e in the RP 4 - space. In this context it is interested to note the relatio n with the Petrovsky-Landis theor y of limit c ycles of the equation dy dx = a 0 + a 1 x + a 2 y + a 11 x 2 + a 12 xy + a 22 y 2 b 0 + b 1 x + b 2 y + b 11 x 2 + b 12 xy + b 22 y 2 . In the famous ar ticle of P − L was dev elo ped approa ch to the studying of the pr oblem of the limit cycles of the first order q uadratic equation. Let us recall the basic facts of the Petrovsky-Landis theory . F or s tudying of the closed curves of quadra tic the first order eq ua tion is considered the equation dy dx = − y ( y − 1) x ( x − 1) with so lutio n y ( x ) = C ( x − 1) ( x − C ) . As it was showed for the clo sed curves the par ameter C satisfies the a lg ebraic equations X a n ( µ i ) C n = 0 where the co efficients a n ( µ i ) ar e dependent from the parameters of the quadra tic equation. Such t yp e of equation arises from the c o ndition Z c ( x − C ) 2 [ x ( x − 1) P ( x, y ) + y ( y − 1) Q ( x, y )] x 3 ( x − 1) 3 dx = 0 , which lead to deter mination o f the clo sed integral curves of the fir st or der system of equations dy dx = a 0 + a 1 x + a 2 y + a 11 x 2 + a 12 xy + a 22 y 2 b 0 + b 1 x + b 2 y + b 11 x 2 + b 12 xy + b 22 y 2 . As it was shown by Petrovsky-Landis in this case the para meter C is determined from ca lc ula tions of the residues of integral after substitution of the express ion y = C ( x − 1) ( x − C ) in it. The simples t of them are 5 THE SE COND ORDER ODE ’S CUBIC O N THE FIRST DERIV A TIVE 8 ( b2 + 2 b0 + a12 + a1 + a2 + 2 a0 + b12 + b1 ) C 2 + + ( − 2 a0 − b12 − a2 − b1 + 2 b22 − 2 b0 ) C − 2 b22 − b2 = 0 , x = 0 , ( − b2 − 2 b0 − a12 − a1 − a2 − 2 a0 − b12 − b1 ) C 2 + + ( a2 + b1 + a12 + 2 b0 + 2 a0 − 2 a11 ) C + 2 a11 + a1 = 0 , x = 1 , ( b12 − 2 b22 ) C + b2 + 2 b22 = 0 , x = C. ( − a12 + 2 a11 ) C − 2 a11 − a1 = 0 , x = ∞ . Others eq ua tions on C fro m the co m binations of these conditions a re follow ed. According the results of ( P − L ) g eneral qua n tity of the v alues C defined by a such type of equations is equal 1 4 a nd this num b er coincides with the quantit y of clo sed solutions determined b y the quadratic equation. On o ther side 11 curves fro m 14 can be transformed into the small neighbor ho o d of the essen tial singular p oints of the equation dy dx = − y ( y − 1) x ( x − 1) . They a re: (0 , 1 ), (1 , 0), (0 , 0), (1 , 1 ) and so on... As result only three closed curves do no t be transfor med in to the neighbo rho o d o f the singular p oints and so the quantit y o f the limit cycles defined by the e quation is equal thr ee. It is interested to note that a some conditions on the para meter C are a pp ear ed in context of the second or der ODE. In fact the result of joint consider ation of the first order equations y ′ = Q ( x, y ) P ( x, y ) , y ′ = y ( y − 1) x ( x − 1) the function y ( x ) = C ( x − 1) ( x − C ) and the second or der ODE (13 ) we get the conditions α ( x, y ) C 5 + β ( x, y ) C 4 + γ ( x, y ) C 3 + δ ( x, y ) C 2 + ǫ ( x, y ) C + µ ( x, y ) = 0 , where α ( x, y ) = ( a12 + b12 ) y 2 + + ( b1 + b2 + (2 a22 + 2 a11 + 2 b22 + 2 b11 ) x + a2 + a1 ) y + + ( a12 + b12 ) x 2 + ( a1 + b2 + b1 + a2 ) x + 2 a0 + 2 b0 , β ( x, y ) = 2 b22 y 3 + (( − 5 a12 − b12 − 2 b22 ) x + 2 b2 − b12 ) y 2 + + (( − 4 a22 − 3 b2 − 5 a1 − 3 a2 − 2 b11 − 6 b22 − b1 ) x − a2 ) + +  ( − 4 b11 + a12 − 10 a11 − b1 2 − 4 a22 ) x 2 + 2 b0 − 2 b2 − b1  y + +( − 2 a12 + 2 a11 ) x 3 + ( − 2 a12 − 2 b1 − 2 a1 − 3 b12 − 2 a2 ) x 2 − 4 b0 + +( − 6 a0 − 2 b1 − 4 b0 − 3 b2 − a1 − 2 a2 ) x − 2 a0 , µ ( x, y ) = − y 2 x 5 a12 +  − 2 x 6 a11 + ( a12 − a1 ) x 5  y + 2 x 6 a11 + x 5 a1 . 6 THREE DIMENSIONAL HOMOGENEO US SYSTEM 9 F rom these rela tio n we g et the r elations b etw een the parameter C and c o e fficien ts a ij , b ij , a i , b i As exa mple at the v alues x = 0 and x = 1 we g et conditions o n the v alue C ( a2 + a12 + b1 + b12 + b2 + 2 a0 + a1 + 2 b0 ) C 2 + + ( − 2 b0 − 2 a0 − b1 − a2 − b12 + 2 b22 ) C − b2 − 2 b22 = 0 , ( a2 + a12 + b1 + b12 + b2 + 2 a0 + a1 + 2 b0 ) C 2 + + (2 a11 − 2 b0 − b1 − 2 a0 − a12 − a2 ) C − 2 a11 − a1 = 0 . The subs titution ( x = C , y = 1 − C ) lead to the conditions on the v alue C ( b12 − 2 b22 ) C + 2 b22 + b2 = 0 . After substitution ( y = 1 − x, x = 1 , C = 1 /C 1) we get ( a1 + 2 a11 ) C1 2 + ( a12 − 2 a11 + b1 + a2 + 2 a0 + 2 b0 ) C1 − − a1 − a2 − a12 − 2 b0 − b12 − b1 − b2 − 2 a0 = 0 , and the substitution ( y = 1 − x, x = 0 , C = 1 /C 1) lea d to the condition ( − 2 b22 − b2 ) C1 2 + ( − 2 b0 − b12 − b1 − a2 − 2 a0 + 2 b22 ) C1 + + a1 + a2 + a12 + 2 b0 + b12 + b1 + b2 + 2 a0 = 0 . In result of a such t ype consideration we ha ve got the conditions of Petrovskii-Landis ar ticle on parameter C . 6 Three dimensional h omogeneous system As it was shown in article (V.Dryuma,2 0 06) b et ween the pla nar sys tem dx dt = a 0 + a 1 x + a 2 y + a 11 x 2 + a 12 xy + a 22 y 2 , dy dt = b 0 + b 1 x + b 2 y + b 11 x 2 + b 12 xy + b 22 y 2 , and a spatial ho mogeneous quadr atic system of equations dx dt = P ( x, y , z ) , dy dt = Q ( x, y , z ) , dz dt = Q ( x, y , z ) of the form dx dt = 4 a 0 z 2 + 4 a 2 y z + (3 a 1 − b 2 ) xz + 4 a 22 y 2 + (3 a 12 − 2 b 22 ) xy + (2 a 11 − b 12 ) x 2 , dy dt = 4 b 0 z 2 + 4 b 1 xz + (3 b 2 − a 1 ) y z + 4 b 11 x 2 + (3 b 12 − 2 a 11 ) xy + (2 b 22 − a 12 ) y 2 , dz dt = − ( a 1 + b 2 ) z 2 − (2 b 22 + a 12 ) y z − (2 a 11 + b 12 ) xz exists so me connections. F or a such system the condition ∂ P ( x, y , z ) ∂ x + ∂ Q ( x, y , z ) ∂ y + ∂ R ( x, y , z ) ∂ z = 0 7 SIX-DIMENSIONAL RIEMANN SP ACE 10 is fulfilled and in v ariables ξ ( t ) = x ( t ) z ( t ) , η ( t ) = y ( t ) z ( t ) it takes the for m of the equation dξ dη = a 0 + a 1 ξ + a 2 η + a 11 ξ 2 + a 12 ξ η + a 22 η 2 b 0 + b 1 ξ + b 2 η + b 11 ξ 2 + b 12 ξ η + b 22 η 2 equiv alen t the planar sy stem . It is significant tha t the spatial sys tem in the v ariables x ( t ) = dX ( t ) dt , y ( t ) = d Y ( t ) dt , z ( t ) = d Z ( t ) dt takes the form of geo desic equations of a three dimensional s pace d 2 X i dt 2 + Γ i j k dX j dt dX k dt = 0 (15) in lo cal co ordinates X i = ( X ( t ) , Y ( t ) , Z ( t ). In doing so the co efficient Γ i j k are constant and dep end on the parameter s a i , a ij , b i , b ij . The set of such co efficients can b e considere d as the co efficients of affine connection o n a three dimensional spa ce H 3 in lo ca l co ordinates X i . F rom geometr ic al p oint of view throug h the equa tions (5) on the space H 3 the structure of affinely connected space with co nstant co efficients of c o nnection Γ i j k is determined. 7 Six-dimensional Riemann space F or the n -dimensional space equipp ed with affine connection Γ i j k the metrics of the Riemann extension V 2 n has the form 2 n ds 2 = − 2Γ i j k dX j dX k dξ i − 2 dξ i dX i (16) where χ i are a n additional co ordinates. Non zer o co efficients o f affine connections of the s pace H 3 are Γ 1 33 = − 4 a 0 , Γ 1 23 = − 2 a 2 , Γ 1 13 = 1 2 ( b 2 − 3 a 1 ) , Γ 1 22 = − 4 a 22 , Γ 1 12 = 1 2 (2 b 22 − 3 a 12 ) , Γ 1 11 = b 12 − 2 a 11 , Γ 2 33 = − 4 b 0 , Γ 2 13 = − 2 b 1 , Γ 2 23 = 1 2 ( a 1 − 3 b 2 ) , Γ 2 11 = − 4 b 11 , Γ 2 12 = 1 2 (2 a 11 − 3 b 12 ) , Γ 2 22 = a 12 − 2 b 22 , Γ 3 33 = a 1 + b 2 , Γ 3 23 = 1 2 ( a 12 + 2 b 22 ) , Γ 3 13 = 1 2 (2 a 11 + b 12 ) . According with the (16) the Riemann metric of s ix dimensional extended spa ce V 6 has the form 6 ds 2 = ( − 2 U b12 + 4 U a11 + 8 b11 V ) dx 2 + (8 a22 U − 2 V a12 + 4 V b22 ) dy 2 + + (8 b0 V − 2 W b2 + 8 a0 U − 2 W a1 ) dz 2 + ( − 4 U b22 − 4 V a11 + 6 V b12 + 6 U a12 ) dx dy + + (6 U a1 − 2 U b2 − 4 W a11 − 2 W b12 + 8 b1 V ) dx dz + + ( − 2 W a12 + 8 a2 U + 6 V b2 − 2 V a1 − 4 W b22 ) dy dz + 2 dx dU + 2 dy dV + 2 dz dW . 7 SIX-DIMENSIONAL RIEMANN SP ACE 11 Remark that we use the deno tes ( x, y , z ) for the co o rdinates ( X , Y , Z ). Geo desic of the metric consist from tw o par ts. Nonlinear system o f coupled equatio ns on co ordina tes ( x, y , z ) d 2 ds 2 x ( s ) + ( − 2 a11 + b12 )  d ds x ( s )  2 + 2 ( − 3 / 2 a12 + b22 )  d ds x ( s )  d ds y ( s )+ +2 ( − 3 / 2 a1 + 1 / 2 b2 )  d ds x ( s )  d ds z ( s ) − 4 a22  d ds y ( s )  2 − 4 a2  d ds y ( s )  d ds z ( s ) − 4 a0  d ds z ( s )  2 = 0 , d 2 ds 2 y ( s ) − 4 b11  d ds x ( s )  2 + 2 ( − 3 / 2 b12 + a11 )  d ds x ( s )  d ds y ( s ) − 4 b1  d ds x ( s )  d ds z ( s )+ + ( − 2 b22 + a12 )  d ds y ( s )  2 + 2 ( − 3 / 2 b2 + 1 / 2 a1 )  d ds y ( s )  d ds z ( s ) − 4 b0  d ds z ( s )  2 = 0 , d 2 ds 2 z ( s )+ 2 ( a11 + 1 / 2 b12 )  d ds x ( s )  d ds z ( s )+ 2 (1 / 2 a12 + b22 )  d ds y ( s )  d ds z ( s )+ ( a1 + b2 )  d ds z ( s )  2 = 0 . And the linear sy stem of equations on co ordinates ( U, V , W ) d 2 ds 2 U ( s ) + A1 d ds U ( s ) + B1 d ds V ( s ) + C1 d ds W ( s ) + E1 U ( s ) + F1 V ( s ) + H1 W ( s ) = 0 d 2 ds 2 V ( s ) + A2 d ds U ( s ) + B2 d ds V ( s ) + C2 d ds W ( s ) + E2 U ( s ) + F2 V ( s ) + H2 W ( s ) = 0 d 2 ds 2 W ( s ) + A3 d ds U ( s ) + B3 d ds V ( s ) + C3 d ds W ( s ) + E3 U ( s ) + F3 V ( s ) + H3 W ( s ) = 0 , where the co efficients ( Ai, B i , C i ) are depe nded from the pa r ameters ( a, b ) and der iv ativ es ( ˙ x, ˙ y , ˙ z ) with res pect to the parameter s . T aking in co nsideration x, y , z -equa tions the U , V , W - equations can b e one time in tegr ated and take the form d ds U ( s ) = (2 a11 + b12 ) z ( s ) W ( s )+ + (( − 3 b12 + 2 a11 ) y ( s ) − 8 b11 x ( s ) − 4 b1 z ( s )) V ( s )+ + ((2 b12 − 4 a11 ) x ( s ) + ( − 3 a12 + 2 b22 ) y ( s ) + ( b2 − 3 a1 ) z ( s )) U ( s ) , d ds V ( s ) = ( a12 + 2 b22 ) z ( s ) W ( s )+ + (( − 3 a12 + 2 b22 ) x ( s ) − 8 a22 y ( s ) − 4 a2 z ( s )) U ( s )+ + (( − 3 b12 + 2 a11 ) x ( s ) + (2 a12 − 4 b22 ) y ( s ) + ( − 3 b2 + a1 ) z ( s )) V ( s ) , d ds W ( s ) = (( b2 − 3 a1 ) x ( s ) − 8 a0 z ( s ) − 4 a2 y ( s )) U ( s )+ + (( − 3 b2 + a1 ) y ( s ) − 8 b0 z ( s ) − 4 b1 x ( s )) V ( s )+ + ((2 a11 + b12 ) x ( s ) + ( a12 + 2 b22 ) y ( s ) + 2 ( a1 + b2 ) z ( s )) W ( s ) . In r esult w e hav e got a six-dimensio nal Riemann space asso ciated with a qua dratic the first order system of equations. An inv estigation the pro pe r ties of the metric at the change of para meters a , b may b e useful for the theory of a such t yp e of the systems. In particular a study of the Killing pr op erties of the metric allow us to get information on par ticular int egr als of geo des ic equations. 8 ON THE SURF ACES DEFINED BY SP A TIAL SYSTEM OF EQUA TIONS 12 8 On the sur faces defined b y spatial system of equations The equation of surfaces z = z ( x, y ) defined by the spatial system o f eq ua tions (4) has the form of the first or der p.d.e z x  4 a0 ( z ) 2 + (4 a2 y + (3 a1 − b2 ) x ) z + 4 a22 y 2  + z x  (3 a12 − 2 b22 ) xy + (2 a11 − b12 ) x 2  + + z y  4 b 0 z 2 + (3 b2 − a1 ) y z + 4 b11 x 2 + 4 b1 xz  + z y  (2 b22 − a12 ) y 2 + ( − 2 a11 + 3 b12 ) xy  + + ( b2 + a1 ) ( z ) 2 + ((2 b22 + a12 ) y + (2 a11 + b12 ) x ) z . (17) An e xamples of solutions of this equation were obtained by the method of ( u, v )- tr ansformation developed ea r lier by a uthor. T o integrate the partial nonlinear first order differential equation F ( x, y , z ( x, y ) , z x , z y ) = 0 (18) we use a following change o f the functions a nd v ariables z ( x, y ) → u ( x, t ) , y → v ( x, t ) , z x → u x − v x v t u t , v y → u t v t . (19) In result instea d of the equation (18) one get the r elation b etw een the new v ariables u ( x, t ) and v ( x, t ) and their partial deriv atives Φ( u, v , u x , u t , v x , v t ) = 0 . (20) In many cases the s olution o f last equation is a more simple pro blem than so lution o f the equation (18). In result of a pplication of the ( u, v )-tra nsformation the equation for z = z ( x, y ) takes the form of (20). Using the substitution u ( x, t ) = t ∂ ∂ t ω ( x, t ) − ω ( x, t ) , v ( x, t ) = ∂ ∂ t ω ( x, t ) , where ω ( x, t ) = xA ( t ) , we find fr om a given rela tion the equation for the function A ( t )  − t 2 b2 − t b22 + a22 A ( t ) − t 3 b0 + a0 A ( t ) t 2 + a2 t A ( t )   d dt A ( t )  2 + +  − 2 a0 ( A ( t )) 2 t − t b12 − a2 ( A ( t )) 2 − t 2 b1 + t a1 A ( t )  d dt A ( t )+ +  2 t 2 b0 A ( t ) + t b2 A ( t ) + a12 A ( t )  d dt A ( t ) − − t b11 − t b0 ( A ( t )) 2 − a1 ( A ( t )) 2 + t b1 A ( t ) + a11 A ( t ) + a0 ( A ( t )) 3 = 0 . Solutions of this equa tion depend from the para meters a, b and can play a key v alue in theory of the qua dr atic systems. Another t yp e of substitution v ( x, t ) = t ∂ ∂ t ω ( x, t ) − ω ( x, t ) , u ( x, t ) = ∂ ∂ t ω ( x, t ) , where ω ( x, t ) = xB ( t ) , lead to the equation o n the function B ( t )  a22 B ( t ) t 2 + a2 B ( t ) t + a0 B ( t ) + b22 t 2 + t b2 + b0   d dt B ( t )  2 + 9 EIGHT-DIMENSIONAL RIEMANN SP ACE F OR THE LO RENZ SYSTEM OF EQUA TIO NS 13 +  t b12 − a2 ( B ( t )) 2 + b1 + a1 B ( t ) − 2 b22 tB ( t ) − 2 a22 ( B ( t )) 2 t + t a12 B ( t ) − b2 B ( t )  d dt B ( t ) − − a12 ( B ( t )) 2 − b12 B ( t ) + a11 B ( t ) + a22 ( B ( t )) 3 + b22 ( B ( t )) 2 + b11 = 0 (21) which is a mor e simple than previo us and also can b e useful in theory of quadra tic s ystems. Let us consider s ome examples. In the v ariables B ( t ) = S, dB ( t ) dt = T the equa tion (21) deter mines algebr a ic curve F ( S, T , t, a, b ) = 0 having genus g = 1 or g = 0 in dep ending on the parameter s a, b . As exa mple at the conditions b 0 = 0 , a 0 = 0 , b 2 = 0 , a 1 = 0 , a 2 = 0 , a 22 = 0 , b 11 = 0 , a 12 = 0 , a 1 1 = b 12 , b 1 = 0 the equa tion (21) takes the for m b22 t 2  d dt A ( t )  2 + ( − 2 b22 tA ( t ) + t b12 ) d dt A ( t ) + b22 ( A ( t )) 2 = 0 and defines algebra ic curve of the ge nus g = 0 . Int egr al of this equation is defined b y the relation ln( t ) + ln( q − 4 b22 B ( t ) b12 + b12 2 − b12 ) − b12 q − 4 b22 B ( t ) b12 + b12 2 − b12 − − ln( q − 4 b22 B ( t ) b12 + b12 2 + b12 ) − b12 q − 4 b22 B ( t ) b12 + b12 2 + b12 − 1 / 2 b12 b22 B ( t ) − ln( B ( t )) − − C1 = 0 . After the inv erse ( u, v )- tr ansformation with the help of the function B ( t ) can b e find the s olution of the equation (17) at indica ted ab ov e the v alues of pa rameters. 9 Eigh t-dimensional Riemann space for the L orenz system of equations T o inv estig a tion of the pr op erties of classica l Lorenz equations dx ds = σ ( y − x ) , dy ds = r x − y − xz , dz ds = − bz + xy (22) we use its pre s en tatio n in the fo r m d ds ξ ( s ) = 1 / 5 ( b − 4 σ + 1) ξ ρ + σ η ρ, d ds η ( s ) = − ξ θ + r ξ ρ + 1 / 5 ( b + σ − 4) η ρ, d ds θ ( s ) = ξ η + 1 / 5 ( σ − 4 b + 1) ρ θ , 9 EIGHT-DIMENSIONAL RIEMANN SP ACE F OR THE LO RENZ SYSTEM OF EQUA TIO NS 14 d ds ρ ( s ) = 1 / 5 ( σ + 1 + b ) ρ 2 . The r elation b e t ween b oth systems is defined b y the conditions x ( s ) = ξ ρ , y ( s ) = η ρ , z ( s ) = θ ρ . F our dimensio nal system can b e presented in the form d 2 X i ds 2 + Γ i j k dX j ds dX k ds = 0 , which allow us to consider it as geo desic equa tions of the space w ith constant affine connectio n. Nonzero c ompo nent s of connection a re Γ 1 14 = 4 σ − b − 2 10 , Γ 1 24 = − σ 2 , Γ 2 13 = 1 2 , Γ 2 14 = − r 2 , Γ 2 34 = 4 − σ − b 10 , Γ 3 34 = 4 b − σ − 1 10 , Γ 3 12 = − 1 2 , Γ 4 44 = − σ + b + 1 5 The metric of a sso ciated space is 8 ds 2 = 2 / 5 ( b + σ + 1) du 2 V + + (2 / 5 dz du + 2 dx dy − 8 / 5 dz du b + 2 / 5 dz du σ ) U + + (2 r dx du + 2 / 5 dz du σ − 2 dx dz − 8 / 5 dz du + 2 / 5 dz du b ) Q + + ( − 8 / 5 dx du σ + 2 σ dy du + 2 / 5 dx du b + 2 / 5 dx du ) P + +2 dy dQ + 2 dz dU + 2 du dV + 2 dx dP . (23) After integration geo desic of additio nal co ordina tes take the form d dt P ( t ) − ( z ( t ) − r u ( t )) Q ( t ) − 1 5 ( − b − 1 + 4 σ ) u ( t ) P ( t ) + y ( t ) U ( t ) = 0 , d dt Q ( t ) + x ( t ) U ( t ) + σ u ( t ) P ( t ) = 0 , d dt U ( t ) + 1 5 (( σ + b − 4) u ( t ) − x ( t )) Q ( t ) + 1 5 ( − 4 b + 1 + σ ) u ( t ) U ( t ) = 0 , d dt V ( t ) + 1 / 5 ( b + σ + 1) u ( t ) V ( t ) − x ( t ) y ( t ) U ( t ) u ( t ) + x ( t ) Q ( t ) z ( t ) u ( t ) = 0 . Prop erties of this s ystem from the par ameters are dep enden t a nd can b e inv estigated with the he lp of the Wilczynski inv ariants. 10 LAPLACE O PERA TOR 15 10 Laplace op erator In theory of Riemann spaces the equation Lψ = g ij ( ∂ 2 ∂ x i ∂ x j − Γ k ij ∂ ∂ x k ) ψ ( x ) = 0 (24) can b e used to the study of the prop erties of the spa ce. F or the eight-dimensional space with metric (23 ) corresp onded the Lorenz system we get the equation on the function ψ ( x, y , z , u, P , Q, U , V ) − 2 U ∂ 2 ∂ P ∂ Q ψ + 1 / 5  ∂ ∂ V ψ  σ + 2 Q ∂ 2 ∂ P ∂ U ψ + 1 / 5  ∂ ∂ V ψ  b + 8 / 5  ∂ 2 ∂ P ∂ V ψ  P σ − − 2 / 5  ∂ 2 ∂ P ∂ V ψ  P − 2 σ P ∂ 2 ∂ Q∂ V ψ − 2  ∂ 2 ∂ P ∂ V ψ  rQ + 8 / 5  ∂ 2 ∂ U ∂ V ψ  Q − 2 / 5  ∂ 2 ∂ U ∂ V ψ  U σ + +8 / 5  ∂ 2 ∂ U ∂ V ψ  U b − 2 / 5  ∂ 2 ∂ U ∂ V ψ  U + 2 ∂ 2 ∂ P ∂ x ψ − 4 / 5 ∂ ∂ V ψ + 2 ∂ 2 ∂ Q∂ y ψ + 2 ∂ 2 ∂ U ∂ z ψ + 2 ∂ 2 ∂ V ∂ u ψ − − 2 / 5  ∂ 2 ∂ P ∂ V ψ  P b − 2 / 5 V  ∂ 2 ∂ V 2 ψ  b − 2 / 5 V  ∂ 2 ∂ V 2 ψ  σ − 2 / 5 V ∂ 2 ∂ V 2 ψ − 2 / 5  ∂ 2 ∂ U ∂ V ψ  Qb − − 2 / 5  ∂ 2 ∂ U ∂ V ψ  Qσ = 0 . This eq uation has v aries t yp e of pa rticular solutions. As exa mple ψ ( P, Q, U, V ) = H ( P, Q, U ) + V P where the function ψ ( P, Q, U, V ) sa tisfies the equation 2 Q ∂ 2 ∂ P ∂ U H ( P , Q, U ) + 9 / 5 P σ − 2 U ∂ 2 ∂ P ∂ Q H ( P , Q, U ) − 1 / 5 P b − 2 r Q − 6 / 5 P = 0 . Its so lution is in form H ( P , Q, U ) =  − 3 / 10 − 1 / 20 b + 9 20 σ  arctan( Q U ) P 2 + P rU + F1 ( Q, U ) + F2 ( P , Q 2 + U 2 ) where Fi are arbitrar y functions. A t the co ndition b = 9 σ − 6 this so lution takes a more s imple form. In the case ψ ( P, Q, U, V ) = H ( P, Q ) + V we ge t the solutio n H ( P , Q ) = F2 ( P ) + F1 ( Q ) + (1 / 10 b − 2 / 5 + 1 / 1 0 σ ) P Q U for which the re la tion b = 4 − σ is a s pecia l. 11 EIKONAL E QUA TION 16 11 Eik onal equation Solution of the eikonal equation g i j ∂ F ∂ x i ∂ F ∂ x j = 0 (25) also give us useful information a bo ut the prop erties of Riemann s pace. In the case of the spa ce with the metric (23) we get the equation 2  ∂ ∂ x φ  ∂ ∂ P φ + 2  ∂ ∂ y φ  ∂ ∂ Q φ + 2  ∂ ∂ z φ  ∂ ∂ U φ + 2  ∂ ∂ u φ  ∂ ∂ V φ − 2 U  ∂ ∂ P φ  ∂ ∂ Q φ + +2 Q  ∂ ∂ P φ  ∂ ∂ U φ + 8 / 5  ∂ ∂ P φ   ∂ ∂ V φ  P σ − 2 / 5  ∂ ∂ P φ   ∂ ∂ V φ  P b − 2 / 5  ∂ ∂ P φ   ∂ ∂ V φ  P − − 2  ∂ ∂ P φ   ∂ ∂ V φ  rQ − 2 σ P  ∂ ∂ Q φ  ∂ ∂ V φ + 8 / 5  ∂ ∂ U φ   ∂ ∂ V φ  Q − 2 / 5  ∂ ∂ U φ   ∂ ∂ V φ  Qσ − − 2 / 5  ∂ ∂ U φ   ∂ ∂ V φ  Qb +8 / 5  ∂ ∂ U φ   ∂ ∂ V φ  U b − 2 / 5  ∂ ∂ U φ   ∂ ∂ V φ  U σ − 2 / 5  ∂ ∂ U φ   ∂ ∂ V φ  U − − 2 / 5 V  ∂ ∂ V φ  2 b − 2 / 5 V  ∂ ∂ V φ  2 σ − 2 / 5 V  ∂ ∂ V φ  2 = 0 . In par ticular case φ ( x, y , z , u, P , Q, U, V ) = A ( Q, U, V ) + P U one get the equatio ns to determinatio n of the function A ( Q, U, V ) − 2 U 2 ∂ ∂ Q A ( Q, U, V )+2 QU ∂ ∂ U A ( Q, U, V )+2 QU P +6 / 5 U  ∂ ∂ V A ( Q, U, V )  P σ +6 / 5 U  ∂ ∂ V A ( Q, U, V )  P b − − 4 / 5 U  ∂ ∂ V A ( Q, U, V )  P − 2 U  ∂ ∂ V A ( Q, U, V )  rQ − 2 σ P  ∂ ∂ Q A ( Q, U, V )  ∂ ∂ V A ( Q, U, V )+ +8 / 5  ∂ ∂ U A ( Q, U, V )   ∂ ∂ V A ( Q, U, V )  Q + 8 / 5  ∂ ∂ V A ( Q, U, V )  QP − − 2 / 5  ∂ ∂ U A ( Q, U, V )   ∂ ∂ V A ( Q, U, V )  Qσ − 2 / 5  ∂ ∂ V A ( Q, U, V )  Qσ P − − 2 / 5  ∂ ∂ U A ( Q, U, V )   ∂ ∂ V A ( Q, U, V )  Qb − 2 / 5  ∂ ∂ V A ( Q, U, V )  QbP + − 8 / 5  ∂ ∂ U A ( Q, U, V )   ∂ ∂ V A ( Q, U, V )  U b − 2 / 5  ∂ ∂ U A ( Q, U, V )   ∂ ∂ V A ( Q, U, V )  U σ − − 2 / 5  ∂ ∂ U A ( Q, U, V )   ∂ ∂ V A ( Q, U, V )  U − 2 / 5 V  ∂ ∂ V A ( Q, U, V )  2 b − 2 / 5 V  ∂ ∂ V A ( Q, U, V )  2 σ − − 2 / 5 V  ∂ ∂ V A ( Q, U, V )  2 = 0 The solution of this equation is A ( Q, U, V ) = = F00 ( U ) c 3 V +1 / 2 Q 2 U c 3 F00 ( U ) σ +3 / 5 QU +3 / 5 bU Q σ − 2 / 5 QU σ +2 / 5 Q 2 σ − 1 / 10 Q 2 − 1 / 10 Q 2 b σ + F5 ( U ) REFERENCES 17 where F00 ( U ) = 5 U − 4 c 3 + σ c 3 + c 3 b , F5 ( U ) = 1 / 2 U 2 ( − 2 + σ ) 1 + σ + C1 and b = 2 / 3 + 2 / 3 σ, r = 1 / 3 + 1 / 3 σ. References [1] I.G.Petrovsky , E.M.Landis, O n the quantit y of limit cycle s of the equiation dy /dx = P ( x, y ) /Q ( x, y ), where P ( x, y ) and Q ( x, y ) are the p olinomials of the se cond order degr ee, Matematicheskii sb ornik , (In Russia n), 37(79), No 2, p.20 9–250 . [2] Dryuma V., The Riemann Extension in theory of differential eq uations and their applica- tions, Matematicheskaya fizika, analiz, ge ometriya , 2 0 03,v.10 , No.3, 1–19. [3] Dryuma V., The Inv aria n ts, the Riema nn and Einstein-W eyl ge o metries in theo r y of ODE’s, their ap- plications and all that, New T rends in Integrability and Partial Solv ability , p. 11 5-156 , (ed. A.B.Shabat et al.), Kluw er Academic Publishers, (ArXiv: nlin: SI/0 30302 3, 11 March, 200 3, 1–37). [4] Dryuma V., Applications o f Riemannian a nd Einstein-W eyl Geometr y in the theory of s econd order ordinary differe ntial equations , The or etic al and Mathematic al Physics , 200 1, V.128, N 1 , 8 4 5–855 . [5] V alery Dryuma, On theory of sur faces defined by the first o rder sustems of equations Buletinul A c ademiei de Stiinte a R epublicii Moldov a , (matematica), 1(5 6), 200 8, p.161–1 75. [6] V alery Dryuma, Riemann extensions in theo ry o f the first order systems of differential eq ua tions ArXiv: math. DG/05105 26 v1 25 Oct 2005 , 1–21.