On Geometry of the R"ossler system of equations

1 ON GEOMETR Y OF THE R ¨ OSSLER SYSTEM 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 spaces of consta nt affine connection is prop os ed to study the R¨ ossler system of equations dx ds = − ( y + z ) , dy ds = x + αy , dz ds = β + xz − ν z . After its presentation in a homogeneous form d ds ξ ( s ) + 1 / 5 ξ 2 − 1 / 5 ξ ρ ν + 1 / 5 ξ ρ α + η ρ + θ ρ = 0 d ds η ( s ) + 1 / 5 ξ η − ξ ρ − 1 / 5 η ρ ν − 4 / 5 η ρ α = 0 d ds θ ( s ) − 4 / 5 ξ θ + 4 / 5 θ ρ ν + 1 / 5 θ ρ α − β ρ 2 = 0 d ds ρ ( s ) + 1 / 5 ξ ρ − 1 / 5 ρ 2 ν + 1 / 5 ρ 2 α = 0 , where x = ξ ρ , y = η ρ , z = θ ρ , it can be considered as geodesic equations d 2 X i ds 2 + Π i j k dX j ds dX k ds = 0 . of four dimensio nal space M 4 of constant affine connection with the comp onents Π i j k = Π i j k ( α, β , ν ) depe nding on parameters. 1 F rom the first order system of equati ons to the second order systems of ODE The systems of the first order differential equations dx i ds = c i + a i j x j + b i j k x j x k (1) depe nding on the parameters a, b, c are not suitable ob ject of considera tio n f r o m usually p oint of the Riemann geometry . 2 FR OM AF FINEL Y C ONNECTED SP ACE TO TH E RIEMANN SP ACE 2 The systems of the second order differential equations in form d 2 x i ds 2 + Π i kj ( x ) dx k ds dx j ds = 0 (2) are best s uited to do that. They can b e considered as geo des ic of 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 on M k . With the help of such co efficie nts can be co nstructed curv ature tenso r and others geometrical o b jects defined on v ariet y M k . 2 F rom affinely connected space to the Riemann space W e shall construct the Riemann space s ta rting fr om a given affinely connected spa c e defined by the systems of the s econd o rder O DE’s. With this aim we use the no tion o f the Riemann extension of nonriemannian space which w as used earlier in the articles of author. Remind the basic prop erties of this construction. With help of the co efficients of affine connection of a giv en n-dimensional space can b e in tro duced 2n-dimensional Riemann space D 2 n in lo ca l co o r dinates ( x i , Ψ i ) having the metric of form 2 n ds 2 = − 2Π k ij ( x l )Ψ k dx i dx j + 2 d Ψ k dx k (3) where Ψ k are a n additional co ordinates . The impor tant pr op erty of s uch type metric is that the geo desic equations of metr ic (3) decomp ose s int o tw o parts ¨ x k + Π k ij ˙ x i ˙ x j = 0 , (4) and δ 2 Ψ k ds 2 + R l kj i ˙ x j ˙ x i Ψ l = 0 , (5) 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 tenso r o f n-dimensional spa ce with a giv en a ffine connection. The fir s t part (4) of the full system is the system of equations for ge o desic of basic spa ce with lo cal co ordinates x i and it do not c o ntains the supplementary co ordinates Ψ k . The s econd part (5 ) of the system ha s the form of linear N × N matr ix system of second o r der ODE’s for supplementary co ordinates Ψ k d 2 ~ Ψ ds 2 + A ( s ) d ~ Ψ ds + B ( s ) ~ Ψ = 0 . (6) Remark that the full system of geo desic s has the first integral − 2Π k ij ( x l )Ψ k dx i ds dx j ds + 2 d Ψ k ds dx k ds = ν (7) which is equiv alen t to the relation 2Ψ k dx k ds = ν s + µ (8) where µ, ν are parameters . The geo metry of extended space connects with geo metry of basic space. F or exa mple the property of the s pa ce to b e Ricci-fla t R ij = 0 or symmetrica l R ij k l ; m = 0 keeps also for an extended space. 3 EIGHT-DIMENSIONAL RIEMANN SP ACE FOR THE R ¨ OSSLER SYSTEM OF EQUA TIONS 3 It is imp ortant to note that for extended space having the met r ic (3) a ll sc a lar curv ature inv a riants are v anished. As consequence the prop er ties o f linear sys tem of equation (5-6) dep ending from the the inv ariants of N × N matrix-function E = B − 1 2 dA ds − 1 4 A 2 under change of the coordina tes Ψ k can b e of used for that. First applications the notion of extended spa ces for the studying of nonlinea r second order O DE’s connected with nonlinear dynamical systems hav e b een c onsidered by author (V.Dryuma 200 0-200 8). 3 Eigh t-dimensional Riemann space for the R¨ ossler sy stem of equations T o inv estigation the pr op erties of the R¨ o ssler s ystem equations dx ds = − ( y + z ) , dy ds = x + αy , dz ds = β + xz − ν z (9) we use its pr esentation in ho mogeneous form d ds ξ ( s ) + 1 / 5 ξ 2 − 1 / 5 ξ ρ ν + 1 / 5 ξ ρ α + η ρ + θ ρ = 0 d ds η ( s ) + 1 / 5 ξ η − ξ ρ − 1 / 5 η ρ ν − 4 / 5 η ρ α = 0 d ds θ ( s ) − 4 / 5 ξ θ + 4 / 5 θ ρ ν + 1 / 5 θ ρ α − β ρ 2 = 0 d ds ρ ( s ) + 1 / 5 ξ ρ − 1 / 5 ρ 2 ν + 1 / 5 ρ 2 α = 0 , where x = ξ ρ , y = η ρ , z = θ ρ . The relation betw een b o th s ystems is defined by the conditions x ( s ) = ξ ρ , y ( s ) = η ρ , z ( s ) = θ ρ . Remark that for a given system ˙ ξ = P, ˙ η = Q, ˙ θ = R, ˙ ρ = T the condition P ξ + Q η + R θ + T ρ = 0 is fulfield. Such type of the system can b e rewr iten in the form d 2 X i ds 2 + Π i j k dX j ds dX k ds = 0 , which allow us to consider it as geo desic equations of the space with cons tant affine connectio n. In our case no nzero comp onents of connection are Π 1 11 = 1 5 , Π 1 14 = α − ν 10 , Π 1 24 = 1 2 , Π 1 34 = 1 2 , Π 2 12 = 1 10 , 3 EIGHT-DIMENSIONAL RIEMANN SP ACE FOR THE R ¨ OSSLER SYSTEM OF EQUA TIONS 4 Π 2 14 = − 1 2 , Π 2 24 = − 4 α + ν 10 , Π 3 34 = 4 ν + α 10 , Π 3 12 = − 4 10 , Π 3 44 = − β , Π 4 14 = 1 10 , Π 4 44 = α − ν 5 . The metric of corr esp onding Riemann space is 8 ds 2 = − 2 Π 1 11 P dx 2 + 2  − 2 Π 2 12 Q − 2 Π 3 12 U  dxdy + 2  − 2 Π 1 14 P − 2 Π 2 14 Q − 2 Π 4 14 V  dxdu + +2  − 2 Π 1 24 P − 2 Π 2 24 Q  dy du + 2  − 2 Π 1 34 P − 2 Π 3 34 U  dz du + +  − 2 Π 3 44 U − 2 Π 4 44 V  du 2 + 2 dxdP + 2 dy dQ + 2 dz dU + 2 dudV , (10) where ( P, Q, U, V ) a re an a dditional c o ordinates. Geo desic of the metric (10) for co or dinates ( x, y , z , u ) are d 2 ds 2 x ( s )+1 / 5  d ds x ( s )  2 +1 / 5  d ds u ( s )   d ds x ( s )  α − 1 / 5  d ds u ( s )   d ds x ( s )  ν +  d ds u ( s )  d ds y ( s )+ +  d ds u ( s )  d ds z ( s ) = 0 , d 2 ds 2 y ( s )+1 / 5  d ds y ( s )  d ds x ( s ) −  d ds u ( s )  d ds x ( s ) − 4 / 5  d ds u ( s )   d ds y ( s )  α − 1 / 5  d ds u ( s )   d ds y ( s )  ν = 0 , d 2 ds 2 z ( s ) − 4 / 5  d ds y ( s )  d ds x ( s )+4 / 5  d ds u ( s )   d ds z ( s )  ν +1 / 5  d ds u ( s )   d ds z ( s )  α − β  d ds u ( s )  2 = 0 , d 2 ds 2 u ( s ) + 1 / 5  d ds u ( s )  d ds x ( s ) + 1 / 5  d ds u ( s )  2 α − 1 / 5  d ds u ( s )  2 ν = 0 and they ha ve a form of homogeneous R¨ ossler system in the v ariables ξ = dx ds , η = dy ds , θ = dz ds , ρ = du ds . The system of second order differential equa tions for additional coo rdinates can b e reduced to the linear sy s tem of the fir st order equations with v ariable co e fficie n ts d dt P ( t ) + A 1 P ( t ) + B 1 Q ( t ) + C 1 U ( t ) + E 1 V ( t ) = 0 , d dt Q ( t ) + A 2 P ( t ) + B 2 Q ( t ) + C 2 U ( t ) + E 2 V ( t ) = 0 , d dt U ( t ) + A 3 P ( t ) + B 3 Q ( t ) + C 3 U ( t ) + E 3 V ( t ) = 0 , d dt V ( t ) + A 4 P ( t ) + B 4 Q ( t ) + C 4 U ( t ) + E 4 V ( t ) = 0 , where A i , B i , C i , E i are the functions of the v ariables x, y , z , u ). Prop erties of such type of the systems can b e inv estiga ted with help of the Wilczynski inv ar iants. 4 LAPLACE OPERA TO R 5 4 Laplace op erator In theor y o f Riemann spa ces the equation Lψ = g ij ( ∂ 2 ∂ x i ∂ x j − Γ k ij ∂ ∂ x k ) ψ ( x ) = 0 (11) can b e us ed to the study of the prop erties of spaces. F or the eight-dimensional space with the metric (10) co rresp onded the R¨ o ssler system we get the equation on the function ψ ( x, y , z , u, P, Q, U, V ) = θ ( P , Q, U, V ) 4 / 5 ∂ ∂ P θ ( P , Q, U, V ) + 2 / 5 P ∂ 2 ∂ P 2 θ ( P , Q, U, V ) − 8 / 5  ∂ 2 ∂ P ∂ Q θ ( P , Q, U, V )  U + +2 / 5  ∂ 2 ∂ P ∂ Q θ ( P , Q, U, V )  Q − 2  ∂ 2 ∂ P ∂ V θ ( P , Q, U, V )  Q + 2 / 5  ∂ 2 ∂ P ∂ V θ ( P , Q, U, V )  V + +2  ∂ 2 ∂ Q∂ V θ ( P , Q, U, V )  P + 2  ∂ 2 ∂ U ∂ V θ ( P , Q, U, V )  P + 8 / 5  ∂ ∂ V θ ( P , Q, U, V )  α + +2 / 5  ∂ ∂ V θ ( P , Q, U, V )  ν − 2 / 5  ∂ 2 ∂ V 2 θ ( P , Q, U, V )  V ν + 2 / 5  ∂ 2 ∂ V 2 θ ( P , Q, U, V )  V α + +2 / 5  ∂ 2 ∂ U ∂ V θ ( P , Q, U, V )  U α + 2 / 5  ∂ 2 ∂ Q∂ V θ ( P , Q, U, V )  Qν + 8 / 5  ∂ 2 ∂ Q∂ V θ ( P , Q, U, V )  Qα − − 2 / 5  ∂ 2 ∂ P ∂ V θ ( P , Q, U, V )  P ν − 2  ∂ 2 ∂ V 2 θ ( P , Q, U, V )  β U + 8 / 5  ∂ 2 ∂ U ∂ V θ ( P , Q, U, V )  U ν + +2 / 5  ∂ 2 ∂ P ∂ V θ ( P , Q, U, V )  P α = 0 . This equation has v aries type o f particular solutions. A simplest one is ψ ( x, y , z , u, P, Q, U, V ) = e ( ν − α ) P − 5 Q +5 U + V at the co nditions o n parameter s of the R¨ ossler system β = 25 2 ν, α = − 3 / 2 ν. As examples obtained by direct substitutions we get the quadr atic s olution θ ( P , Q, U, V ) = = 1 / 1 8 9 Q 2 + 30 QP α − 18 QV + 25 P 2 α 2 − 30 P α V + 9 V 2 + 15 P 2 + 60 U α P + 4 5 β U P c 2 with conditions ν = 8 / 3 α, β = arbi trar y . Cubic so lution θ ( P , Q, U, V ) = − 1 240  − 96 α l3 − 2880 α 2 l1  V 3 β 2 − 1 240 ( − 48 β l3 − 1 440 β l1 α ) U V 2 β 2 + +  − 1 240 (60 β l3 + 1800 β l1 α ) P 2 β 2 + l1 U 2  V + k1 U 3 + ( l3 P + l2 Q ) U 2 − 5 EIKONAL EQUA TION 6 − 1 240 (225 0 β l1 α + 7 5 β l3 ) P 2 Q β 2 + m2 Q 2 − 1 240  400 β 2 l1 + 3000 β l1 α + 100 β l3  P 2 β 2 ! U + nQ 3 at the co ndition ν = 6 α and a rbitrary co efficients l , m , k , n . A po lynomial solution of degre e four θ ( P , Q, U, V ) = rQ 4 + k1 U 4 − 2 / 5 m3 U 3 V + l2 U Q 3 + m2 U 2 Q 2 + n2 U 3 Q + m3 U 2 P 2 at the co ndition ν = − 7 13 α. Remark that the proper ties of of suc h t yp e of solutions depend o n parameters and ma y be highly diversified. More complicated so lutions o f the Laplace e q uation can b e obtained by a pplication of the method of ( u, v )- tr ansformation developed in the works of author. 5 Eik onal equation Solutions of eikonal equation g i j ∂ F ∂ x i ∂ F ∂ x j = 0 (12) also g ives useful information a bo ut the prop erties of Riemann space. In particular the condition F ( x 1 , x 2 , ..., x i ) = 0 where function F ( x i ) satisfies the equatio n (12), deter mines ( N − 1)-dimensio nal hypers urface with normals forming an is otropic vector field. F or the space with the metric (10) the eikonal equation on the function ψ ( x, y , z , u, P, Q, U, V ) = η ( P, Q, U, V ) takes the form 2 / 5 P  ∂ ∂ P η ( P, Q, U, V )  2 − 8 / 5  ∂ ∂ P η ( P, Q, U, V )   ∂ ∂ Q η ( P, Q, U, V )  U + +2 / 5  ∂ ∂ P η ( P, Q, U, V )   ∂ ∂ Q η ( P, Q, U, V )  Q − 2  ∂ ∂ P η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  Q + +2 / 5  ∂ ∂ P η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  P α − 2 / 5  ∂ ∂ P η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  P ν + +2 / 5  ∂ ∂ P η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  V + 2  ∂ ∂ Q η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  P − − 8 / 5  ∂ ∂ Q η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  Qα − 2 / 5  ∂ ∂ Q η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  Qν + +2  ∂ ∂ U η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  P + 8 / 5  ∂ ∂ U η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  U ν + +2 / 5  ∂ ∂ U η ( P, Q, U, V )   ∂ ∂ V η ( P, Q, U, V )  U α + 2 / 5  ∂ ∂ V η ( P, Q, U, V )  2 V α − − 2 / 5  ∂ ∂ V η ( P, Q, U, V )  2 V ν − 2  ∂ ∂ V η ( P, Q, U, V )  2 β U = 0 . (13) 5 EIKONAL EQUA TION 7 A simplest solution of this equation is η ( P, Q, U, V ) = − Q α + V ν − α + U α + P with condition on the c o efficien ts of the R¨ oss ler system − 5 β α − 1 1 α ν + 8 ν 2 + 3 α 2 = 0 . (14) F rom here we find ν = 11 16 α + 1 / 16 p 25 α 2 + 160 β α T o provide a more complicated solutions of the eq uation (12) we use the metho d of ( u, v )-tra nsformation. F or the sake of conv enience we rewrite the equation (12) in the form 2 / 5 x  ∂ ∂ x η ( x, y , z , p )  2 − 8 / 5  ∂ ∂ x η ( x, y , z , p )   ∂ ∂ y η ( x, y , z , p )  z + +2 / 5  ∂ ∂ x η ( x, y , z , p )   ∂ ∂ y η ( x, y , z , p )  y − 2  ∂ ∂ x η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  y + +2 / 5  ∂ ∂ x η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  xα − 2 / 5  ∂ ∂ x η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  xν + +2 / 5  ∂ ∂ x η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  p + 2  ∂ ∂ y η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  x − − 8 / 5  ∂ ∂ y η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  y α − 2 / 5  ∂ ∂ y η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  y ν + +2  ∂ ∂ z η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  x + 8 / 5  ∂ ∂ z η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  z ν + +2 / 5  ∂ ∂ z η ( x, y , z , p )   ∂ ∂ p η ( x, y , z , p )  z α + 2 / 5  ∂ ∂ p η ( x, y , z , p )  2 pα − 2 / 5  ∂ ∂ p η ( x, y , z , p )  2 pν − − 2  ∂ ∂ p η ( x, y , z , p )  2 β z = 0 (15) Now after change of the function and deriv atives in a ccordance with the rules η ( x, y , z , p ) → u ( x, t, z , p ) , y → v ( x, t, z , p, ∂ η ( x, y , z , p ) ∂ x → ∂ u ( x, t, z , p ) ∂ x − ∂ u ( x,t,z , p ) ∂ t ∂ v ( x,t,z ,p )) ∂ t ∂ v ( x, t, z , p ) ∂ x , ∂ η ( x, y , z , p ) ∂ z → ∂ u ( x, t, z , p ) ∂ z − ∂ u ( x,t,z , p ) ∂ t ∂ v ( x,t,z ,p )) ∂ t ∂ v ( x, t, z , p ) ∂ z , ∂ η ( x, y , z , p ) ∂ p → ∂ u ( x, t, z , p ) ∂ p − ∂ u ( x,t,z , p ) ∂ t ∂ v ( x,t,z ,p )) ∂ t ∂ v ( x, t, z , p ) ∂ p , ∂ η ( x, y , z , p ) ∂ y → ∂ u ( x,t,z , p ) ∂ t ∂ v ( x,t,z ,p )) ∂ t , where u ( x, t, z , p ) = t ∂ ∂ t ω ( x, t, z , p ) − ω ( x, t, z , p ) , v ( x, t, z , p ) = ∂ ∂ t ω ( x, t, z , p ) 5 EIKONAL EQUA TION 8 we find the equation on the function ω ( x, t, z , p )  ∂ ∂ x ω ( x, t, z , p )  2 x − 5 tx ∂ ∂ p ω ( x, t, z , p ) − pν  ∂ ∂ p ω ( x, t, z , p )  2 − 5 β z  ∂ ∂ p ω ( x, t, z , p )  2 + + p  ∂ ∂ x ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p ) + 4 z ν  ∂ ∂ z ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p ) + 4  ∂ ∂ x ω ( x, t, z , p )  z t − −  ∂ ∂ x ω ( x, t, z , p )   ∂ ∂ t ω ( x, t, z , p )  t − 5  ∂ ∂ t ω ( x, t, z , p )   ∂ ∂ x ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p )+ + pα  ∂ ∂ p ω ( x, t, z , p )  2 + 5 x  ∂ ∂ z ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p ) + z α  ∂ ∂ z ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p )+ +4 t  ∂ ∂ t ω ( x, t, z , p )  α ∂ ∂ p ω ( x, t, z , p ) + t  ∂ ∂ t ω ( x, t, z , p )  ν ∂ ∂ p ω ( x, t, z , p ) − − xν  ∂ ∂ x ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p ) + xα  ∂ ∂ x ω ( x, t, z , p )  ∂ ∂ p ω ( x, t, z , p ) = 0 . (16) In spite of the fact that this equation lo ok s not a simple than equa tion (15 ) its pa rticular s o lutions can b e find w itho ut trouble. As example the substitution of the form ω ( x, t, z , p ) = B ( x, t, z ) + k pt, int o the equatio n (16 ) lead to expr ession o n the function B ( x, t, z ) B ( x, t, z ) = α tx + tz + A ( t ) , with arbitrar y function A ( t ) and the conditions on co efficients of the R¨ ossler sy s tem β = 1 / 5 α (8 + 5 k ) k 2 , ν = α (1 + k ) k depe nding from arbitra ry parameter k . Using the function ω ( x, t, z , p ) w e ca n obtain the so lution of the equation (13) by eliminatio n of the parameter t fro m the relations η ( x, y , z , p ) − t ∂ ∂ t ω ( x, t, z , p ) + ω ( x, t, z , p ) = 0 , y − ∂ ∂ t ω ( x, t, z , p ) = 0 . As example at the choice ( A ( t ) = 1 + t 2 we get η ( x, y , z , p ) = 1 / 4 k 2 p 2 +( − 1 / 2 y k + 1 / 2 z k + 1 / 2 α xk ) p +1 / 4 z 2 +(1 / 2 α x − 1 / 2 y ) z − 1+1 / 4 y 2 − 1 / 2 y α x +1 / 4 α 2 x 2 . A t the co ndition A ( t ) = ln( t ) we get η ( x, y , z , p ) = 1 − ln( − ( − y + α x + z + k p ) − 1 ) . Remark that para meter s α, β , ν in these case s the relatio n (14) is satisfied. T o cite another example. Substitution the expre s sion ω ( x, t, z , p ) = A ( t ) x + tz + C ( t ) p int o the equatio n (16 ) lead to the sy s tem of equa tions on the functions A ( t ) , C ( t ) ( tν C ( t ) + 4 tα C ( t ) − 5 A ( t ) C ( t ) − A ( t ) t ) d dt A ( t ) + ( A ( t )) 2 − ν A ( t ) C ( t ) + α A ( t ) C ( t ) = 0 , REFERENCES 9 ( tν C ( t ) + 4 tα C ( t ) − 5 A ( t ) C ( t ) − A ( t ) t ) d dt C ( t ) + A ( t ) C ( t ) − ν ( C ( t )) 2 + α ( C ( t )) 2 = 0 , − 5 β z ( C ( t )) 2 + (5 z α t − 5 A ( t ) z + 5 z ν t ) C ( t ) + 3 A ( t ) z t = 0 . F rom this system of equations we find the condition on parameters ν = 3 / 5 β − α (17) and ex pression o n the function A ( t ) A ( t ) = − β C ( t ) . F unction C ( t ) in th is case sa tisfies the eq uation (25 β C ( t ) + 8 β t + 1 5 tα ) d dt C ( t ) − 8 β C ( t ) + 10 α C ( t ) = 0 . Its solution is defined by the rela tion − tα + ( C ( t )) − 1 / 2 8 β +15 α − 4 β +5 α C1 α − β C ( t ) = 0 . Using the expression on the function C ( t ) we can find the function ω ( x, t, z , p ) and after elimination of the pa rameter t from corresp onding relations it is po ssible to g et the solution of the eikonal equation at the co ndition (17 ) on parameters of the R¨ ossler sys tem of equa tions. References [1] Dryuma V., The Riemann Extension in theory of differential equations and their applica- tions, Matematicheskaya fi zika, analiz, ge ometriya , 2003 ,v.10, No.3, 1–1 9. [2] Dryuma V., The Inv a riants, the Riemann and E instein-W eyl g eometries in theo r y o f O DE’s, their ap- plications and all that, New T rends in Integrabilit y and Partial Solv ability , p. 115- 156, (ed. A.B.Sha bat et al.), K luw er Aca demic P ublis he r s, (Ar Xiv: nlin: SI/030 3 023, 11 March, 2003, 1–37 ). [3] Dryuma V., Applications of Riemannia n and Einstein-W eyl Geometry in the theor y of second order ordinary differential equations, The or etic al and Mathematic al Physics , 20 0 1, V.128 , N 1, 84 5–855 . [4] V aler y Dryuma, On theo r y of surfaces defined b y the first order sustems of equa tions Buletinul A c ademiei de Stiinte a R epublicii Moldova , (matematica), 1(5 6), 2 008, p.1 61–17 5. [5] V aler y Dryuma , Riemann extensions in theory of the firs t o rder systems of differential eq ua tions ArXiv: math. DG/05105 26 v1 25 O ct 2005 , 1–2 1. [6] V aler y Dr y uma, Riemann g eometry in theo ry of the fir st order s y stems of differen tial equations ArXiv: 0807.0117 8 v1 [nlin.SI] 1 Jul 2008 , 1– 17.