THREE-DIMENSIONAL DYNAMICAL SYSTEMS WITH FOUR-DIMENSIONAL VESSIOT-GULDBERG-LIE ALGEBRAS

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1 Journal of Applied Analysis and Computation Volume 7, Number 3, August 207, Website: DOI:0.948/ THREE-DIMENSIONAL DYNAMICAL SYSTEMS WITH FOUR-DIMENSIONAL VESSIOT-GULDBERG-LIE ALGEBRAS Nail H. Ibragimov,2, and Aliya A. Gainetdinova 2 Abstract Dynamical systems attract much attention due to their wide applications. Many significant results have been obtained in this field from various points of view. The present paper is devoted to an algebraic method of integration of three-dimensional nonlinear time dependent dynamical systems admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras L 4. The invariance of the relation between a dynamical system admitting nonlinear superposition and its Vessiot-Guldberg-Lie algebra is the core of the integration method. It allows to simplify the dynamical systems in question by reducing them to standard forms. We reduce the three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras to 98 standard types and show that 86 of them are integrable by quadratures. Keywords Time dependent dynamical system, nonlinear superposition of solutions, Vessiot-Guldberg-Lie algebra L 4, standard forms of L 4. MSC(200) 34A, 34A34, 70G65.. Introduction We consider time dependent nonlinear dynamical systems given by first-order ordinary differential equations dx i dt = f i (t, x), i =,..., n, (.) with n > dependent variables x,..., x n. We denoted by x the n-dimensional vector x = (x,..., x n ) and refer to the system (.) as an n-dimensional dynamical system. A major obstacle in investigating nonlinear dynamical systems is that they do not obey the usual superposition principle which provides powerful tools in dealing with linear systems. Furthermore, integration methods based on Lie group analysis of differential equations are not effective for dealing with the system (.) because the corresponding author. address: nib@bth.se (N. H. Ibragimov) Research Centre ALGA: Advances in Lie Group Analysis, Department of Mathematics and Natural Sciences, Blekinge Institute of Technology, SE Karlskrona, Sweden 2 Laboratory Group analysis of mathematical models in natural and engineering sciences, Ufa State Aviation Technical University, Karl Marx Str. 2, Ufa, Russia

2 Three-dimensional dynamical systems 873 determining equations for calculating Lie symmetries are not over-determined in the case of first-order ordinary differential equations. Therefore only narrow categories of particular nonlinear dynamical systems can be solved analytically. Consequently, nonlinear systems are mostly analyzed by numerical methods or by approximating them by linear systems using, e.g. perturbation theory. In this paper we are concerned with constructing wide classes of integrable threedimensional dynamical systems (.) using, instead of the linear superposition principle, the more general concept of nonlinear superpositions introduced in 893 by Vessiot [], Guldberg [2] and Lie [3]. S. Lie [4] noticed in 885 that the key features of linear ordinary differential equations dx i dt = ai k(t)x k, i =,..., n, are based on the fact that the differential operators X ik = x i, i, k =,..., n, xk generate a finite continuous group, namely the linear homogeneous group with n variables x i. This observation led him (see [4], 8) to believe that the main properties of the linear equations can be extended to the nonlinear equations having the form of generalized separation of variables: dx i dt = T (t)ξ i (x) + + T r (t)ξ i r(x), i =,..., n, (.2) provided that the linear span L r of the first-order differential operators is closed under the commutator: X α = ξα(x) i, α =,..., r, (.3) xi [X α, X β ] = c γ αβ X γ. (.4) It means that L r is a finite-dimensional Lie algebra. The coefficients T α (t) in Equations (.2) are any smooth functions of the variable t. S. Lie showed (see [4, p.28]) that the general solution of his system (.2) can be expressed via a certain finite number m of particular solutions x = (x,..., x n ),..., x m = (x m,..., x n m) (.5) of the system (.2) and that the expression (nonlinear superposition) x = ϕ(x,..., x m, C,..., C n ) (.6) for the general solution x = (x,..., x n ) as a function of the particular solutions (.5) and arbitrary constants C,..., C n is obtained by solving the equations J i (x, x,..., x m ) = C i, i =,..., n, (.7) with respect to x = (x,..., x n ), where J i are invariants of m+ points x, x,..., x m with respect to the group with the basic generators X,..., X r. Later E. Vessiot [] and A. Guldberg [2] came to a lucky idea to look for all systems of ordinary differential equations possessing fundamental systems of integrals, or in modern terminology, admitting nonlinear superpositions.

3 874 N. H. Ibragimov, A. A. Gainetdinova Definition.. A dynamical system (.) admits a nonlinear superposition if the general solution of the system (.) can be written as a vector function (.6) of a finite number of its particular solutions (.5) and n arbitrary constants C,..., C n. The solution to Vessiot-Guldberg s problem was given by S. Lie. He announced in [3] the following statement (the detailed proof is published in [5, Chapter 24, pp ]; see also [6, Sect. 6.7]). Theorem.. The system (.) admits a nonlinear superposition if and only if it has the form of generalized separation of variables (.2). The number m of necessary particular solutions (.5) is estimated by nm r. (.8) The nonlinear superposition (.6) for the system (.2) is given by the equation (.7). The algebra L r spanned by the operators (.3) is called the Vessiot-Guldberg-Lie algebra for the dynamical system (.2). A renowned example of a first-order ordinary differential equation admitting a nonlinear superposition is provided by the general Riccati equation dx dt = P (t) + Q(t)x + R(t)x2. In this example n =, r = 3, the nonlinear superposition (.6) is given by the cross-ratio theorem stating that any four solutions x, x 2, x 3 and x of the Riccati equation are connected by the equation (x x 2 )(x 3 x ) = C, C = const. (x x)(x 2 x 3 ) The operators (.3) of the Vessiot-Guldberg-Lie algebra have the form X = x, X 2 = x x, X 3 = x 2 x, and the number of necessary particular solutions is m = 3, hence the estimation (.8) is satisfied with the equality sign. We have demonstrated in [7] that there are 3 standard forms of time dependent three-dimensional dynamical systems (.) admitting nonlinear superpositions with three-dimensional Vessiot-Guldberg-Lie algebras L 3. The solvable L 3 provide 24 standard forms that are integrable by quadratures. The purpose of the present paper is to enumerate standard forms of threedimensional nonlinear dynamical systems admitting nonlinear superpositions with four-dimensional Vessiot-Guldberg-Lie algebras L 4 and to single out the integrable systems. We adopt here the notation and terminology used in [7]. 2. Integration method A method for integration of dynamical systems admitting nonlinear superposition has been suggested in [8, Section.2]. The method is based on the fact that dynamical systems admitting nonlinear superposition and their Vessiot-Guldberg- Lie algebras behave coherently under any change of the dependent variables x i = x i (x), i =,..., n. (2.)

4 Three-dimensional dynamical systems 875 Table. Non-isomorphic structures of four-dimensional real Lie algebras Type [X, X 2 ] [X 2, X 3 ] [X, X 3 ] [X, X 4 ] [X 2, X 4 ] [X 3, X 4 ] I II X III X X 3 IV 0 X V 0 X + X 2 X VI 0 X 2 X VII 0 hx 2 X VIII 0 X + bx 2 bx X IX X X 3 2X X X 3 X X XI X X 2 XII ax X 2 X 2 + X 3 XIII X 0 X 2 XIV X X + X 2 X 2 + X 3 XV ax bx 2 cx 3 XVI ax bx 2 X X 2 + bx 3 XVII 0 X 0 2X X 2 X 2 + X 3 XVIII 0 X 0 ( + h)x X 2 hx 3 XIX 0 X 0 2bX bx 2 X 3 X 2 + bx 3 XX 0 X 2 X X 2 X 0 Namely, it can be shown ( [9], see also [7]) that the system (.2) is written in the new variables (2.) in the form d x i dt = T (t) ξ i ( x) + + T r (t) ξ i r( x), i =,..., n, (2.2) where ξ α = (ξ α,..., ξ n α) are transformed according to the vector law ξ i α = xi (x) x j ξ j α, i =,..., n, (2.3) and the coefficients T α (t) in Equations (2.2) are the same as in Equations (.2). According to the above property, one can simplify dynamical systems admitting nonlinear superposition by reducing bases ξ α of the Vessiot-Guldberg-Lie algebras to simple (standard) forms using appropriate changes of variables (2.). 3. Four-dimensional Lie algebras in the real domain Recall that the non-isomorphic structures of four-dimensional Lie algebras L 4 in the complex domain were described by S. Lie [0]. We will use the classification in the real domain given in []. It is presented in our paper in Table. Note that the constants in Table satisfy the conditions a 0, h, c 0, b 0. For our integration purposes we need realizations of Lie algebras L 4 by firstorder linear partial differential operators in the three-dimensional space. The enumeration of non-similar realizations of four-dimensional real Lie algebras in the three-dimensional space can be found in recent publications [2, 3]. The paper [2] deals with classification of systems of second-order ordinary differential equations. Paper [3] contains a useful short review of results and an extensive list of literature on classification of low-dimensional Lie algebras. In Tables 2 and 3 we use the realizations given in [3].

5 876 N. H. Ibragimov, A. A. Gainetdinova 4. Standard forms of L 4 and associated dynamical systems We write the three-dimensional dynamical systems (.2) admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras in the form x = ξ (x, y, z)t (t) + ξ 2(x, y, z)t 2 (t) + ξ 3(x, y, z)t 3 (t) + ξ 4(x, y, z)t 4 (t), y = ξ 2 (x, y, z)t (t) + ξ 2 2(x, y, z)t 2 (t) + ξ 2 3(x, y, z)t 3 (t) + ξ 2 4(x, y, z)t 4 (t), z = ξ 3 (x, y, z)t (t) + ξ 3 2(x, y, z)t 2 (t) + ξ 3 3(x, y, z)t 3 (t) + ξ 3 4(x, y, z)t 4 (t), (4.) where x, y, z denote the derivatives of the dependent variables x, y, z with respect to t. We reduce all systems (4.) into standard forms by using the realizations of non-isomorphic real Lie algebras L 4. The result is given in Tables 2 and 3. Table 2 contains the standard forms of the system (4.) associated with solvable Vessiot- Guldberg-Lie algebras L 4. In this table ϕ(z), ψ(z), θ(z) and θ(y, z) are arbitrary functions. Table 3 contains the standard forms of the system (4.) associated with nonsolvable Vessiot-Guldberg-Lie algebras L Discussion of Tables 2 and 3 Tables 2 and 3 show that there exist 98 distinctly different classes of systems (4.). Applying an arbitrary change of the dependent variables x, y, z to each class of the equations from Tables 2 and 3 one obtains an infinite set of three-dimensional dynamical systems admitting nonlinear superposition with four-dimensional Vessiot- Guldberg-Lie algebras. After a detailed inspection of Tables 2 and 3 we conclude that 86 classes of the systems (4.) with solvable L 4, namely, the systems -84, 86 and 89 from Table 2 are integrable by quadratures. Note that the system 32 can be integrated reducing it to the system x = T ( t) + bx + y, y = T 2 ( t) + by y, z = T 3 ( t) by introducing the new independent variable t = T 3 (t)dt. The systems 33, 68 and 84 are integrated likewise. The systems 85, 87 and 88 from Table 2 provide linear systems with variable coefficients and, in general, are not integrable by quadratures. The systems with nonsolvable L 4 (Table 3) are nonlinear and not integrable by quadratures. 6. Conclusions The three-dimensional dynamical systems (4.) admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras are mapped to standard forms. They are classified into 98 classes presented in Tables 2 and 3. Each class is a representative of an infinite set of equations involving three arbitrary functions of three variables. Systems associated with solvable four-dimensional Vessiot- Guldberg-Lie algebras are classified into 89 classes presented in Table 2, among them 86 are integrable by quadratures.

6 Three-dimensional dynamical systems 877 References [] E. Vessiot, Sur une classe d équations différentielles Ann. Sci. Ecole Norm. Sup., 893, 0. [2] A. Guldberg, Sur les équations différentielles ordinaire qui possèdent un système fundamental d intégrales, C.R. Acad. Sci. Paris, 893, 6. [3] S. Lie, Sur une classe d équations différentialles qui possèdent des systèmes fundamentaux d intégrales, C.R. Acad. Sci. Paris, 893, 6. Reprinted in: S. Lie, Ges. Abhandl. vol. 4, B.G. Teubner, Leipzig, 929. [4] S. Lie, Allgemeine Untersuchungen über Differentialgleichungen, die eine continuirliche, endliche Gruppe gestatten, Mathematische Annalen, 25, Heft, 885. Reprinted in: S. Lie, Ges. Abhandl. vol. 6, Teubner, Leipzig, 927. [5] S. Lie, Vorlesungen über continuerliche Gruppen mit geometrischen und anderen Anwendungen, B.G. Teubner, Leipzig, 893. [6] N.H. Ibragimov, A practical course in differential equations and mathemtical modelling, Higher Education Press, Beijing, [7] N.H. Ibragimov, A.A. Gainetdinova, Three-dimensional dynamical systems admitting nonliner superposition with three-dimensional Vessiot-Guldberg-Lie algebras, Appl. Math. Letters, 206, 52. [8] N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, Chichester, 999. [9] N.H. Ibragimov, Utilization of canonical variables for integration of systems of first-order differential equations, in: Archives of ALGA, vol. 6, ALGA Publications, Blekinge Institute of Technology, Sweden, Karlskrona, $file/archives6.pdf ( ) [0] S. Lie, Theorie der Transformationsgruppen, vol. 3, B.G. Teubner, Leipzig, 893. [] G.M. Mubarakzyanov, On solvable Lie algebras, Izv. Vyssh. Uchebn. Zaved. Mat., 963, (32). [2] C. Wafo Soh, F.M;. Mahomed, Canonical forms for systems of two second-order ordinary differential equations, J. Phys A: Math. Gen., 200, 34. [3] R.O. Popovich, V.M. Boyko, M.O. Nesterenko, M.W. Lutfullin, 2003 Realizations of real low-dimensional Lie algebras, J. Phys A: Math. Gen., 2003, 36.

7 878 N. H. Ibragimov, A. A. Gainetdinova Tables Table 2. Dynamical systems with solvable L 4 I 2 3 II III IV 3 4 V x, y, z x + ϕ(z) y, θ(z) x + ψ(z) y x, y x, z, θ(y, z) x x x, y x, ϕ(y) x, ψ(y) x x, x x, y, x, x x + z z, z x, x x +ϕ(z) y, y, z x y, z y x, x x + y y + z z, y x, z x x, x x + z, y, y y + C z x, x x + z y, z z x, x x, y, y y y, y y + x, x + x y y, y x, y y + z x, x x +y y, y x, y y x, x, x, z, z x, y z, z x +ϕ(y) z, y x y, (x+y) x +y y, z x,, (x + y) y x + y ( ) y + z, ez z x + y x,, (x + y) y x + y y + z, ez x x = T (t) + zt 3 (t) + θ(z)t 4 (t), y = T 2 (t) + ϕ(z)t 3 (t) + ψ(z)t 4 (t), x = T (t) + yt 2 (t) + zt 3 (t) + θ(y, z)t 4 (t), y = 0, x = T (t) + yt 2 (t) + ϕ(y)t 3 (t) + ψ(y)t 4 (t), y = 0, x = T (t) + xt 2 (t), y = T 3 (t), x = T (t) + xt 2 (t) + zt 4 (t), y = T 3 (t), z = zt 2 (t) x = T (t) + xt 2 (t), y = ϕ(z)t 2 (t) + T 3 (t) + zt 4 (t), x = T (t) + xt 2 (t) + yt 3 (t) + zt 4 (t), y = yt 2 (t), z = zt 2 (t) x = T (t) + xt 2 (t), y = T 3 (t) + yt 4 (t), z = T 2 (t) + CT 4 (t) x = T (t) + xt 2 (t), y = zt 2 (t) + T 3 (t) + yt 4 (t), z = zt 4 (t) x = T (t) + xt 2 (t), y = T 3 (t) + yt 4 (t), x = T (t) + xt 2 (t) + yt 3 (t), y = y[t 2 (t) T 4 (t)], x = T (t) + xt 2 (t) + yt 3 (t), y = y[t 2 (t) T 4 (t)], x = T (t) + zt 3 (t), y = T 4 (t), z = T 2 (t) x = T (t) + zt 3 (t) + yt 4 (t), y = 0, z = T 2 (t) + ϕ(y)t 3 (t) x = T (t) + (x + y)t 3 (t), y = T 2 (t) + yt 3 (t), x = T (t) + (x + y)t 3 (t) + ze z T 4 (t), y = T 2 (t) + yt 3 (t) + e z T 4 (t), x = T (t) + (x + y)t 3 (t) + e z T 4 (t), y = T 2 (t) + yt 3 (t),

8 Three-dimensional dynamical systems 879 Table 2 (Continued) VI VII VIII x, y x, x x y, z x, y x,x x y,ze y x x, y x, x x y, e y x x, y, x x + y y, x, y, x x e z x x, x, x, z + y y + z, y, x x + z, ϕ(y) z y, x x + z, ez x y, x x + ay y, z x, y, x x + ay y + z, e z x + eaz y x, y, x x + ay y + z, e z x x, y x, x x z x, y x, x x z y a x x, y x, x x y a x x, y,x x e az y + ( a)y y, + ( a)y y, + ( a)y y, + ay y + z, x,, (bx + y) y x + (by x) y, z x,, (bx + y) y x + (by x) y + z, ebz (cos(z) x sin(z) y ) x, y, (b y)x x x ( + y 2 ) y, z x = T (t) + yt 2 (t) + xt 3 (t), y = T 2 (t), x = T (t) + yt 2 (t) + xt 3 (t) + ze y T 4 (t), y = T 2 (t), x = T (t) + yt 2 (t) + xt 3 (t) + e y T 4 (t), y = T 2 (t), x = T (t) + xt 3 (t), y = T 2 (t) + yt 3 (t), x = T (t) + xt 3 (t) + e z T 4 (t), y = T 2 (t) + yt 3 (t), x = T (t) + xt 3 (t), y = T 2 (t), + ϕ(y)t 4 (t) x = T (t) + xt 3 (t) + e z T 4 (t), y = T 2 (t), x = T (t) + xt 3 (t), y = T 2 (t) + ayt 3 (t), x = T (t) + xt 3 (t) + e z T 4 (t), y = T 2 (t) + ayt 3 (t) + e az T 4 (t), x = T (t) + xt 3 (t) + e z T 4 (t), y = T 2 (t) + ayt 3 (t), x = T (t) + yt 2 (t) + xt 3 (t), y = ( a)yt 3 (t), x = T (t) + yt 2 (t) + xt 3 (t) + z y a T 4 (t), y = ( a)yt 3 (t), x = T (t) + yt 2 (t) + xt 3 (t) + z y a T 4 (t), y = ( a)yt 3 (t), x = T (t) + xt 3 (t) + e z T 4 (t), y = T 2 (t) + ayt 3 (t) + e az T 4 (t), x = T (t) + (bx + y)t 3 (t), y = T 2 (t) + (by x)t 3 (t), x = T (t) + (bx + y)t 3 (t) + e bz cos(z)t 4 (t), y = T 2 (t) + (by x)t 3 (t) e bz sin(z)t 4 (t), x = T (t) + yt 2 (t) + (b y)xt 3 (t), y = ( + y 2 )T 3 (t),

9 880 N. H. Ibragimov, A. A. Gainetdinova Table 2 (Continued) 35 x, y x, x = T (t) + yt 2 (t) + (b y)xt 3 (t) (b y)x x ( + y2 ) y, +z y 2 exp( b arctan(y))t 4 (t), z + y 2 exp( b arctan(y)) y = ( + y 2 )T 3 (t), x 36 x, y x, x = T (t) + yt 2 (t) + (b y)xt 3 (t) (b y)x x ( + y2 ) y, + y 2 exp( b arctan(y))t 4 (t), + y 2 exp( b arctan(y)) y = ( + y 2 )T 3 (t), x x = T (t) + yt 4 (t), XI 37 x, y, z, y x + z y y = T 2 (t) + zt 4 (t), XII XIII x, y, 2 z2 x + z y, y x z x, y x, z, yz x y x, y x, z x, y y z x, y x, 2 y2 x, y x, bx x x, z x, y y, z, + (y + z) y + z z y, z y, bx x + y y x, (b ) y y + z z, (bx + yz) z x + x, y x, z x, bx x + (b )y y + ((b )z y) z x, y, e( b)z x bx x + y y z + z y, x, y x, y ln y b x, bx x + (b )y y x, y, z, x x + z y x, y, εe z x +z y, x x z x, y x, y y, (x + yz) z x + x, y x, z x, x x + y y + (z y) z x = T (t) 2 z2 T 3 (t) + yt 4 (t), y = T 2 (t) + zt 3 (t), z = T 4 (t) x = T (t) + yt 2 (t) + yzt 4 (t), y = T 4 (t), x = T (t) + yt 2 (t) + zt 4 (t), y = T 4 (t), z = yt 4 (t) x = T (t) + yt 2 (t) + 2 y2 T 3 (t), y = T 4 (t), x = T (t) + bxt 4 (t), y = T 2 (t) + (y + z)t 4 (t), + zt 4 (t) x = T (t) + bxt 4 (t), y = T 2 (t) + zt 3 (t) + yt 4 (t), z = T 4 (t) x = T (t) + yt 2 (t) + (bx + yz)t 4 (t), y = (b )yt 4 (t), + zt 4 (t) x = T (t) + yt 2 (t) + zt 3 (t) + bxt 4 (t), y = (b )yt 4 (t), z = ((b )z y) T 4 (t) x = T (t) + e ( b)z T 3 (t) + bxt 4 (t), y = T 2 (t) + zt 3 (t) + yt 4 (t), z = T 4 (t) x = T (t) + yt 2 (t) + y = (b )yt 4 (t), x = T (t) + xt 4 (t), y = T 2 (t) + zt 4 (t), x = T (t) + εe z T 3 (t) + xt 4 (t), y = T 2 (t) + zt 3 (t), z = T 4 (t) y b ln y T 3(t) + bxt 4 (t), x = T (t) + yt 2 (y) + (x + yz)t 4 (t), y = yt 4 (t), x = T (t) + yt 2 (t) + zt 3 (t) + xt 4 (t), y = yt 4 (t), z = (z y)t 4 (t)

10 Three-dimensional dynamical systems 88 Table 2 (Continued) 52 XIV XV XVI 68 x, y, y ln y x x, x x + y y x, y,, (x + y) z x + (y + z) y + z z x, y, 2 z2 x + z y, (x + y) x + y y z x, y x, y + z z, (x + yz) z x x, y x, z x, x x y y z x, y x, 2 y2 x, x x y x, cz z y, z, ax x + by y + x, y x, z x, ax x + (a b) y y + (a c) z z x, y, z x + ϕ(z) y, x x + y y x, x, y, ϕ(y) x, x x + z y, ϕ(y) x, x x x, y x, z, x x + cz z x, y, exp(( c)z) x, x x + y y + z y, ε exp((a )z) x, ε 2 exp((b )z y, by y + z x, y x, z, (a b) y y + z z x + ax x + ax x + x, exp((a b)y) x, exp((a )y) x, ax x + y x, y, z, ax + x (by + z) y + (bz y) z x = T (t) + yt 2 (t) y ln y T 3 (t) + xt 4 (t), y = yt 4 (t), x = T (t) + (x + y)t 4 (t), y = T 2 (t) + (y + z)t 4 (t), + zt 4 (t) x = T (t) 2 z2 T 3 (t) + (x + y)t 4 (t), y = T 2 (t) + zt 3 (t) + yt 4 (t), z = T 4 (t) x = T (t) + yt 2 (t) + (x + yz)t 4 (t), y = T 4 (t), + zt 4 (t) x = T (t) + yt 2 (t) + zt 3 (t) + xt 4 (t), y = T 4 (t), z = yt 4 (t) x = T (t) + yt 2 (t) + 2 y2 T 3 (t) + xt 4 (t), y = T 4 (t), x = T (t) + axt 4 (t), y = T 2 (t) + byt 4 (t), + czt 4 (t) x = T (t) + yt 2 (t) + zt 3 (t) + axt 4 (t), y = (a b)yt 4 (t), z = (a c)zt 4 (t) x = T (t) + zt 3 (t) + axt 4 (t), y = T 2 (t) + ϕ(z)t 3 (t) + yt 4 (t), x = T (t) + ϕ(y)t 3 (t) + xt 4 (t), y = T 2 (t), x = T (t) + ϕ(y)t 3 (t) + xt 4 (t), y = T 2 (t), x = T (t) + yt 2 (t) + xt 4 (t), y = 0, + czt 4 (t) x = T (t) + e ( c)z T 3 (t) + xt 4 (t), y = T 2 (t) + yt 4 (t), x = T (t) + ε e (a )z T 3 (t) + axt 4 (t), y = T 2 (t) + ε 2 e (b )z T 3 (t) + byt 4 (t), x = T (t) + yt 2 (t) + axt 4 (t), y = (a b)yt 4 (t), + zt 4 (t) x = T (t) + e (a b)y T 2 (t) +e (a )y T 3 (t) + axt 4 (t), y = T 4 (t), x = T (t) + axt 4 (t), y = T 2 (t) + (by + z)t 4 (t), + (bz y)t 4 (t)

11 882 N. H. Ibragimov, A. A. Gainetdinova Table 2 (Continued) x, y, 69 ε exp((b a) arctan(z)) + z 2 x + z y, (ax εy exp((b a) arctan(z)) + z 2 ) + (b z)y x y ( + z 2 ) z 70 7 XVII 72 x, y x, z x, ax x + [(a b)z y] z + [(a b)y + z] y, exp((a b)y) cos(y) x x, exp((a b)y) sin(y) x, ax x + y x, y, y + x z (, 2x + 2 z2) + (y + z) x y + x = T (t) +ε exp((b a) arctan(z)) + z 2 T 3 (t) +(ax εy exp((b a) arctan(z)) + z 2 ) T 4 (t), y = T 2 (t) + zt 3 (t) + (b z)yt 4 (t), z = ( + z 2 )T 4 (t) x = T (t) + yt 2 (t) + zt 3 (t) + axt 4 (t), y = [(a b)y + z] T 4 (t), z = [(a b)z y] T 4 (t) x = T (t) + e (a b)y cos(y)t 2 (t) e (a b)y sin(y)t 3 (t) + axt 4 (t), y = T 4 (t), x = T (t) + yt 3 (t) + ( 2x + 2 z2) T 4 (t), y = T 2 (t) + (y + z)t 4 (t), + zt 4 (t) XVIII z z x, y, y x + z y, 2x x + y y z x, y x, ( y, 2x + 2 y2) x + y y + z x, y, y x + z y, ( + b) x x + y y + ( b) z z x = T (t) + yt 3 (t) + 2xT 4 (t), y = T 2 (t) + zt 3 (t) + yt 4 (t), z = T 4 (t) x = T (t) + yt 3 (t) + ( 2x + 2 y2) T 4 (t), y = T 3 (t) + yt 4 (t), x = T (t) + yt 3 (t) + ( 2x + 2 y2) T 4 (t), y = T 3 (t) + yt 4 (t), x, y x, y (, 2x + 2 y2) x + y y x, y, y x + z, x = T (t) + yt 3 (t) + ( + b)xt 4 (t), ( + b) x x + y = y + T 2 bz (t) + yt 4 (t), z + bzt 4 (t) x = T (t) + yt 3 (t) + ( + b)xt 4 (t), y = T 2 (t) + zt 3 (t) + yt 4 (t), z = ( b)zt 4 (t) x, y, y, ( + b) x x x + y y + z x, y y x, y, y, ( + b) x x x + y, y x, z x + y y x, y x,, ( + b) x y x + by y + z x, y x,, ( + b) x y x + by y x, y, y x + z, x x + y y + C z x = T (t) + yt 3 (t) + ( + b)xt 4 (t), y = T 2 (t) + yt 4 (t), x = T (t) + yt 3 (t) + ( + b)xt 4 (t), y = T 2 (t) + yt 4 (t), x = T (t) + yt 3 (t) + zt 4 (t), y = T 2 (t) + yt 4 (t), x = T (t) + yt 2 (t) + ( + b)xt 4 (t), y = T 3 (t) + byt 4 (t), x = T (t) + yt 2 (t) + ( + b)xt 4 (t), y = T 3 (t) + byt 4 (t), x = T (t) + yt 3 (t) + xt 4 (t), y = T 2 (t) + yt 4 (t), + CT 4 (t)

12 Three-dimensional dynamical systems 883 Table 2 (Continued) x, y, y x + ( z, XIX ax + z 2 y 2) x + (ay + z) y + (az y) z XX x, y, x x + y y + z, y x x y + C z x, y, x x + z, xy x ( + y 2 ) y x, y, x x + y y, y x x y + z x, y, x x +y y, y x x y x, y, x x, xy x ( + y 2) y x = T (t) + yt 3 (t) + 2 (4ax y2 + z 2 )T 4 (t), y = T 2 (t) + (ay + z)t 4 (t), + (az y)t 4 (t) x = T (t) + xt 3 (t) + yt 4 (t), y = T 2 (t) + yt 3 (t) xt 4 (t), + CT 4 (t) x = T (t) + xt 3 (t) xyt 4 (t), y = T 2 (t) ( + y 2 )T 4 (t), x = T (t) + xt 3 (t) + yt 4 (t), y = T 2 (t) + yt 3 (t) xt 4 (t), x = T (t) + xt 3 (t) + yt 4 (t), y = T 2 (t) + yt 3 (t) xt 4 (t), x = T (t) + xt 3 (t) xyt 4 (t), y = T 2 (t) ( + y 2 )T 4 (t), Table 3. Dynamical systems with nonsolvable L 4 IX X x, x x + y y, 2xy y + y z, y ( z 2 + c ) z x2 x + x + 2yz y +, c { ; 0; } x, x x + y ( y, x 2 + y 2) x + 2xy y, x, x x + y ( y, x 2 y 2) x + 2xy y, x, x x + y y, x 2 x + 2xy y, z x, x x + y y, x 2 x + 2xy y, yz y x, x x + y y, x 2 x + 2xy y, y y x, x x, x2 x, y z z sin(x) tan(y) x cos(x) y, x, sin(x) z y cos(x) tan(y) x, sin(x) sec(y) z sin(x) tan(y) x cos(x) sin(x) y + cos(x) sec(y) z, x, z cos(x) tan(y) x y, x = T (t) + xt 2 (t) + x 2 T 3 (t) + yt 4 (t), y = yt 2 (t) + 2xyT 3 (t) + 2yzT 4 (t), z = yt 3 (t) + (z 2 + c)t 4 (t) x = T (t) + xt 2 (t) + (x 2 + y 2 )T 3 (t), y = yt 2 (t) + 2xyT 3 (t), x = T (t) + xt 2 (t) + (x 2 y 2 )T 3 (t), y = yt 2 (t) + 2xyT 3 (t), x = T (t) + xt 2 (t) + x 2 T 3 (t), y = yt 2 (t) + 2xyT 3 (t), x = T (t) + xt 2 (t) + x 2 T 3 (t), y = yt 2 (t) + 2xyT 3 (t) + yzt 4 (t), x = T (t) + xt 2 (t) + x 2 T 3 (t), y = yt 2 (t) + 2xyT 3 (t) + yt 4 (t), x = T (t) + xt 2 (t) + x 2 T 3 (t), y = T 4 (t), x = T 2 (t) sin(x) tan(y)t (t) cos(x) tan(y)t 3 (t), y = sin(x)t 3 (t) cos(x)t (t), x = T 3 (t) sin(x) tan(y)t (t) cos(x) tan(y)t 2 (t), y = sin(x)t 2 (t) cos(x)t (t), + sin(x) sec(y)t (t) + cos(x) sec(y)t 2 (t)