1. INTRODUCTION ROMAI J., 6, 1(2010), 15 28

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1 ROMAI J., 6, 1(2010), AUTOMORPHISMS AND DERIVATIONS OF HOMOGENEOUS QUADRATIC DIFFERENTIAL SYSTEMS Ilie Burdujan Department of Mathematics, University of Agricultural and Veterinary Medicine Ion Ionescu de la Brad Iaşi, Romania burdujan Abstract A binary algebra is associated with any homogeneous quadratic differential system. Derivations and automorphisms of a homogeneous quadratic differential system are just derivations and automorphisms of the algebra associated to it. The real 3-dimensional commutative algebras without nilpotent elements of order two, which have a derivation, are classified up to an isomorphism. Consequently, the corresponding homogeneous quadratic differential systems are classified up to a center-affine equivalence. Keywords: homogeneous quadratic differential system MSC: 37C INTRODUCTION The theory of quadratic differential system (shortly, QDS) is the first step toward a systematic study of nonlinear systems of differential equations. There exist almost exhaustive studies on QDSs with two unknown functions [8], [5], [13], [14]. More exactly, the classification up to an affine equivalence of these systems was achieved. In fact, several classifications were made according with various classification criteria. Among them we quote the classifications which were achieved by Markus [8] (with a completion due to Date [5]) and those due to Vulpe-Sibirschii [13] for quadratic differential systems in plane. Unfortunately, the similar studies for similar systems in R 3 creates important technical difficulties. A direction which had opened large perspectives for the study of systems of quadratic differential equations (and, more general, of systems of polynomial differential equations) was the one revealed by L. Markus [8] in Markus realized that a commutative algebra, which may be a non-associative one (in general), can be naturally associated to each homogeneous quadratic differential system (shortly, HQDS). Along this way, the homogeneous quadratic differential systems can be studied on the multidimensional spaces and on Banach spaces as well. The derivations and the automorphisms of a HQDS (i.e. of the quadratic vector form which defines it) are defined. The existence of such automorphisms entails the existence of certain symmetries that - in their turn - imply the existence of some 15

2 16 Ilie Burdujan suitable coordinates in which the system would take the simplest form (i.e., a large number of system s coefficients would be either 0 or small integers). Notice that any derivation (resp. automorphism) of a HQDS is a derivation (resp. automorphism) of its associated algebra. In the first part of this paper we present several results concerning the HQDSs on Banach spaces. They generalize the similar results that were already proved for HQDSs on R n (see, for example, [6], [7], [9], [14]). The last part of our paper deals with the classification of some HQDSs on R 3. To this end, we firstly remark that the set of important general results, concerning the derivation algebra of a real commutative algebra, is not too large. However, there exist notable progresses on this subject for the so-called NN-algebras, i.e. algebras without nilpotent elements of order 2 (see [1], [2]). Some results concerning the derivation algebra of an NN-algebra are presented and they are used for classifying the real 3-dimensional commutative NN-algebras having at least a derivation. It was proved that there exist nine multiparametric classes of nonisomorphic such NN-algebras. Accordingly, there exist nine multiparametric classes of HQDSs on R 3, whose associated algebras are NN-algebras with a derivation, that are mutually nonequivalent up to an affinity. 2. PRELIMINARIES Let E( 1 ) be a real Banach space. Definition 2.1. Any equation of the form dx dt = F(X) (1) where F : E E is a continuous quadratic vector form on E is called a homogeneous quadratic differential equation (shortly, HQDE) on E. Recall that a vector form F is quadratic if F(sX) = s 2 F(X) for all s R and X E. With any quadratic vector form F is associated its polar form which is a symmetric bilinear vector form G : E E E defined by G(X, Y) = 1 [F(X + Y) F(X) F(Y)], (2) 2 for all X, Y E. The definition of tensor product assures us that there exists a unique linear mapping µ F : E E E such that µ ι E = G (here ι E (x, y) = x y). In its turn, µ F is naturally identified with a (1,2)-tensor. Since G is symmetric, it results that µ F is a covariant symmetric (1,2)-tensor. On the other hand, G allows to define a commutative binary operation on E by x y = G(x, y) for all x, y E. (3) The obtained commutative algebra is denoted by E( ). In general, E( ) is a nonassociative algebra. Taking into account that x x = x 2 = F(x), HQDE (1) can be

3 Automorphisms and derivations of homogeneous quadratic differential systems 17 written in the form dx = X 2, (4) dt for pointing out its connection with algebra E( ). Now, let us consider another HQDE on the real Banach space E, namely dy dt = F 1 (Y), (5) where F 1 : E E is a continuous quadratic vector form. The polar form G 1 of F 1 can be identified with a covariant symmetric (1,2)-tensor on E. The binary operation, defined on E by x y = G 1 (x, y) for all x, y E, (6) organizes E as a commutative algebra denoted by E ( ). Definition 2.2. It is said that HQDE (1) is center-affinely equivalent (or, CA-equivalent) with the HQDE (5) if and only if there exists a continuously invertible linear mapping h : E E such that X = h(y) is a solution for (1) whenever Y is a solution for (5); in this case it is said that h is an equivalence of equation (1) with equation (5). An equivalence of equation (1) with itself is called an automorphism of equation (1) or an automorphism of F. Proposition 2.1. [3] HQDE (1) is CA-equivalent with the HQDE (5) if and only if there exists a continuously invertible linear mapping h : E E such that h F 1 = F h. (7) Since h 1 is continuous and the equality h 1 F = F 1 h 1 holds, it results the following assertion. Corolar 2.1. If (1) is CA-equivalent with (5), then (5) is also CA-equivalent with (1). Theorem 2.1. [3] Equations (1) and (5) are CA-equivalent if and only if the algebras E( ) and E ( ) are continuously isomorphic. The continuous linear mapping h : E E is an automorphism for (1) if and only if it is a continuous automorphism of algebra E( ). If Φ t and Ψ t denote the flows of equations (1) and (5), respectively, then the two equations are CA-equivalent if and only if there exists a continuous invertible linear mapping h : E E such that h Ψ t = Φ t h. Remark 2.1. The existence of a CA-equivalence h : E E allows to identify the spaces E and E. Then, Theorem 2.1 assures us that there exists a 1-to-1 correspondence between the set of classes of CA-equivalent HQDEs on E and the set of classes

4 18 Ilie Burdujan of isomorphic commutative algebras defined on E. Accordingly, there exists a correspondence between certain qualitative properties of a HQDE (1) and the properties invariant up to an isomorphism of the associated algebra. Remark 2.2. The binary CA-equivalence relation defined by Definition 2.2 on the set of HQDEs on a fixed Banach space is an equivalence relation (i.e. it is reflexive, symmetric and transitive). That is why, in what follows, we shall use the term equivalence instead of CA-equivalence. Theorem 2.1 assures us that there exists a correspondence between the affinely invariant properties of any HQDE and the properties invariant up to an isomorphism of the associated algebra. This correspondence is not completely identified. Some of its components are presented in the next Proposition. Proposition 2.2. The following assertions hold, for any HQDE (1) on the Banach space E: 1. the set N(E) of all nilpotents of order two of E( ) is in a bijective correspondence with the set of all steady state solutions of (1), 2. the set I(E) of all idempotents of E( ) is in a bijective correspondence with the set of ray (from or to 0) solutions of (1), 3. E( ) is a nilalgebra if and only if all solutions of (1) are polynomials, 4. E 2 = E E is a proper ideal of E if and only if (1) has at least a linear prime integral, 5. if E( ) is a power-associative algebra then all solutions of (1) are rational functions. In fact, any structural property of E( ) (i.e., a property which is invariant up to an isomorphism) enforces the existence of a property of Eq. (1) (and conversely). For example, if E( ) has no idempotent element then the zero solution of (1) is not asymptotically stable; if (1) has a Liapunov function then E has no idempotent element. NOTE. These results were already proved, in the finite dimensional case, by Kinyon&Sagle, Rörhl, Walcher (see [6], [9], [14]). 3. DERIVATIONS AND AUTOMORPHISMS OF A HQDS Definition 3.1. A derivative of F is any continuous linear transformation D : E E satisfying DF(X) = F (X) DX f or all X E, (8) where F df(x + sy) (X) Y = lim f or all X, Y E. Any derivative of the vector s 0 ds form F is also named a derivation of HQDE (1).

5 Automorphisms and derivations of homogeneous quadratic differential systems 19 Proposition 3.1. D is a derivation of HQDE (1) if and only if it is a derivation of E( ). Proof. Indeed, the equality F G(X + sy, X + sy) G(X, y) (X) Y = lim = 2G(X, Y) s 0 s holds for all X, Y E. It implies that D satisfies to Leibniz rule. We denote by Der F the set of all derivatives of F (i.e. the set of all derivations of (1)) and by Der E the so-called derivation algebra of E( ). Proposition 1.2 asserts that Der F = Der E. Consequently, Der F is a Lie subalgebra of gl(e). Definition 3.2. An automorphism of F is a continuous invertible linear transformation ϕ : E E satisfying to condition F ϕ = ϕ F. Any automorphism of F is also called an automorphism of HQDE (1). Proposition 3.2. ϕ is an automorphism of HQDE (1) if and only if it is an automorphism of E( ). We denote by Aut F the set of all automorphisms of F (i.e. the set of all automorphisms of (1)) and by Aut E the set of all automorphisms of algebra E( ). Proposition 1.3 asserts that Aut F = Aut E. Consequently, Aut F is a Lie subgroup of GL(E). Any solution of (1) is analytic. Let us consider its flow Φ t. Then, for each ϕ Aut F we have Φ t ϕ = ϕ Φ t. Some features of the complex connections between Aut E and Der E are presented in the next Proposition and Example as well. Proposition 3.3. [10] Let D be a derivation of a real non-associative algebra E( ). Then: (i) exp td for t R is a uniparametric group of automorphisms of E, (ii) the Lie algebra of Aut E is Der E. Example. Algebra having in basis B = (e 1, e 2, e 3 ) the multiplication table e 2 1 = 0 e 1 e 2 = e 1 e 1 e 3 = 0 e 2 2 = ae 2 e 2 e 3 = a 3 e 2 3 = e 2 with a R \ {2} has the derivation algebra α 0 β Der E = α, β R 0 0 0,

6 20 Ilie Burdujan and the automorphism group Aut E = α 0 β ±1 α, β R, α THE CLASSIFICATION OF REAL 3-DIMENSIONAL NN-ALGEBRAS WITH DERIVATIONS In [2], [3] it was proved that any real 3-dimensional NN-algebra A( ) has dim Der A {0, 1} and that there exist NN-algebras having a derivation. Consequently, the problem to classify, up to an isomorphism, the real 3-dimensional NNalgebras having derivations is consistent. Let A( ) be a real 3-dimensional NN-algebra with a nonzero derivation. Then there exists a derivation D with the eigenvalues {0, ±i} ([1], [2]). Accordingly, there exists a basis B = (e 1, e 2, e 3 ) such that De 1 = 0, De 2 = e 3, De 3 = e 2, e 2 2 = e2 3, e 2 e 3 = 0. In fact, the natural vector space decomposition A = ker D Im D exists. Since ker D is a 1-dimensional NN-algebra, it contains an idempotent element e 1. Then, the equalities De 2 2 = 2e 2 e 3 = 0 imply e 2 2 = e2 3 = εe 1 with ε 0. We can choose e 2 and e 3 such that ε = ±1. Supposing now that e 1 e 2 = ae 1 + be 2 + ce 3 and applying D to it, it follows e 1 e 3 = ce 2 + be 3. Applying again D to e 1 e 3 it results that e 1 e 2 = be 2 + ce 3. Therefore, the multiplication table of the algebra, in basis B, is T e 2 1 = e 1 e 1 e 2 = be 2 + ce 3 e 1 e 3 = ce 2 + be 3 e 2 2 = εe 1 e 2 3 = εe 1 e 2 e 3 = 0, where either ε = ±1, b 2 + c 2 0 or ε = 1, b = c = 0 (these conditions assure that A is an NN-algebra). We shall denote by A(b, c, ε) the algebra having, in a basis B, the multiplication table T. In order to classify, up to an isomorphism, the real 3-dimensional NN-algebras we need to identify the lattices of their subalgebras. First at all, we determine the idempotents of such algebras, because they identify all the 1-dimensional subalgebras. We denote by I(A) the set of all idempotents of A( ). The condition (xe 1 + ye 2 + ze 3 ) 2 = xe 1 + ye 2 + ze 3 is equivalent to the system x 2 + ε(y 2 + z 2 ) = x (2bx 1)y 2cxz = 0 2cxy + (2bx 1)z = 0. (9)

7 Automorphisms and derivations of homogeneous quadratic differential systems 21 Obviously, this last homogeneous system has the null solution, which is not of any interest in the problem of idempotents; since e 1 is an idempotent, the system (4) has necessarily the solution x = 1, y = z = 0. It remains to look for the solutions with x {0, 1} and y 2 + z 2 0. In this case, necessarily = 2bx 1 2cx 2cx 2bx 1 = (2bx 1)2 + 4c 2 x 2 = 0, i.e. b 0, x = 1, c = 0. This time, system (4) has solutions only when k = 2b ε 2b 1 4b 2 > 0, namely: x = 1 2b, y = k cos θ, z = k sin θ for θ [0, 2π). Therefore, the following four classes of real 3-dimensional NN-algebras arise naturally: C1. A(b, c, ε) with c 0, C2. A(b, 0, ε) with b 0, and k 0, C3. A(b, 0, ε) with b 0, and k > 0, C4. A(0, 0, 1). The following assertions are readily provable: if A belongs to class C1, then I(A) = {e 1 } and L e1 has the eigenvalues {1, b ± ic}, if A belongs to class C2, then I(A) = {e 1 } and L e1 has the eigenvalues {1, b, b}, if A belongs to class C3, then I(A) = {e 1 } {e(b) = 1 2b e 1 + k(e 2 cos θ + e 3 sin θ) θ [0, 2π), b 0}; { L e1 has the eigenvalues {1, b, b} and L e(b) has the eigenvalues 1, 1 2, 1 b }, 2b algebra A(0, 0, ε) has I(A) = {e 1 } and L e1 has the eigenvalues {1, 0, 0}. Taking into account the idempotents and their spectra it results that the four classes C1, C2, C3, C4 provide a partition of the set of all real 3-dimensional NN-algebras with a derivation. Anyone of these classes can be separately analysed for being possibly partitioned in subclasses of non-isomorphic algebras. Proposition 4.1. The set of algebras belonging to class C1 have the properties: (i) Algebras A(b, c, ε) and A(b, c, ε), (with c 0), are always isomorphic, (ii) Algebras A(b, c, ε) and A(b, c, ε), with c > 0 and c > 0, are isomorphic if and only if b = b and c = c. (iii) Any two algebras A(b, c, 1) and A(b, c, 1) with c > 0 and c > 0 are nonisomorphic.

8 22 Ilie Burdujan Proof. (i) It is enough to consider the multiplication table of A(b, c, ε) in basis B = ( f 1 = e 1, f 2 = e 2, f 3 = e 3 ). (ii) If T : A(b, c, ε) A(b, c, ε) is an isomorphism, then necessarily T(e 1 ) = f 1. As e 1 and f 1 must have the same spectrum it results b = b and c = c. (iii) Let us suppose that A(b, c, 1) has, in basis B, the multiplication table T while A(b, c, 1) has, in basis B = ( f 1, f 2, f 3 ), the multiplication table T f 2 1 = f 1 f 1 f 2 = b f 2 + c f 3 f 1 f 3 = c f 2 + b f 3 f 2 2 = f 1 f 2 3 = f 1 f 2 f 3 = 0. If T : A(b, c, ε) A(b, c, ε) is an isomorphism, then necessarily T(e 1 ) = f 1 and a = a, b = b. If T(e 2 ) = α f 1 +β f 2 +γ f 3, then T(e 2 2 ) = T(e 1) = (T(e 2 )) 2 is equivalent to α 2 β 2 γ 2 = 1 bαβ cαγ = 0 cαβ + bαγ = 0. The last two equations imply αβ = αγ = 0. Since the first equation excludes the possibility α = 0, the condition α 0 implies β = γ = 0 and T(αe 1 e 2 ) = 0 (what contradicts the injectivity of T). In a similar way it is proved the following Proposition. Proposition 4.2. The set of algebras belonging to classes C2 and C3 have the properties: (i) Algebras A(b, 0, ε) and A(b, 0, ε) with bb 0 are isomorphic if and only if b = b. (ii) Any two algebras A(b, 0, 1) and A(b, 0, 1) with bb 0 are not isomorphic. Therefore, these results give the classification up to an isomorphism of the real 3-dimensional NN-algebras which have a nonzero derivation. More exactly, it was obtained the following result. Theorem 4.1. If A( ) is a real 3-dimensional commutative NN-algebra which has a nonzero derivation, then there exists a basis in A such that its multiplication table has the form corresponding to one of the following nine classes of non-isomorphic algebras: 1 A(b, c, 1) with b, c R and c > 0, 2 A(b, c, 1) with b, c R and c > 0, 3 A(b, 0, 1) with b R, b 0 and b 1 2, 4 A(b, 0, 1) with b R \ {1} and b 1 2, 5 A(b, 0, 1) with b R \ {1} and b > 1 2, 6 A(b, 0, 1) with b R, b 0 and b < 1 2,

9 Automorphisms and derivations of homogeneous quadratic differential systems 23 7 A(1, 0, 1), 8 A(1, 0, 1), 9 A(0, 0, 1). Proposition 4.3. Every algebra A(b, c, ε), with b 2 + c 2 0, is simple. Proof. Let I A be a nonzero ideal and v = xe 1 + ye 2 + ze 3 I, v 0; then the elements v e 1 = xe 1 + (by cz)e 2 + (cy + bz)e 3, v e 2 = εye 1 + bxe 2 + cxe 3, v e 3 = εze 1 cxe 2 + bxe 3, belong to I. Consequently, if = x(b 2 + c 2 )[x 2 ε(y 2 + z 2 )] is nonzero then v e 1, v e 2 and v e 3 are linearly independent and, necessarily, I A. If x = 0, then the equalities v e 2 = εye 1, v e 3 = εze 1 and v 0 imply e 1, e 1 e 2, e 1 e 3 I, i.e. I A. In the case when x 0 and ax 2 ε(y 2 + z 2 ) = 0, the linear independence of vectors v e 2, v e 3 and (v e 2 ) e 3 = εcxe 1 + εy( ce 2 + be 3 )( I) is equivalent to condition 1 = εx(b 2 + c 2 )[cx 2 εyz] 0; similarly, the linear independence of vectors v e 2, v e 3 and (v e 3 ) e 2 = εcxe 1 + εz(be 2 + ce 3 )( I) is equivalent to condition 2 = εx(b 2 + c 2 )[cx 2 + εyz] 0. Therefore, if then I A. If = 0, then cx2 = εyz = 0 imply necessarily c = 0 and b 0. Thus, the complementary cases y = 0 and, respectively, y 0 must be considered. When y = 0 (necessarily, c = 0 and b 0), then e 2 = 1 bx v e 2 I, e 1 e 2 I and e 1 = 1 εbx (v e 2) e 2 I, e 1 e 3 I, i.e. I A. In its turn, y 0 implies z = 0, i.e. v = xe 1 + ye 2. Then e 3 = 1 bx v e 3 I, e 1 e 3 I and e 1 = 1 εbx (v e 3) e 3 I, e 1 e 3 I, i.e. I A. NOTE. If b = c = 0 then A 2 = Re 1 is a nontrivial ideal. Proposition 4.4. The algebras A(1, 0, ε) are unitary and power-associative as well. According to Theorem [3], Der A = RD. Therefore, {e td t R} is a uniparametric group of automorphisms for A( ); it is isomorphic to the matrix group cos θ sin θ θ [0, 2π). Certainly, Aut A do not contain other uni- 0 sin θ cos θ parametric subgroup of automorphisms. The problem is to establish whether other new automorphisms of A( ) exist or do not exist. The answer is contained in the next Proposition. Proposition 4.5. Aut A coincides with the matrix group cos θ sin θ θ [0, 2π) 0 sin θ cos θ.

10 24 Ilie Burdujan Recall that the trace form of algebra A( ) is the bilinear form g : A A R defined by g(v, w) = trace L v w for all v, w A. Let us consider v = x 1 e 1 + x 2 e 2 + x 3 e 3 and w = y 1 e 1 + y 2 e 2 + y 3 e 3 in A. Since L v = x 1 εx 2 εx 3 bx 2 cx 3 bx 1 cx 1 cx 2 + bx 3 cx 1 bx 1 v w = [x 1 y 1 + ε(x 2 y 2 + x 3 y 3 )]e 1 + [b(x 1 y 2 + x 2 y 1 ) c(x 1 y 3 + x 3 y 1 )]e 2 + [c(x 1 y 2 + x 2 y 1 ) + b(x 1 y 3 + x 3 y 1 )]e 3 and trace L v = (2b + 1)x 1, it results that g(v, w) = (2b + 1)[x 1 y 1 + ε(x 2 y 2 + x 3 y 3 )]. This bilinear form vanishes identically when b = 1/2 and it is nondegenerate whenever b 1/2. In what follows we try to solve the HQDSs corresponding to these NN-algebras. The before presented results assures that, for a HQDS with three unknown functions whose associated algebra is an NN-algebra with a nonzero derivation, which belongs to class C 1, there exists a change of variables (i.e., of unknown functions) such that the system becomes dx 1 = x dt ε(x2 2 + x2 3 ) dx 2 = 2bx dt 1 x 2 2cx 1 x 3 (10) dx 3 = 2cx dt 1 x 2 + 2bx 1 x 3 with ε = ±1 and b, c R and c > 0. The next equation is obtained adding the last two equations, after their multiplication with x 2 and respectively x 3,, d(x x2 3 ) dt = 4bx 1 (x2 2 + x2 3 ). (11) Using the change of unknown functions y 2 = x2 2 + x2 3, z 2 = x 2 x, the system (5) 3 becomes dx 1 = x dt εy 2 dy 2 = 4bx dt 1 y 2 (12) dz 2 = 2cx dt 1 (1 + z 2 2 ). This last system allows us to determine the following independent prime integrals for

11 Automorphisms and derivations of homogeneous quadratic differential systems 25 (5): ( f 1 (x 1, x 2, x 3 ) = ln 2b x bε ) 1 2b ln ( x 1 2b 2b ) x2, x2 2 + x2 3 f 2 (x 1, x 2, x 3 ) = c ln ( x2 2 + x 3) x2 + 2b arctg 2 x 3 if b {0, 1/2}. In the case b = 0, c > 0, ε = 1 the system (5) has the following two independent prime integrals: f 1 (x 1, x 2, x 3 ) = x2 2 + x2 3, 1 f 2 (t, x 1, x 2, x 3 ) = arctg x 1 t. x2 2 + x2 3 x2 2 + x2 3 If b = 0, c > 0, ε = 1 then the system (5) has the following two independent prime integrals: f 1 (x 1, x 2, x 3 ) = x2 2 + x2 3, 1 f 2 (t, x 1, x 2, x 3 ) = 2(x2 2 + x2 3 ) ln x 1 x2 2 + x2 3 t. x 1 + x2 2 + x2 3 In case b = 1/2, c > 0, ε = 1, system (5) has the following prime integrals: f 1 (x 1, x 2, x 3 ) = c ln (x2 2 + x2 3 ) + arctg x 2 x, 3 2 x1 f 2 (x 1, x 2, x 3 ) = (x2 2 + x2 3 ) e x2 2 + x2 3. Finally, in the case b = 1/2, c > 0, ε = 1 the system (5) has the following two independent prime integrals: f 1 (x 1, x 2, x 3 ) = c ln(x2 2 + x2 3 ) + arctg x 2 x, 3 f 2 (x 1, x 2, x 3 ) = x 2 1 x ln(x x x2 3 ). 3 The HQDS corresponding to an algebra of type C 2 or C 3 has the form dx 1 = x dt ε(x2 2 + x2 3 ) dx 2 = 2bx dt 1 x 2 dx 3 = 2bx dt 1 x 3. It has the following two independent prime integrals: (13) f 1 (x 1, x 2, x 3 ) = x 2 x 3, f 2 (x 1, x 2, x 3 ) = [ (1 2b)x ε(x2 2 + x2 3 )] x 1/b 3.

12 26 Ilie Burdujan Let us solve the Cauchy problem with the initial data x 1 (0) = α, x 2 (0) = β, x 3 (0) = γ. Considering the complex function z = x 2 + ix 3, the last two equations in (5) leads to dz dt = 2(b + ic)x 1z. A straightforward computation gives us x 2 (t) = ke 2bθ(t) cos (2cθ(t) + ϕ 0 ) x 3 (t) = ke 2bθ(t) sin (2cθ(t) + ϕ 0 ) where ϕ 0 = arctg γ β, k2 = β 2 + γ 2 and θ(t) = t 0 x 1(τ) dτ. In its turn, the equation (6) gives x x2 3 = k2 e 4bθ(t). Therefore, the first equation in (5) may be transformed into the following integrodifferential equation: dx 1 dt = x εk2 e 4bθ(t). (14) Differentiating (14), it follows d 2 x 1 dt 2 2(1 + 2b)x dx bx1 3 = 0. (15) dt In the case b 0, this nonlinear equation can be analysed by the method of Lie symmetries. System (5) can be solved in the case b = 0. Indeed, for ε = 1, the Cauchy problem, with the initial data x 1 (0) = α, x 2 (0) = β, x 3 (0) = γ, has the solution ksin kt + αcos kt x 1 (t) = k kcos kt αsin kt x 2 (t) = k cos (2cθ(t) + ϕ 0 ) x 3 (t) = k sin (2cθ(t) + ϕ 0 ) (16) where θ(t) = 1 k ln a sin kt k cos kt. In the case ε = 1, the solution is k sinh kt + α cosh kt x 1 (t) = k k cosh kt α sinh kt x 2 (t) = k cos (2cθ(t) + ϕ 0 ) x 3 (t) = k sin (2cθ(t) + ϕ 0 ), (17)

13 Automorphisms and derivations of homogeneous quadratic differential systems 27 where ( θ(t) = 1 + α ) ln kx tanh 2 1 ( 1 α ) ln kx tanh k k α k k + α ( ) 2 ln kx +( α2 k + k) k tanh 2a tanh kx k (k2 + α 2 ) ln k (k + α)(k α) k(k 2 α 2 ). According to Proposition 1.3 [7], the orbit e td (P) of P is a solution to HQDS (5) if and only if P is a nonzero solution of equation P 2 = DP. Let us find the solution of HQDS (5) that are orbits of the Lie group Aut A. If P = α e 1 + β e 2 + γ e 3, then the equation P 2 = DP becomes α 2 + ε(β 2 + γ 2 ) = 0 2bαβ 2cαγ = γ 2cαβ + 2bαγ = β. Equation (18) has a nonzero solution when ε = 1, only. Any such a solution must have α 0. Therefore, the last two equations in (18) can be written in the form { 2bα β (2cα 1)γ = 0 (2cα 1)β + 2bα γ = 0. This homogeneous system, with the unknowns β and γ, has a nonzero solution if and only if 2bα (2cα 1) 2cα 1 2bα = 0 4b2 α 2 + (2cα 1) 2 = 0. (18) Therefore, the solutions P to DP = P 2 in A( ) lie in the plane x 1 = 1 2c with center ( 1 2c, 0, 0) and radius R = 1 2c. More exactly, they are on the circle P(ω) = 1 2c e 1 + R[e 2 cos ω + e 2 sin ω]. Then, the solution which is the trajectory through P(ω) is X ω (t) = e td P(ω) = 1 2c e 1 + R[e 2 cos (t + ω) + e 3 sin (t + ω)]. It is a periodic solution of the least period 2π. Moreover, X ω (t) is isolated in the set of all periodic solutions with period 2π. Following Kinyon& Sagle [6], let us denote by S 1 (c) and S 2 (c) the sets S 1 (c) = {v trace L v = c} and S 2 (c) = {v trace L v 2 = c}, respectively. Then, the solution e td P

14 28 Ilie Burdujan is on the plane S 1 (c), where c = trace L P. Moreover, any periodic solution lies on S 1 (c) S 2 (0), what agrees with Proposition 5.23 [6]. NOTE. Equation (15) is of the form d2 x 1 dt 2 + f (x 1) dx 1 + g(x dt 1 ) = 0 which contains as particular cases van der Pol s equation and Duffing s equation, as well. We shall study it in a forthcoming paper. References [1] BENKART G., Osborn M. J., The derivation algebra of a real division algebra, Amer. J. of Math., 103, 6(1981), [2] BURDUJAN I., On derivation algebra of a real algebra without nilpotents of order two, Ital. J. of Pure and Applied Math., 8(2000), [3] BURDUJAN I., Quadratic differential systems, 2008, Ed. PIM-Iaşi. (in Romanian) [4] BURDUJAN I., Infinitesimal groups associated with quadratic dynamical systems, ROMAI Journal, 1, 1(2005), [5] DATE T., Classifications and Analysis of Two-dimensional Real Homogeneous Quadratic Differential Equation Systems, J. Diff. Eqs., 32(1979), [6] KINYON K. M., SAGLE A. A., Quadratic Dynamical Systems and Algebras, J. of Diff. Eqs, 117(1995), [7] KINYON K. M., SAGLE A. A., Automorphisms and Derivations of Differential Equations and Algebras, Rocky Mountain J. of Math., 24, 1(1994), [8] MARKUS L., Quadratic Differential Equations and Non-associative Algebras, in Contributions to the Theory of Nonlinear Oscillations, Annals of Mathematics Studies, 45, Princeton University Press, Princeton, N. Y., [9] RÖHRL H., Algebras and differential equations, Nagoya Math. J., 68(1977), [10] SAGLE, A. A., WALDE, R. E., Introduction to LIE groups and LIE algebras, Academic Press, New York and London, [11] SCHLOMIUK, DANA, Algebraic particular integrals, integrability and the problem of the center, Trans. A.M.S., 338(1979), [12] SCHLOMIUK, DANA, Basic Algebro-Geometric concepts in the study of planar polynomial vector fields, Publicacions Mathemàtiques, 41(1997), [13] VULPE, N. I.,SIBIRSKIĭ, K. S., Geometrical Classification of Quadratic differential systems, Differentialnye Uravnenje, 13, 5(1977), (in Russian) [14] WALKER, S., Algebras and differential equations, Hadronic Press, Palm Harbor, 1991.