On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

1 Facultad de Ciencias, Universidad Autónoma de Baja California, Km. 103 Carretera Tijuana-Ensenada, 22860 Ensenada, BC, Mexico 2 Grupo Alximia SA de CV, Departamento de Investigación, Ryerson 1268, Zona Centro, 22800 Ensenada, BC, Mexico 3 Departamento de Matemáticas, Universidad de Sonora, 83000 Hermosillo, SON, Mexico 4 División Multidisciplinaria de la UACJ en Cuauhtémoc, Universidad Autónoma de Ciudad Juárez, 32310 Ciudad Juárez, CHIH, Mexico


Introduction
Throughout this work, K will stand for a field, usually the real R or the complex field C.
Consider the autonomous ordinary differential equation with f : Ω ⊂ K n → K n having certain regularity conditions.

International Journal of Mathematics and Mathematical Sciences
In general the solution is not easy to obtain since this is usually a system of coupled differential equations.There is a vast literature regarding the solution of ordinary differential equations by different means and in particular by techniques utilizing generalized analytic functions, see for instance 1-14 .These include applications to the three-dimensional Stokes problem, solutions of planar elliptic vector fields with degeneracies, the Dirichlet problem, multidimensional stationary Schr ödinger equation, among others 3, 5, 6, 12, 13 .In particular, the technique that we present is of interest for people working on vector fields with singularities.For instance, in order to gain insight into the behaviour of analytic vector fields, correct visualization of vector fields in the vicinity of their singular set is required, in the case of visualization of two-dimensional complex analytic vector fields with essential singularities the usual methods only provide partial results see 15-17 , whilst the technique which we promote provides accurate and correct solutions 18, 19 .These questions arise naturally in discrete and continuous dynamical systems see 20-22 .As a first step in obtaining a solution to 1.1 , we notice that, if K n can be given the structure of a certain algebra A, it is possible to reduce this system to a single autonomous differential equation with g being a function A-differentiable with respect to the variable ζ in the algebra A.
Having done so, we proceed to show that it is possible to solve 1.2 by extending a geometric technique introduced in 18, 19 in the context of complex analytic vector fields related to Newton vector fields which are first studied by 23 .This technique is based upon the construction of two functions, which, are respectively, constant and linear on the trajectories which are the solutions of 1.2 .
The paper is organized as follows.In Section 2 the algebras in question are introduced, in particular we introduce the notion of normed, associative, commutative, and with a unit, finite dimensional algebra A over K, showing in Section 2.1 that these have a first fundamental representation into the algebra of n × n matrices over K, M n, K .Furthermore in Section 2.2 normal algebras are defined and their corresponding tensor products are constructed.This is standard material which can be found in 24-26 but is presented here for completeness.
In Section 3 we give the definition of A-differentiability, and proceed to show that if the family of matrices {Jf x : x ∈ Ω} is linearly equivalent to a subset of an algebra B in M n, K , that is, the image of the first fundamental representation of an algebra A with respect to the canonical basis of 27 , where the conditions that are given ensure the existence of an algebra A such that the set of relations are the generalized Cauchy-Riemann equations for A, 27, 28 show that these equations give a criterion for A-differentiability .Furthermore 29 considers the case when K C, where he proves that every analytic map f in the usual sense which is A-differentiable has an expansion in power series see 29, pp.646 and 653 .
In Section 4 we show that for normal algebras it is possible to express the function f x in terms of a single variable ζ in the algebra, and hence there is a function g ζ that represents f x .The analogy being the algebra of complex numbers, where z x iy is the variable and the A-differentiable functions being the analytic functions.We also show that there is a differentiable operator ∂/∂ζ which has the property that ∂ g/∂ζ g , where g is the Lorch derivative of g see 26 , hence providing a framework for the usual calculus of one variable.
In Section 5 we start by showing that in this context the differential equation 1.1 takes the form with g being a function A-differentiable with respect to the variable ζ in the algebra A, and S being a certain singular set, where the solutions are not defined.
We then proceed to show that the geometric technique, introduced in 18, 19 , of finding two functions h 1 and h 2 which are constant and linear on the trajectories ζ t which are solutions of the differential equation, can be extended to the case of 1.3 .We end the section and the paper with an example.
Definition 2.1.A K-algebra or algebra over K is a finite dimensional K-linear space A on which is defined a bilinear map A × A → A that is associative and commutative, and there is a unit element e e A in A that satisfies ex xe x for all x ∈ A.
An element a ∈ A is called regular if there exist a unique element in A denoted by a −1 ∈ A called inverse of a such that a −1 a aa −1 e.An element a ∈ A which is not regular is called singular.If a, b ∈ A and b is regular, the quotient a/b will mean ab −1 .

Algebras and Their Fundamental Representations
We define the first fundamental representation.
If B {β 1 , . . ., β n } is an ordered basis of an algebra A, the product between the elements of B is given by where c ijk ∈ K for i, j, k ∈ {1, 2, . . ., n} are called the structure constants of A. The first fundamental representation of A associated to B is the isomorphism R : A → M n, K defined by where R i is the matrix whose entry j, k is c ijk for i 1, 2, . . ., n.The commutativity and the associativity of A are equivalent to the identities

2.4
The product between the elements of B define the structure constants c ijk for i, j, k ∈ {1, 2, 3}.So, R 3 with the product given is an algebra A.

Normal Algebras and Their Tensor Products
Let B 1 , B 2 , . . ., B l be matrices B i ∈ M k i , K , each B i of one the following four types: where In this case we will say that the matrix B given by is in its normal form.We will associate to B an algebra of matrices of dimension n over K which contains B and use the following nomenclature: The first block will be called real simple block, the second real Jordan block, the third simple complex block, and the fourth complex Jordan block.
For i 1, 2, . . ., l, let σ i : M k i , K → M n, K be the linear sections defined by substituting the matrix M ∈ M k i , K in the block B i of the matrix B and taking the other entries as zero.The real Jordan block may be written in the following way B i a i D i N i , where D i is the identity and N i is a nilpotent matrix of order k i .The simple complex block may be written in the form B i a i D i b i J i , where D i is a diagonal matrix and J i is a matrix with J 2 i −D i .The complex Jordan blocks may be written in the form B i a i D i b i J i N i , where D i is the identity and J i is a matrix with We define the matrices {β i,j : 1 , and Observe that the product of the matrices β i 1 ,j 1 and β i 2 ,j 2 is the zero matrix if i 1 / i 2 .The products of β i,j 1 and β i,j 2 for i 1, . . ., l are as follows.
ii If B i is a real Jordan block, then iii If B i is a complex simple block, then 2.9 International Journal of Mathematics and Mathematical Sciences iv If B i is a complex Jordan block, then , where β i,j 0 when j ≥ 2k i 1.
The commutativity of the elements in the set with respect to the matrix product, follows from the well-known result: For a Jordan canonical form D N the diagonal matrix D commutes with the nilpotent matrix N.Moreover, the K-linear space spanned by B is an K-algebra, as is claimed in the following proposition.
Proposition 2.3.The set B : {β 1,1 , . . ., β 1,k 1 , . . ., β l,1 , . . ., β l,k l } is a base for an n-dimensional linear space which is an algebra A with respect to the matrix product, and its first fundamental representation R with respect to B is the identity isomorphism.
Proof.Let R be the first fundamental representation of A associated to B. As the algebra A is generated by β i,1 , β i,2 considered only when this exists, i.e., k i ≥ 2 , and β i,3 considered only in the case when B i is a Jordan complex block , in order to prove that R is the identity, we only need to prove that R β i,j β i,j , but for these three cases the equality is trivial.So, R is the identity isomorphism.Definition 2.4.Given a matrix B ∈ M n, K in its normal form, one will call the algebra in M n, K , as constructed above, an K-normal algebra (containing B).Definition 2.5.Two matrix algebras A 1 and A 2 in M n, K are linearly equivalent if there exists an invertible matrix For a proof of the following result, see 30 .

Proposition 2.6. Let A and B be algebras. There exists a product in
where a 1 a 2 and b 1 b 2 denote the products in A and B, respectively.The product is associative and e A ⊗ e B e A⊗B .
Therefore, the finite tensor product of algebras is an algebra.
Definition 2.7.The complexification of an R-algebra A is the C-algebra A ⊗ C. One calls an algebra which is the complexification of an R-normal algebra a C-normal algebra.
As usual, if the context is clear, one will drop the C from the name and refer to the C-normal algebra just as a normal algebra.The following proposition and its corollary follow from a straightforward calculation.Proposition 2.8.Let A and B be pand q-dimensional matrix algebras in M n, K and M m, K , respectively, and P ∈ M n, K and Q ∈ M m, K be invertible matrices.Then, one has

2.13
Corollary 2.9.The tensor product of matrix algebras linearly equivalent to normal algebras is algebras which are linearly equivalent to the tensor product of normal algebras.
So, by Corollary 2.9 the algebras linearly equivalent to the tensor product of normal algebras are closed under the tensor product.
The following result shows that the first fundamental representation, with respect to an appropriate base, of a tensor product of normal algebras is the inclusion of the algebra in the corresponding matrix space.Proposition 2.10.Let A and B be pand q-dimensional K-algebras, and R 1 : A → M p, K and R 2 : B → M q, K be first fundamental representations associated to the basis A {α 1 , . . ., α p } and B {β 1 , . . ., β q }, respectively.Then, R : Proof.Let {α 1 , . . ., α p } be a base of A and let {β 1 , . . ., β q } be a base of B. We use the notations C i and G j for the matrices R 1 α i and R 2 β j , respectively, for 1 ≤ i ≤ p, 1 ≤ j ≤ q.The set 14 is a base for A ⊗ B see 30 .In order to find the structure constants of A ⊗ B we take the products

2.15
from which we obtain a first fundamental representation R of A ⊗ B, where R α i ⊗ β l is the matrix H il whose entry kl, js is given by h il,kt,js : c ijk d lst , where c ijk and d lst are the entries k, j and t, s of C i and G l for 1 ≤ i ≤ p and 1 ≤ l ≤ q, respectively.

International Journal of Mathematics and Mathematical Sciences
On the other hand we have that R The tensor product of C i and G l is given by from which we see that in the position kt, js the element c ijk d lst appears.Thus, we have the equalities of matrices Corollary 2.11.If A is a matrix algebra in M n, K which is a tensor product of K-normal algebras, then there exists a base B of A in which the corresponding first fundamental representation R : A → M n, K is the identity isomorphism, that is, R x x for all x ∈ A.
Proof.We have that A , where B i is a K-normal algebra for every i ∈ {1, . . ., m}.Obviously, for every i ∈ {1, . . ., m} we can consider a base for B i as that given in Proposition 2.3, and taking the corresponding tensor products of these basis we obtain a base B for A. By Propositions 2.3 and 2.10 we have that the first fundamental representation of A associated to B is the identity.

Differentiability on Algebras
In a paper published in 1893, Sheffers laid a foundation for a theory of analytic functions on algebras, see 27, 29 and references therein.
Differentiability on algebras is a stronger concept than the usual differentiability over in the open set U, we denote by Jf x its Jacobian matrix at the point x, in the standard base of K n .We also use the notation J B f x for the Jacobian matrix at the point x of f with respect to a base B.

International Journal of Mathematics and Mathematical Sciences 9
The following definition was introduced in 26 .Definition 3.1.Let A be an algebra and let f : Ω ⊂ A → A be a map defined in the open set Ω.One says that f is A-differentiable at x 0 ∈ A if there exists an element f x 0 ∈ A, which one calls the A-derivative of f at x 0 , satisfying lim where f x 0 h denotes the product in A of f x 0 with h.If f is A-differentiable at all the points of Ω, one says that f is A-differentiable on Ω and one calls the map f assigning f x to the point x ∈ Ω the A-derivative of f, or Lorch derivative of f.
It follows see, e.g., 26-29 that a map f : Ω ⊂ A → A is A-differentiable at x 0 if and only if J B f x 0 ∈ R A and is continuous as a function of x 0 , where B : {e 1 , . . ., e n } is a base of A and R : A → M n, K is the first fundamental representation of the algebra A associated to B.
In fact, in this case, where R i R e i and u i : Ω → K for i 1, 2, . . ., n.
If Ω : R Ω , B : R A , and g : Ω ⊂ B → B is defined by then g is B-differentiable and its differential at y is given by g y n i 1 u i R −1 y R i , thus, the relation between the Jacobian matrix of f and the B-differential of g is J B f x g R x .The matrix equation 3.

International Journal of Mathematics and Mathematical Sciences
Suppose we consider a basis A {α 1 , . . ., α n } of A, where α i n j 1 s ji e j for i 1, . . ., n, s ji ∈ K, it can be proved that J A f S −1 Jf S, where S s ij .If we denote by S i the image of α i under the first fundamental representation of A associated to A, we have for i 1, . . ., n, that S i S −1 n j 1 s ji R j S.
Remark 3.3.For the differentiation of algebras the usual properties of differentiation of functions from R n to R m remain true.Furthermore, the usual rules of differentiation of functions of one variable are satisfied in the case of algebras, therefore polynomial functions, rational functions, and those expressed by means of convergent power series as the exponential, trigonometric, and other usual functions are differentiable in algebras.
The following theorem gives conditions that ensure the existence of an algebra A in which f is A-differentiable.

differentiable for an algebra A if and only if the set of matrices
is a subset of an algebra B which is linearly equivalent to an algebra T which is a finite tensor product of normal algebras in M n, K .Moreover, A is a K-linear space K n and has a base B such that the image of the first fundamental representation of A associated to B is R A T.
Proof.Let A {α 1 , . . ., α n } be a base for T as given in Corollary 2.11.Then B {Bα 1 B −1 , . . ., Bα n B −1 } is a base for B, where B is a matrix such that B BTB −1 .Because Jf x ∈ B for every x ∈ Ω, we have Jf x n i 1 u i x β i , where u i : Ω → K are functions and β i Bα i B −1 .Now consider the base G {γ 1 . . ., γ n } of K n defined by γ i n j 1 b ji e j , where B b ij and {e 1 , . . ., e n } is the standard base of K n .Then, we have that in other words J G f x ∈ T, which means that if we define a product between the elements of G using the structure constants of the products of the elements of A, we have that K n is an algebra A such that its first fundamental representation associated to G is that given by R γ i α i for i 1, . . ., n. Therefore f is A-differentiable.

Reduction to a Variable in the Algebra
Sometimes f f 1 , . . ., f n : Ω ⊂ K n → K n can be expressed as a function of a variable ζ n i 1 x i e i of an algebra A whose image under the first fundamental representation is the tensor product of normal algebras as in the previous sections .In this section we show some necessary conditions for this to be true and also allow for the expression of f in terms of ζ.

. , δ n } of invertible elements of the algebra A such that the following diagram commutes
where A is the matrix associated to the change of basis E → D, where E {e 1 , . . ., e n } is the canonical basis of K n .
Proof.By a change of basis one has the following commutative diagram where B is the matrix associated to the change of basis from E to B, which was used in the previous theorem, and g B • f • B −1 .Now consider a base D {δ 1 , . . ., δ n } of regular elements of A, where without loss of generality δ 1 e.Then if A is the matrix associated to the change of basis from

4.3
Then we have and since Furthermore one can define the differential operators

4.9
If and since 4.13 In the following example we show how we can reduce the variables of a map by substituting for variables in an algebra.
and I is the identity, the multiplication in the associated algebra R A is given by multiplication of matrices I, N, and N 2 representing, respectively, R e we see that the matrix associated to the change of basis from E to D is

4.20
Then so by substituting this in g and simplifying we obtain the function in the variable of the algebra.

Solving Systems of Ordinary Differential Equations by Reduction to One Variable on an Algebra
By following 18, 19 , we obtain, as direct corollaries of Theorem 3.4, Proposition 4.1, and Theorem 4.2, the following results.
Corollary 5.1.Let be as in Theorem 4.2.Then there exists Φ : Ω ⊂ A → A which is A-differentiable on Ω \ S, where S is a singular set, such that Proof.We need to show that there exists Φ : Remark 5.3.As a special case one notices that if A C then 1/ g will be a complex analytic function on Ω \ S hence S consists of isolated points the isolated singularities of 1/ g .This has been studied in 18, 19 .are the level curves of h x, y .These are presented in Figure 1.In this case we have plotted level bands of h x, y , so that the trajectories are in fact the boundaries between the colored bands.We have also plotted, in white, the trajectory that passes through a singular point corresponding to a pole of the complex analytic function g z .To exemplify the parametrization, we have plotted in black the points corresponding to t 0, 1, 2, 3 see Table 1 .
In order to parametrize the solution we need to calculate h 2 x, y which turns out to be

5.27
According to Corollary 5.8 we proceeded to calculate the intersection the level curves h x, y −1.68195 and h 2 x, y 2.26841 − t for t 0, 1, 2, 3, 4. The results are shown in Table 1 and as black dots in Figure 1.

Theorem 4 . 2 .
Let f : Ω ⊂ K n → K n be a A-differentiable map over an algebra A with first fundamental representation R A , then g A • f • A −1 can be expressed in a variable ζ n j 1 y j δ j , for some regular basis D {δ 1 , . . ., δ n }.Moreover there is a partial differential operator ∂/∂ζ such that g ζ ∂ g/∂ζ ζ .Proof.By Proposition 4.1 there is a basis D {δ 1 , . . ., δ n } of regular elements of the algebra A, with δ 1 e, without loss of generality we may assume that ||δ j || 1. Introducing the variable ζ in the base D as ζ n i 1 y i δ i n i 1 x i e i , we then have f x 1 , . . ., x n g ζ , with g A • f • A −1 , where A is the matrix associated to the change of basis E to D. Denote by ζ k the k-conjugate of ζ related to D, which is defined for k 2, . . ., n by

Remark 5 . 4 .
In case that S / Ω , then Ω \ S is an open dense set in Ω .This is true since the set A * is an open dense set in A hence for a continuous g one has that g −1 A * is an open dense subset of Ω .Remark 5.5.By the previous remark, in case that S / Ω , then Φ ζ exists for ζ ∈ Ω \ S even though it might be a multivalued function defined on each component of ζ ∈ Ω \ S.

Figure 1 :
Figure 1: Visualization of the trajectories of 5.26 .In this case we have plotted level bands of h x, y , so that the trajectories are in fact the boundaries between the colored bands.We have also plotted, in white, the trajectory that passes through a singular point corresponding to a pole of the complex analytic function g z .To exemplify the parametrization, we have plotted in black the points corresponding to t 0, 1, 2, 3 see Table1.
we assign to A the norm induced from the operator norm in M n, K see 24 .In this way each algebra is a normed algebra, that is, there exists a norm • : A → R satisfying xy ≤ x y for all x, y ∈ A and e 1. 2 y, e 3 z e 1 u, e 2 v, e 3 w xue 1 e 1 xve 1 e 2 xwe 1 e 3 yue 1 e 2 yve 2 e 2 ywe 2 e 3 zue 1 e 3 zve 2 e 3 xwe 3 e 3 xu − 2yv − yw − zv − zw e 1 xv − yu − zyv − yw zv e 2 zw zu yv yw zv zw e 3 .
2 is equivalent to the n 2 equations Remark 3.2.It should be noted that in the context of algebras equations 3.5 play the same role as the Cauchy-Riemann equations in the case of one complex variable and thus serve as a criterion for analyticity, see27, 28, 31-33 .
i does not depend on the conjugated variables.In this way we obtain an expression g ζ depending only on ζ, and not on ζ k for k 2, . . ., n.
12 so the right hand expression of g ζ n i 1 g i ζ δ 1 , R e 2 , and R e 3 , where E {e 1 , e 2 , e 3 } is the canonical basis of R 3 .It is easy to see that e 2 and e 3 do not have inverse in the algebra A. Consider the basis of regular elements D {δ 1 e 1 , δ 2 e 1 e 2 , δ 3 e 1 e 3 }, 4.16 δ 1 y 2 δ 2 y 3 δ 3 4.19 International Journal of Mathematics and Mathematical Sciences is the variable in the algebra A. Now consider the conjugates ζ 2 y 1 δ 1 − y 2 δ 2 y 3 δ 3 , ζ 3 y 1 δ 1 y 2 δ 2 − y 3 δ 3 .
On the other hand, if g ζ is the polynomial function g ζ c n ζ n • • • c 1 ζ c 0 with c n singular, then g ζ may be regular for all ζ ∈ A, see 26, p. 418 , in which case S ∅.
* being the regular elements of A.Remark 5.2.Note that S can be many things.For instance if g ζ cζ with c / ∈ A * , then S A.
Remark 5.6.Further characterization and properties of S will be studied elsewhere.In what follows we assume that S / Ω .Let f : Ω ⊂ R n → R n be A-differentiable in an algebra with first fundamental representation R A .Then the solutions to the differential equation, Moreover one can find explicitly the point x t 1 , for t 1 ∈ R, as the intersection of the level curve h 1 x h 1 x 0 and the hypersurface h 2 x h 2 x 0 − t 1 t 0 .×48− 96x − 24x 2 − 16x 3 − 9x 4 2x 5 2 −3 2x 4 x 2 y 2 3 2x y 4 .Jf A −1 x, y • A −1 belongs to the normal algebra C, so the previous results are available.Note that in R 2 one has Π x, y exp iArg x, y hence the level curves of Π • are in correspondence with the level curves of Arg • .So constant of motion h associated to g z and the one associated to f x, y is the result follows immediately by applying Π • and log • to 5.12 .Corollary 5.8.In particular, to visualize the trajectory that passes through the point x 0 ∈ Ω\A −1 S at time t 0 ∈ R, one needs only to plot the level curve {x ∈ Ω : h 1 x h 1 x 0 }.J g x, y A• Jf A −1 x, y • A −1 ∈ R C ,