Algebrization of Nonautonomous Differential Equations

Abstract

Given a planar system of nonautonomous ordinary differential equations, , conditions are given for the existence of an associative commutative unital algebra with unit and a function on an open set such that and the maps and are Lorch differentiable with respect to for all , where and represent variables in . Under these conditions the solutions of the differential equation over define solutions of the planar system.

1. Introduction

The theory of analytic functions on algebras is based on Lorch analyticity; see [1–5]. Results of classical function theory have been extended to finite dimensional associative commutative unital algebras:(i)The Cauchy integral theorem is satisfied for analytical functions in algebras, and the Cauchy integral formula has an analogous version in algebras.(ii)The classical theorems on Taylor power series are easily established, and Laurent expansion may be defined in several disjoint regions in each one of which it may define a different analytic function.(iii)Analyticity of functions of variables in algebras is characterized by the generalized Cauchy-Riemann equations, which is a set of first-order linear partial differential equations.

This theory allows us to consider differential equations over algebras, which can be used to solve family planar systems having the form

For this work and hereinafter any algebra will be assumed to be associative, commutative, and unital with unit , and will denote the linear space endowed with an algebra structure.

In this paper a planar vector field is said to be -algebrizable or -differentiable if there exists an algebra for the which is Lorch differentiable (see Section 2 for definitions). In the same way, we say that a planar autonomous system of ordinary differential equations is algebrizable if is -algebrizable.

Definition 1. Let be an algebra. We say that a function defined in an open set has an -differentiable lifting if is a function defined in an open set such that (i) the maps and are -differentiable functions with respect to for all , where and represent variables in , (ii) for all , and (iii) for all .

A nonautonomous differential equation over an algebra is denoted by where is a function defined in an open set . For every point , a solution to the equation through consists of an -differentiable function defined in a neighborhood of , with and -derivative with respect to satisfies for all . If has an -differentiable lifting , we say that planar system (1) is algebrizable and that (2) is an algebrization of (1). A theorem of existence and uniqueness of solutions for differential equations over algebras is proved in [6], and a technique for visualization of solutions is given in [7].

The classical differential equation has solutions . Some differential systems of autonomous differential equations can be written in this form by using variables in algebras. For example, the algebrization of the planar differential system is the differential equation over the algebra defined by the linear space endowed with the product The solutions are given by ; hence the solutions of the planar system are given by , where denotes the unit of . Using the first fundamental representation of (see Section 2) the following expression for the solution of the system is obtained:

Consider now the planar nonautonomous differential system whose algebrization is the Riccati differential equation over : where is the algebra defined by and the product . By the classical Lie methods for solving differential equations (see [8–10]), the solutions of the Riccati equation have the form The functions , where is the unit of , are solutions of the planar system, which can be obtained via the first fundamental representation of :

We consider differential systems having the form (1), where are functions defined in an open set . The aim of this work is to give a family of functions having -differentiable liftings over some algebra . When they exist, the solutions of the differential equation over define solutions of system (1) which can be obtained via the first fundamental representation of .

The paper is organized as follows. In Section 2 definitions of algebra, algebrizability of planar vector fields, and differentiability on modules over algebras, a characterization of the algebrizability of planar vector fields and the form of all the quadratic vector fields which are algebrizable, are given. In Section 3 the definition of algebrizable liftings of functions is presented. It is shown that the class of all of these functions defines an infinite dimensional algebra and the form of a family of these functions is given. In Section 4 it is proved that the solutions of planar systems (1) can be obtained from the solutions of their algebrization; a theorem containing conditions under which a planar system like (1) which is polynomial is algebrizable is given, and it is shown that the class of all the planar systems (1) having an algebrization (2) defines an infinite dimensional algebra. In Section 5 the case of quadratic systems is considered and their algebrizations are given, which are Riccati equations over algebras. The results presented in Sections 3, 4, and 5 are the main contributions of this paper.

2. Algebras and Lorch Analyticity

2.1. Algebras

Definition 2 (see [11]). An algebra is a -linear space endowed with a bilinear product , denoted by , which is associative and commutative for all , and has a unit satisfying for all

An element is called regular if there exists called inverse of such that . If is not regular, then is called singular. If and is regular, the quotient means .

In all the algebras considered in this paper it will be the case that , unless otherwise stated.

Consider an algebra . If is the standard basis of , the product between the elements of is given by , where the coefficients , , are called structure constants of . The first fundamental representation of is the injective linear homomorphism defined by , where is the matrix whose entry is , for .

2.2. Differentiability on Algebras

In this subsection the definition of Lorch differentiability is recalled, which in this paper is called -algebrizability or -differentiability to denote the dependence of the Lorch differential over an algebra .

Let be the norm on defined by (here the vector is represented as a matrix in order for the product to make sense), where is the first fundamental representation of and the Euclidean norm in . For this norm we have for all . Thus, we consider in this work that every algebra is a Banach algebra under the norm .

Definition 3. Let be an algebra and a function on an open set . We say that is -algebrizable or -differentiable on if there exists a function , called the -derivative of on , satisfyingfor all , where denotes the product in of with .

A vector field is -algebrizable on if and only if the Jacobian matrix of is contained in the first fundamental representation of ; that is, for all ; see [5]. It can be shown that the notion of -algebrizability coincides with the holomorphicity when is the complex field.

A method for determining whether a given planar vector field is algebrizable is the following. is algebrizable if and only if for some of the following three types of pairs of matrices(I), ,(II), ,(III), ,

the condition is satisfied for and for all in the domain of definition of , where denotes the usual inner product in . The algebra for each type of pair of matrices is defined by the following corresponding product table of the standard basis vectors of :(I)(II)(III)

Consider a planar autonomous system of quadratic ordinary differential equations in the variables and . If this system is algebrizable for an algebra with Type (I) product, then it can be represented by equations of the formfor some real constants . In the case of algebrizability for an algebra with Type (II) product, the system isfor some real constants , , , , , , and . For Type (III) product the system can be represented by equations of the formfor some real constants , , . Moreover, conditions on the components of vector fields can be given for constructing scalar functions , which we call algebrizante factors, such that are algebrizable vector fields. Inverse integrating factors (see [12, 13]) are constructed for these vector fields.

2.3. Differentiability on Modules over Algebras

In this subsection we give the definition of -differentiability of functions with domain in and image in .

The Cartesian product defines a normed -module with respect to the norm , . This norm satisfies(i) for all and if and only if ,(ii) for all and ,(iii) for all .

In the following definition denotes the norm given above on the -module .

Definition 4. Let be an algebra and a function, where is an open set. We say that is -differentiable on if there exists a module homeomorphism , which we call the differential homomorphism of at , which satisfies the condition We denote by . We say that is -differentiable on if is -differentiable on all the points of .

A function is -differentiable at if and only if the usual Jacobian matrix of at satisfies . The differential homomorphism is represented by a matrix in with respect to the standard basis of , where is the -module of all the matrices of one row and two columns with entries in ; see [14].

3. On Algebrizable Liftings of Functions

In this section are considered functions defined in open intervals , and conditions for the existence of algebras and -algebrizable functions such that will be determined, where is the unit of .

Definition 5. Let be a function defined in an open interval and an algebra with unit . We will say that is an -algebrizable lifting of if (a) is an open set on which is -algebrizable, (b) , and (c) for all .

As a consequence of the following proposition, the family of all the functions having algebrizable liftings is an infinite dimensional algebra.

Proposition 6. Let be an algebra and let be functions, where is an interval. (a)Every constant function admits the -algebrizable lifting .(b) admits the -algebrizable lifting , where is the unit of .(c)If and admit -algebrizable liftings and with respect to , respectively, and are constants in , then and (all products with respect to ) admit algebrizable liftings and , respectively.(d)If has an -algebrizable lifting and is an -algebrizable function with , then is an algebrizable lifting of .

Proof. Identity and constant functions are -differentiable for any algebra . Thus, (a) and (b) hold.
Let and be functions with -algebrizable liftings being and constants. The functions and are -algebrizable and satisfy Therefore and are algebrizable liftings of and , respectively.
If has an -algebrizable lifting and is an -differentiable function with , then . Therefore is an -algebrizable lifting of .

Corollary 7. Let be an algebra. Then the following functions admit -algebrizable liftings: polynomial functions, rational functions, trigonometric functions, exponential functions, and all of those functions which can be defined by linear combinations, products, quotients, and compositions of functions admitting algebrizable liftings.

Every function with polynomial components and has an -algebrizable lifting. The following proposition gives a wider class of these functions.

Proposition 8. Let be an algebra. Any function with components of the form , has an -algebrizable lifting. An -algebrizable lifting is given by

Proof. Consider given by the above expression; then holds for , so , where is the unit of . Therefore, is an -algebrizable lifting of .

In particular, every function with quadratic components and admits the -algebrizable lifting .

4. On Algebrizable Liftings of Planar Systems

Solutions of every algebrizable planar system can be found by solving an algebrization of the system, as it is seen in the following proposition.

Proposition 9. If (2) is an algebrization of (1) and a solution of (2), then is a solution of (1), where denotes the unit of the corresponding algebra.

Proof. Let be a solution of (2). The derivative of with respect to is given by Thus, is a solution of system (1).

As a consequence of the following proposition, the family of all the functions having algebrizable liftings defines an infinite dimensional algebra.

Proposition 10. Let be an algebra with unit . In the following statements and denote functions defined on open sets and they have values in . (a) admits the -differentiable lifting .(b) admits the -differentiable lifting .(c) admits the -differentiable lifting .(d)If and admit -differentiable liftings and , respectively, and are constants in , then and (all products with respect to ) have -differentiable liftings and , respectively.(e)Every function having the form , where and admit -differentiable liftings and , admits an -differentiable lifting given by .(f)Every function having the form , where has an -differentiable lifting and is an -differentiable function, admits an -differentiable lifting .

Proof. The proofs of (a), (b), and (c) are trivial. Let and be the algebrizable liftings of and and then , , and are -algebrizable and (i),(ii),(iii),(iv).Thus, the proof is complete.

Corollary 11. Let be an algebra. The following functions admit -differentiable liftings: polynomial functions, rational functions, trigonometric functions, exponential functions, and all of those functions which can be defined by linear combinations, products, quotients, and compositions of functions admitting -differentiable liftings.

Given a function , where are polynomial functions of the variables , the goal of the paper is to determine if has an -differentiable lifting. As a consequence of the following theorem, every function which is polynomial of the variables , , and , has an -differentiable lifting when is -differentiable for all .

Theorem 12. Let be an algebra with unit and , , where , , and , are homogeneous polynomials of degree in the variables , and for an open interval . Then the following statements are equivalent. (a) has an -differentiable lifting .(b)The map is -algebrizable for all and the functions given by have -algebrizable liftings, for .(c) is an -differentiable lifting of , where are -algebrizable liftings of .

Proof. Obviously (a) implies (b) and (c) implies (a). We now show that (b) implies (c). Suppose that and are homogenous polynomials and of the variables defined on the set as above. Taking the partial derivatives of with respect to and we obtain Let be the first fundamental representation of . Since is -algebrizable for all , then for all . Thus for all and thenis in for all . If and , then Following the same idea, for If is -algebrizable for an algebra with Type (I) product (given in Section 2), then or equivalently for . Thus, the functions are determined by for . Since , then .
Therefore, has an -differentiable lifting , where is an -algebrizable lifting of .
If is -algebrizable for an algebra with Type (II) product (given in Section 2), then or equivalently for . Thus, and is determined by , for . Since , then .
Therefore, has an -differentiable lifting , where is an -algebrizable lifting of .
If is -algebrizable for an algebra with Type (III) product (given in Section 2), then and ; that is, , , for . Thus, and is determined by , for . Since , then .
Therefore, has an -differentiable lifting , where is an -algebrizable lifting of .
Thus, if and are homogenous polynomial of degree in the variables and , then in each of the cases of algebras defined by products of Types (I), (II), and (III) given in Section 2. Since a polynomial function in the variables and can be seen as the finite addition of homogenous polynomial in the variables and , by (d) of Proposition 10, (b) implies (c).

Example 13. Consider the planar system (6); then The Jacobian is given by Thus, is orthogonal to the matrices for all ; that is, for . The map is -algebrizable for an algebra of Type (I) with constants and ; see Section 2. The function can be written as Thus, the functions and of Theorem 12 are given by , , , , , and .
The unit of is and then , , and have -algebrizable liftings , , and , respectively.
From the form of , must be

5. The Case of Second-Degree Polynomials in the Variables , , and

If and in (1) are quadratic polynomials in three variables , , and and is an algebra with respect to which the map is -algebrizable, it will be showed that the -differentiable lifting of is a polynomial in two variables of . Under these conditions (1) has an algebrization which is a Riccati equation over having the form where , , and are polynomials over of degrees two, one, and zero, respectively.

Consider system (1) where are second-degree polynomials of three variables , , and ; that is,

All the quadratic vector fields which are algebrizable with respect to algebras with Type (I) products have the form where are real constants; see Section 2.2. The algebrizability of with respect to algebras with Type (I) products can be verified, by considering as a constant.

The following theorems give conditions that characterize the algebrizability of planar systems like (1) when and are quadratic polynomials. The algebrizability of nonautonomous quadratic systems with respect to algebras with Type (I) products is given in the following theorem.

Theorem 14. Let be an algebra with Type (I) product defined by constants and and the polynomials (35). The following statements are equivalent. (1)The map is -algebrizable.(2)The functions and have the form (3)The function of the variables and , defined by is an -differentiable lifting of , where

Proof. Writing yields The function is -algebrizable if and only if where and . Thus, statements and are equivalent.
The function is polynomial in the variables and of ; hence is -differentiable. satisfies , where is the unit of . So, is an -differentiable lifting of . Thus, statement implies statement .
Since is -differentiable and , then is -algebrizable for all . Thus, statement implies statement .

All the quadratic vector fields which are algebrizable with respect to algebras with Type (II) products have the form where are real constants; see Section 2.2. The algebrizability of with respect to algebras with Type (II) products can be verified, by considering as a constant.

The algebrizability of nonautonomous quadratic systems with respect to algebras with Type (II) products is given in the following theorem.

Theorem 15. Let be an algebra with Type (II) product defined by the constant and let be the polynomials (35). The following statements are equivalent. (1)The map is -algebrizable.(2)The functions and are given by (3)The function of the variables and , defined by is an -differentiable lifting of , where

Proof. Writing yields and the map is -algebrizable if and only if where . Thus, statements and are equivalent.
The rest of the proof is similar to that of Theorem 14.

All the quadratic vector fields which are differentiable with respect to algebras with Type (III) products have the form where are real constants; see Section 2.2. The algebrizability of with respect to algebras with Type (III) products can be verified, by considering as a constant.

The algebrizability of nonautonomous quadratic systems with respect to algebras with Type (III) products is given in the following theorem.

Theorem 16. Let be an algebra with Type (III) product defined by the constant and let be the polynomials (35). The following statements are equivalent. (1)The map is -algebrizable.(2)The functions and are given by (3)The function of the variables and , defined by is an -differentiable lifting of , where , , , , , and .

Proof. Function is -algebrizable if and only if Thus, first and second statements are equivalents. The function given by , where , , , , , , , and , satisfies . Thus, second and third statements are equivalents.

By Theorems 14, 15, and 16, an algebrization of quadratic systems is a Riccati equation over an algebra , where , , and . In the following example is given a nonautonomous quadratic system for the which an algebrization is found by using Theorem 14.

Example 17. Consider system (1) given by The matrices satisfy for . Thus, is -algebrizable for an algebra with Type (I) product defined by the constants and .
The conditions of Theorem 14 are satisfied; then the -differentiable lifting of can be written as where , , , , , and .

Disclosure

The authors declare having no financial affiliation with any organization regarding the material discussed here.

Conflict of Interests

The authors declare that there is no conflict of interests concerning this text. The issues discussed in this paper do not have any secondary interest for any of the authors.

Acknowledgments

The authors wish to acknowledge the support of Grants Promep/103.5/13/ and CB-2010/150532 Conacyt.