# Algebrization of Nonautonomous Differential Equations

Authors: María Aracelia Alcorta-García; Martín Eduardo Frías-Armenta; María Esther Grimaldo-Reyna; Elifalet López-González
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

Given a planar system of nonautonomous ordinary differential equations,dw/dt=F(t,w), conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w)=H(te,w) and the maps H1(τ)=H(τ,ξ) and H2(ξ)=H(τ,ξ) are Lorch differentiable with respect to A for all (τ,ξ)∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ) of the differential equation dξ/dτ=H(τ,ξ) over A define solutions (x(t),y(t))=ξ(te) of the planar system.


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## Body

## 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(1)dxdt=ft,x,y,dydt=gt,x,y.For this work and hereinafter any algebra will be assumed to be associative, commutative, and unital with unite, and A will denote the linear space R2 endowed with an algebra structure.In this paper a planar vector fieldF is said to be A-algebrizable or A-differentiable if there exists an algebra A for the which F is Lorch differentiable (see Section 2 for definitions). In the same way, we say that a planar autonomous system of ordinary differential equations dw/dt=F(w) isalgebrizable if F is A-algebrizable.Definition 1.
LetA be an algebra. We say that a function F:U⊂R3→R2 defined in an open set U has an H(A)-differentiable lifting H:Ω⊂R2×R2→R2 if H is a function defined in an open set Ω such that (i) the maps H1(τ)=H(τ,b) and H2(ξ)=H(a,ξ) are A-differentiable functions with respect to A for all (a,b)∈Ω, where τ and ξ represent variables in A, (ii) (te,x,y)∈Ω for all (t,x,y)∈U, and (iii) F(t,x,y)=H(te,x,y) for all (t,x,y)∈U.A nonautonomousdifferential equation over an algebraA is denoted by (2)dξdτ=Hτ,ξ,where H:Ω⊂A×A→A is a function defined in an open set Ω. For every point (τ0,ξ0)∈Ω, a solution to the equation through (τ0,ξ0) consists of an A-differentiable function ξ:N(τ0)⊂A→A defined in a neighborhood N(τ0) of τ0, with ξ(τ0)=ξ0 and A-derivativedξ/dτ with respect to A satisfies dξ(τ)/dτ=H(τ,ξ(τ)) for all τ∈N(τ0). If F=(f,g) has an H(A)-differentiable lifting H, we say that planar system (1) isalgebrizable and that (2) is analgebrization 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 equationdw/dt=w2 has solutions w(t)=-(t+c)-1. 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 (3)dxdt=3x2+2xy-y2,dydt=3y2+2xy-x2,is the differential equation dξ/dτ=ξ2 over the algebra A defined by the linear space R2 endowed with the product (4)x1,y1x2,y2≔3x1x2+x1y2+y1x2-y1y2,3y1y2+x1y2+y1x2-x1x2.The solutions ξ are given by ξ(τ)=-(τ+c)-1; hence the solutions of the planar system are given by (x(t),y(t))=ξ(te), where e denotes the unit of A. Using the first fundamental representation of A (see Section 2) the following expression for the solution (x(t),y(t)) of the system is obtained: (5)xt,yt=-t-y042t+x0+y02,-t-x042t+x0+y02,x0,y0=-y0,x04x0+y02.Consider now the planar nonautonomous differential system(6)dxdt=-1t3-2xt+tx2-ty2,dydt=-2yt+2txy+ty2,whose algebrization is the Riccati differential equation over A: (7)dξdτ=-1τ3-2ξτ+τξ2,where A is the algebra defined by R2 and the product (u,v)(x,y)=(ux-vy,xy+vx+vy). By the classical Lie methods for solving differential equations (see [8–10]), the solutions of the Riccati equation have the form (8)ξτ=C+τ2τ2C-τ2,C=a,b.The functions (x(t),y(t))=ξ(te), where e is the unit of A, are solutions of the planar system, which can be obtained via the first fundamental representation of A: (9)xt,yt=b2+a+b-t2a+t2a2+ab+b2t2-2a+bt4+t6,-2ba2+ab+b2-2a+bt2+t4.We consider differential systems having the form (1), where f,g:U⊂R3→R are C1 functions defined in an open set U. The aim of this work is to give a family of functions F=(f,g) having H(A)-differentiable liftings H over some algebra A. When they exist, the solutions ξ(τ) of the differential equation dξ/dτ=H(τ,ξ) over A define solutions (x(t),y(t))=ξ(te) of system (1) which can be obtained via the first fundamental representation of A.The paper is organized as follows. In Section2 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 p:I⊂R→R2 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 p 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 algebraA is a R-linear space E endowed with a bilinear product A×A→A, denoted by (x,y)↦xy, which is associative x(yz)=(xy)z and commutative xy=yx for all x,y,z∈A, and has a unit e∈A satisfying ex=xe=x for all x∈A.An elementa∈A is calledregular if there exists a-1∈A calledinverse of a such that a-1·a=a·a-1=e. If a∈A is not regular, then a is calledsingular. If a,b∈A and b is regular, the quotient a/b means a·b-1.In all the algebras considered in this paper it will be the case thatE=R2, unless otherwise stated.Consider an algebraA. If β={e1,e2} is the standard basis of R2, the product between the elements of β is given by eiej=∑k=12cijkek, where the coefficients cijk∈R, i,j,k∈{1,2}, are calledstructure constants of A. Thefirst fundamental representation of A is the injective linear homomorphism R:A→M(2,R) defined by R:ei↦Ri, where Ri is the matrix whose j,k entry is cikj, for i=1,2.


### 2.2. Differentiability on Algebras

In this subsection the definition of Lorch differentiability is recalled, which in this paper is calledA-algebrizability or A-differentiability to denote the dependence of the Lorch differential over an algebra A.Let· be the norm on A defined by |a|=max{XRa:X∈R2,X=1} (here the vector X is represented as a 1×2 matrix in order for the product XR(a) to make sense), where R:A→M(2,R) is the first fundamental representation of A and · the Euclidean norm in R2. For this norm we have ab≤|a||b| for all a,b∈M(2,R). Thus, we consider in this work that every algebra A is a Banach algebra under the norm ·.Definition 3.
LetA be an algebra and F:V⊂A→A a function on an open set V. We say that F is A-algebrizable or A-differentiable on V if there exists a function F′:V⊂A→A, called the A-derivative of F on V, satisfying(10)limh→0⁡Fa+h-Fa-F′ahh=0,for all a∈V, where F′(a)h denotes the product in A of F′(a) with h.A vector fieldF:V⊂A→A is A-algebrizable on V if and only if the Jacobian matrix of F is contained in the first fundamental representation of A; that is, JF(a)∈R(A) for all a∈V; see [5]. It can be shown that the notion of A-algebrizability coincides with the holomorphicity when A is the complex field.A method for determining whether a given planar vector fieldF is algebrizable is the following. F is algebrizable if and only if for some of the following three types of pairs of matrices(I)
B1=10a-1, B2=01b0,(II)
B1=1a0-1, B2=0010,(III)
B1=0100, B2=0010,the condition〈Bi,∂F(x,y)/∂(x,y)〉=0 is satisfied for i=1,2 and for all (x,y) in the domain of definition of F, where 〈·,·〉 denotes the usual inner product in M(2,R). The algebra for each type of pair of matrices is defined by the following corresponding product table of the standard basis vectors e1,e2 of R2:(I)
·e1e2e1e1e2e2e2-be1+ae2(II)
·e1e2e1-ae1e1e2e1e2∑eeeeeeeeeeee(III)
·e1e2e1e10e20e2∑eeeeeeeeeeeeConsider a planar autonomous system of quadratic ordinary differential equations in the variablesx and y. If this system is algebrizable for an algebra with Type (I) product, then it can be represented by equations of the form(11)dxdt=a0+b2-ab1x-bb1y+12b4-ab3x2-2bb3xy-12bb4y2,dydt=b0+b1x+b2y+b3x2+b4xy+12ab4-bb3y2,for some real constants a,b,a0,b0,b1,…,b4. In the case of algebrizability for an algebra with Type (II) product, the system is(12)dxdt=a0+a1x+a2y-12aa3x2+a3xy+a4y2,dydt=b0+a1+aa2y+12a3+aa4y2,for some real constants a, b0, a0, a1, a2, a3, and a4. For Type (III) product the system can be represented by equations of the form(13)dxdt=a0+a1x+a2x2,dydt=b0+b1y+b2y2,for some real constants ai, bi, i=0,1,2. Moreover, conditions on the components of vector fields F can be given for constructing scalar functions α(x,y), which we callalgebrizante factors, such that αF 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 ofA-differentiability of functions with domain in A×A and image in A.The Cartesian productA×A defines anormedA-module with respect to thenorm·:A×A→R, (a1,a2)=|a1|2+|a2|2. This norm satisfies(i)
X≥0 for all X∈A×A and X=0 if and only if X=(0,0),(ii)
aX≤aX for all a∈A and X∈A×A,(iii)
X+Y≤X+Y for all X,Y∈A×A.In the following definition· denotes the norm given above on the A-module A×A.Definition 4.
LetA be an algebra and H:Ω⊂A×A→A a function, where Ω is an open set. We say that H is H(A)-differentiable on X0∈Ω if there exists a module homeomorphism M:A×A→A, which we call the differential homomorphism of H at X0, which satisfies the condition (14)limX→X0⁡HX-HX0-MX-X0X-X0=0.We denote M by DH(X0). We say that H is H(A)-differentiable on Ω if H is H(A)-differentiable on all the points of Ω.A functionH:Ω⊂A×A→A is H(A)-differentiable at X0 if and only if the usual Jacobian matrix of H:Ω⊂R2×R2→R2 at X0 satisfies JH(X0)∈R(A)×R(A). The differential homomorphism DH(X0) is represented by a matrix in M1,2(A) with respect to the standard basis of A×A, where M1,2(A) is the A-module of all the matrices of one row and two columns with entries in A; see [14].


## 2.1. Algebras

Definition 2 (see [11]).
An algebraA is a R-linear space E endowed with a bilinear product A×A→A, denoted by (x,y)↦xy, which is associative x(yz)=(xy)z and commutative xy=yx for all x,y,z∈A, and has a unit e∈A satisfying ex=xe=x for all x∈A.An elementa∈A is calledregular if there exists a-1∈A calledinverse of a such that a-1·a=a·a-1=e. If a∈A is not regular, then a is calledsingular. If a,b∈A and b is regular, the quotient a/b means a·b-1.In all the algebras considered in this paper it will be the case thatE=R2, unless otherwise stated.Consider an algebraA. If β={e1,e2} is the standard basis of R2, the product between the elements of β is given by eiej=∑k=12cijkek, where the coefficients cijk∈R, i,j,k∈{1,2}, are calledstructure constants of A. Thefirst fundamental representation of A is the injective linear homomorphism R:A→M(2,R) defined by R:ei↦Ri, where Ri is the matrix whose j,k entry is cikj, for i=1,2.


## 2.2. Differentiability on Algebras

In this subsection the definition of Lorch differentiability is recalled, which in this paper is calledA-algebrizability or A-differentiability to denote the dependence of the Lorch differential over an algebra A.Let· be the norm on A defined by |a|=max{XRa:X∈R2,X=1} (here the vector X is represented as a 1×2 matrix in order for the product XR(a) to make sense), where R:A→M(2,R) is the first fundamental representation of A and · the Euclidean norm in R2. For this norm we have ab≤|a||b| for all a,b∈M(2,R). Thus, we consider in this work that every algebra A is a Banach algebra under the norm ·.Definition 3.
LetA be an algebra and F:V⊂A→A a function on an open set V. We say that F is A-algebrizable or A-differentiable on V if there exists a function F′:V⊂A→A, called the A-derivative of F on V, satisfying(10)limh→0⁡Fa+h-Fa-F′ahh=0,for all a∈V, where F′(a)h denotes the product in A of F′(a) with h.A vector fieldF:V⊂A→A is A-algebrizable on V if and only if the Jacobian matrix of F is contained in the first fundamental representation of A; that is, JF(a)∈R(A) for all a∈V; see [5]. It can be shown that the notion of A-algebrizability coincides with the holomorphicity when A is the complex field.A method for determining whether a given planar vector fieldF is algebrizable is the following. F is algebrizable if and only if for some of the following three types of pairs of matrices(I)
B1=10a-1, B2=01b0,(II)
B1=1a0-1, B2=0010,(III)
B1=0100, B2=0010,the condition〈Bi,∂F(x,y)/∂(x,y)〉=0 is satisfied for i=1,2 and for all (x,y) in the domain of definition of F, where 〈·,·〉 denotes the usual inner product in M(2,R). The algebra for each type of pair of matrices is defined by the following corresponding product table of the standard basis vectors e1,e2 of R2:(I)
·e1e2e1e1e2e2e2-be1+ae2(II)
·e1e2e1-ae1e1e2e1e2∑eeeeeeeeeeee(III)
·e1e2e1e10e20e2∑eeeeeeeeeeeeConsider a planar autonomous system of quadratic ordinary differential equations in the variablesx and y. If this system is algebrizable for an algebra with Type (I) product, then it can be represented by equations of the form(11)dxdt=a0+b2-ab1x-bb1y+12b4-ab3x2-2bb3xy-12bb4y2,dydt=b0+b1x+b2y+b3x2+b4xy+12ab4-bb3y2,for some real constants a,b,a0,b0,b1,…,b4. In the case of algebrizability for an algebra with Type (II) product, the system is(12)dxdt=a0+a1x+a2y-12aa3x2+a3xy+a4y2,dydt=b0+a1+aa2y+12a3+aa4y2,for some real constants a, b0, a0, a1, a2, a3, and a4. For Type (III) product the system can be represented by equations of the form(13)dxdt=a0+a1x+a2x2,dydt=b0+b1y+b2y2,for some real constants ai, bi, i=0,1,2. Moreover, conditions on the components of vector fields F can be given for constructing scalar functions α(x,y), which we callalgebrizante factors, such that αF 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 ofA-differentiability of functions with domain in A×A and image in A.The Cartesian productA×A defines anormedA-module with respect to thenorm·:A×A→R, (a1,a2)=|a1|2+|a2|2. This norm satisfies(i)
X≥0 for all X∈A×A and X=0 if and only if X=(0,0),(ii)
aX≤aX for all a∈A and X∈A×A,(iii)
X+Y≤X+Y for all X,Y∈A×A.In the following definition· denotes the norm given above on the A-module A×A.Definition 4.
LetA be an algebra and H:Ω⊂A×A→A a function, where Ω is an open set. We say that H is H(A)-differentiable on X0∈Ω if there exists a module homeomorphism M:A×A→A, which we call the differential homomorphism of H at X0, which satisfies the condition (14)limX→X0⁡HX-HX0-MX-X0X-X0=0.We denote M by DH(X0). We say that H is H(A)-differentiable on Ω if H is H(A)-differentiable on all the points of Ω.A functionH:Ω⊂A×A→A is H(A)-differentiable at X0 if and only if the usual Jacobian matrix of H:Ω⊂R2×R2→R2 at X0 satisfies JH(X0)∈R(A)×R(A). The differential homomorphism DH(X0) is represented by a matrix in M1,2(A) with respect to the standard basis of A×A, where M1,2(A) is the A-module of all the matrices of one row and two columns with entries in A; see [14].


## 3. On Algebrizable Liftings of Functionsp:I⊂R→R2

In this section are considered functionsp:I⊂R→R2 defined in open intervals I, and conditions for the existence of algebras A and A-algebrizable functions P such that p(t)=P(te) will be determined, where e is the unit of A.Definition 5.
Letp:I⊂R→R2 be a function defined in an open interval I and A an algebra with unit e. We will say that P:V⊂R2→R2 is an A-algebrizable lifting of p if (a) V is an open set on which P is A-algebrizable, (b) {te:t∈I}⊂V, and (c) p(t)=P(te) for all t∈I.As a consequence of the following proposition, the family of all the functionsp:I⊂R→R2 having algebrizable liftings is an infinite dimensional algebra.Proposition 6.
LetA be an algebra and let p,q:I⊂R→R2 be functions, where I is an interval. (a)
Every constant functionp(t)=c admits the A-algebrizable lifting P(τ)=c.(b)
p(t)=te admits the A-algebrizable lifting P(τ)=τ, where e∈A is the unit of A.(c)
Ifp and q admit A-algebrizable liftings P and Q with respect to A, respectively, and a,b are constants in A, then ap+bq and pq (all products with respect to A) admit algebrizable liftings aP+bQ and PQ, respectively.(d)
Ifp has an A-algebrizable lifting P and Q is an A-algebrizable function with Im(P)⊂Dom(Q), then Q∘P is an algebrizable lifting of Q∘p.Proof.
Identity and constant functions areA-differentiable for any algebra A. Thus, (a) and (b) hold.
Letp and q be functions with A-algebrizable liftings P being Q and a,b∈A constants. The functions S=aP+bQ and T=PQ are A-algebrizable and satisfy (15)Ste=aPte+bQte=apt+bqt,Tte=PteQte=ptqt.Therefore S and T are algebrizable liftings of ap+bq and pq, respectively.
Ifp has an A-algebrizable lifting P and Q is an A-differentiable function with Im(P)⊂Dom(Q), then Q∘P(te)=Q∘p(t). Therefore Q∘P is an A-algebrizable lifting of Q∘p.Corollary 7.
LetA be an algebra. Then the following functions admit A-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 functiont↦(h(t),k(t)) with polynomial components h(t) and k(t) has an A-algebrizable lifting. The following proposition gives a wider class of these functions.Proposition 8.
LetA be an algebra. Any function t↦(h(t),k(t)) with components of the form (16)ht=amtm+am-1tm-1+⋯+a-1t+a0+a1t+⋯antn,kt=bmtm+bm-1tm-1+⋯+b-1t+b0+b1t+⋯bntn,m,n∈N, has an A-algebrizable lifting. An A-algebrizable lifting is given by (17)τ↦am,bm1τm+⋯+a-1,b-11τ+a0,b0+a1,b1τ+⋯+an,bnτn.Proof.
ConsiderQ given by the above expression; then (18)ak,bktek=ak,bktke=aktk,bktkholds for k=-m,…,n, so Q(te)=(h(t),q(t)), where e is the unit of A. Therefore, Q is an A-algebrizable lifting of t↦(h(t),k(t)).In particular, every functionp=(p1,p2):I⊂R→R2 with quadratic components p1(t)=a0+a1t+a2t2 and p2(t)=b0+b1t+b2t2 admits the A-algebrizable lifting P(τ)=(a0,b0)+(a1,b1)τ+(a2,b2)τ2.


## 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 ξ(τ) a solution of (2), then (x(t),y(t))=ξ(te) is a solution of (1), where e denotes the unit of the corresponding algebra.Proof.
Letξ be a solution of (2). The derivative of (x(t),y(t))=ξ(te) with respect to t is given by (19)ddtxt,yt=ddtξte=dξτ,ξdττ=tedτdt=Hτ,ξττ=tee=Hte,ξte=ft,xt,yt,gt,xt,yt.Thus, (x(t),y(t)) is a solution of system (1).As a consequence of the following proposition, the family of all the functionsF:Ω⊂R3→R2 having algebrizable liftings defines an infinite dimensional algebra.Proposition 10.
LetA be an algebra with unit e. In the following statements F and G denote functions defined on open sets Ω⊂R3 and they have values in R2. (a)
F(t,x,y)=c admits the H(A)-differentiable lifting H(τ,ξ)=c.(b)
F(t,x,y)=te admits the H(A)-differentiable lifting H(τ,ξ)=τ.(c)
F(t,x,y)=(x,y) admits the H(A)-differentiable lifting H(τ,ξ)=ξ.(d)
IfF and G admit H(A)-differentiable liftings HF and HG, respectively, and a,b are constants in A, then aHF+bHG and FG (all products with respect to A) have H(A)-differentiable liftings aHF+bHG and HFHG, respectively.(e)
Every functionE having the form E(t,x,y)=F(t,G(t,x,y)), where F and G admit H(A)-differentiable liftings HF and HG, admits an H(A)-differentiable lifting HE given by HE(τ,ξ)=HF(τ,HG(τ,ξ)).(f)
Every functionE having the form E(t,x,y)=F(t,P(x,y)), where F has an H(A)-differentiable lifting HF and P is an A-differentiable function, admits an H(A)-differentiable lifting HE(τ,ξ)=HF(τ,P(ξ)).Proof.
The proofs of (a), (b), and (c) are trivial. LetHF and HG be the algebrizable liftings of F and G and then HF+HG, HFHG, and HF/HG are A-algebrizable and (i)
(aHF+bHG)(te,x,y)=aHF(te,x,y)+bHG(te,x,y)=aF(t,x,y)+aG(t,x,y),(ii)
(HFHG)(te,x,y)=HF(te,x,y)HG(te,x,y)=F(t,x,y)G(t,x,y),(iii)
HE(te,x,y)=HF(te,HG(te,x,y))=F(t,G(t,x,y)),(iv)
HE(te,x,y)=HF(te,P(e,y))=F(t,P(x,y)).Thus, the proof is complete.Corollary 11.
LetA be an algebra. The following functions admit H(A)-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 H(A)-differentiable liftings.Given a functionF(t,x,y)=(f(t,x,y),g(t,x,y)), where f,g are polynomial functions of the variables x,y, the goal of the paper is to determine if F has an H(A)-differentiable lifting. As a consequence of the following theorem, every function F which is polynomial of the variables t, x, and y, has an H(A)-differentiable lifting when (x,y)↦(f(t,x,y),g(t,x,y)) is A-differentiable for all t.Theorem 12.
LetA be an algebra with unit e and F:Ω⊂R3→R2, F=(f,g), where f=∑k=0mfk, g=∑k=0mgk, and fk(t,x,y), gk(t,x,y) are homogeneous polynomials of degree k in the variables x,y, and Ω=I×R2 for an open interval I. Then the following statements are equivalent. (a)
F has an H(A)-differentiable lifting H.(b)
The map(x,y)↦F(t,x,y) is A-algebrizable for all t∈I and the functions hk given by hk(t)=(fk(t,e),gk(t,e)) have A-algebrizable liftings, for k=0,1,…,m.(c)
H(τ,ξ)=∑k=0mHk(τ)ξk is an H(A)-differentiable lifting of F, where Hk(τ) are A-algebrizable liftings of hk(t)=(fk(t,e),gk(t,e)).Proof.
Obviously (a) implies (b) and (c) implies (a). We now show that (b) implies (c). Suppose thatf(t,x,y) and g(t,x,y) are homogenous polynomials f(t,x,y)=∑k=0npk(t)xkyn-k and g(t,x,y)=∑k=0nqk(t)xkyn-k of the variables x,y defined on the set Ω=I×R2 as above. Taking the partial derivatives of F with respect to x and y we obtain (20)∂f,g∂x,yt,x,y=∑k=1nkpktxk-1yn-k∑k=1nn-k+1pk-1txk-1yn-k∑k=1nkqktxk-1yn-k∑k=1nn-k+1qk-1txk-1yn-k=∑k=1nkpktxk-1yn-k∑k=1nn-k+1pk-1txk-1yn-k∑k=1nkqktxk-1yn-k∑k=1nn-k+1qk-1txk-1yn-k=∑k=1nxk-1yn-kkpktn-k+1pk-1tkqktn-k+1qk-1t.Let R be the first fundamental representation of A. Since (x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable for all t∈I, then (∂f,g/∂x,y)(t,x,y)∈R(A) for all (t,x,y)∈Ω. Thus (21)∂f,g∂x,yt,0,1=p1tnp0tq1tnq0t∈RAfor all t∈I and then(22)∂f,g∂x,yt,x,y-yn-1p1tnp0tq1tnq0t=∑k=2nxk-1yn-kkpktn-k+1pk-1tkqktn-k+1qk-1tis in R(A) for all (t,x,y)∈Ω. If x(s)=1/s and y(s)=s(n-2), then (23)lims→∞⁡∂f,g∂x,yt,xs,ys-ysn-1p1tnp0tq1tnq0t=p2tn-1p1tq2tn-1q1t∈RA.Following the same idea, for k=1,2,…,n(24)kpktn-k+1pk-1tkqktn-k+1qk-1t∈RA.
If(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable for an algebra A with Type (I) product (given in Section 2), then (25)kpk+akqk-n-k+1qk-1=0,bkqk+n-k+1pk-1=0,or equivalently (26)qk-1,pk-1=kpk+akqkn-k+1,-bkqkn-k+1for k=1,…,n. Thus, the functions pk,qk are determined by pn,qn for k=0,1,…,n-1. Since (f(t,1,0),g(t,1,0))=(pn(t),qn(t)), then (f(t,x,y),g(t,x,y))=(pn(t),qn(t))(x,y)n.
Therefore,F has an H(A)-differentiable lifting H(τ,ξ)=Hn(τ)ξn, where Hn(τ) is an A-algebrizable lifting of t↦(pn(t),qn(t)).
If(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable for an algebra A with Type (II) product (given in Section 2), then (27)kpk+an-k+1pk-1-n-k+1qk-1=0,kqk=0,or equivalently (28)pk,qk=-an-k+1pk-1k,0for k=1,…,n. Thus, qk=0 and pk is determined by p0, for k=1,2,…,n. Since (f(t,0,1),g(t,0,1))=(p0(t),q0(t)), then (f(t,x,y),g(t,x,y))=(p0(t),q0(t))(x,y)n.
Therefore,F has an H(A)-differentiable lifting H(τ,ξ)=Hn(τ)ξn, where Hn(τ) is an A-algebrizable lifting of t↦(p0(t),q0(t)).
If(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable for an algebra A with Type (III) product (given in Section 2), then (n-k+1)pk-1=0 and kqk=0; that is, pk-1=0, qk=0, for k=1,…,n. Thus, qk=0 and pk is determined by p0, for k=1,2,…,n. Since (f(t,0,1),g(t,0,1))=(pn(t),q0(t)), then (f(t,x,y),g(t,x,y))=(pn(t),q0(t))(x,y)n.
Therefore,F has an H(A)-differentiable lifting H(τ,ξ)=Hn(τ)ξn, where Hn(τ) is an A-algebrizable lifting of t↦(pn(t),q0(t)).
Thus, iff and g are homogenous polynomial of degree n in the variables x and y, then H(τ,ξ)=Hn(τ)ξn in each of the cases of algebras defined by products of Types (I), (II), and (III) given in Section 2. Since a polynomial function (t,x,y)↦(f(t,x,y),g(t,x,y)) in the variables x and y can be seen as the finite addition of homogenous polynomial in the variables x and y, by (d) of Proposition 10, (b) implies (c).Example 13.
Consider the planar system (6); then (29)Ft,x,y=-1t3-2xt+tx2-ty2,-2yt+2txy+ty2.The Jacobian ∂F/∂(x,y) is given by (30)∂Ft,x,y∂x,y=-2t+2tx-2ty2ty-2t+2tx+2ty.Thus, ∂F/∂(x,y) is orthogonal to the matrices (31)B1=101-1,B2=0110,for all (x,y); that is, 〈∂F/∂(x,y),Bi〉=0 for i=1,2. The map (x,y)↦F(t,x,y) is A-algebrizable for an algebra of Type (I) with constants a=1 and b=1; see Section 2. The function F can be written as (32)Ft,x,y=-1t3,0+-2tx,-2ty+tx2-ty2,ty2.Thus, the functions pi and qi of Theorem 12 are given by p0(t,x,y)=-1/t3, q0(t,x,y)=0, p1(t,x,y)=-(2/t)x, q1(t,x,y)=-(2/t)y, p2(t,x,y)=tx2-ty2, and q2(t,x,y)=ty2.
The unite of A is e=(1,0) and then h0(t)=(-1/t3,0), h1(t)=(-2/t,0), and h2(t)=(t,0) have A-algebrizable liftings H0(τ)=-1/τ3, H1(τ)=-2/τ, and H2(τ)=τ, respectively.
From the form ofF(t,xe)=(-1/t3-(2/t)x-tx2)e, H must be (33)Hτ,ξ=-1τ3-2τξ+τξ2.


## 5. The Case of Second-Degree Polynomials in the Variablest, x, and y

Iff and g in (1) are quadratic polynomials in three variables t, x, and y and A is an algebra with respect to which the map (x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable, it will be showed that the H(A)-differentiable lifting H of F=(f,g) is a polynomial in two variables of A. Under these conditions (1) has an algebrization which is a Riccati equation over A having the form (34)dξdτ=Pτ+Qτξ+Rτξ2,where P, Q, and R are polynomials over A of degrees two, one, and zero, respectively.Consider system (1) where f,g:Ω⊂R3→R2 are second-degree polynomials of three variables t, x, and y; that is, (35)ft,x,y=a0+a1t+a2x+a3y+a4t2+a5tx+a6ty+a7x2+a8xy+a9y2,gt,x,y=b0+b1t+b2x+b3y+b4t2+b5tx+b6ty+b7x2+b8xy+b9y2.All the quadratic vector fields which are algebrizable with respect to algebras with Type (I) products have the form(36)dxdt=A0+B2-aB1x-bB1y+12B4-aB3x2-2bB3xy-12bB4y2,dydt=B0+B1x+B2y+B3x2+B4xy+12aB4-bB3y2,where a,b,A0,B0,B1,…,B5 are real constants; see Section 2.2. The algebrizability of (x,y)↦(f(t,x,y),g(t,x,y)) with respect to algebras with Type (I) products can be verified, by considering t as a constant.The following theorems give conditions that characterize the algebrizability of planar systems like (1) when f and g 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.
LetA be an algebra with Type (I) product defined by constants a and b and f,g the polynomials (35). The following statements are equivalent. (1)
The mapF:(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable.(2)
The functionsf and g have the form (37)ft,x,y=a0+a1t+b3-ab2x-bb2y+a4t2+b6-ab5tx-bb5ty+b82-ab7x2-2bb7xy-bb82y2,gt,x,y=b0+b1t+b2x+b3y+b4t2+b5tx+b6ty+b7x2+b8xy+ab82-bb7y2.(3)
The functionH:A×A→A of the variables τ=(t,s) and ξ=(x,y), defined by (38)Hτ,ξ=A0+A1τ+A2ξ+A3τ2+A4τξ+A5ξ2,is an H(A)-differentiable lifting of F, where (39)A0=a0,b0,A1=a1,b1,A2=b3-ab2,b2,A3=a4,b4,A4=b6-ab5,b5,A5=b82-ab7,b7.Proof.
Writingg(t,x,y)=B0+B1x+B2y+B3x2+B4xy+B5y2 yields (40)B0=b0+b1t+b4t2,B1=b2+b5t,B2=b3+b6t,B3=b7,B4=b8,B5=b9.The function (x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable if and only if (41)ft,x,y=A0+B2-aB1x-bB1y+12B4-aB3x2-2bB3xy-12bB4y2,where A0=a0+a1t+a4t2 and b9=ab8/2-bb7. Thus, statements (1) and (2) are equivalent.
The functionH is polynomial in the variables τ and ξ of A; hence H is H(A)-differentiable. H satisfies H(te,x,y)=(f(t,x,y),g(t,x,y)), where e is the unit of A. So, H is an H(A)-differentiable lifting of F. Thus, statement (2) implies statement (3).
SinceH is H(A)-differentiable and F(t,x,y)=H(te,x,y), then (x,y)↦F(t,x,y) is A-algebrizable for all t. Thus, statement (3) implies statement (1).All the quadratic vector fields which are algebrizable with respect to algebras with Type (II) products have the form(42)x˙=A0+A1x+A2y-12aA3x2+A3xy+A4y2,y˙=B0+A1+aA2y+12A3+aA4y2,where a,A0,…,A4,B0 are real constants; see Section 2.2. The algebrizability of (x,y)↦(f(t,x,y),g(t,x,y)) with respect to algebras with Type (II) products can be verified, by considering t 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.
LetA be an algebra with Type (II) product defined by the constant a and let f,g be the polynomials (35). The following statements are equivalent. (1)
The map(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable.(2)
The functionsf and g are given by (43)ft,x,y=a0+a1t+a2x+a3y+a4t2+a5tx+a6ty-12aa8x2+a8xy+a9y2,gt,x,y=b0+b1t+a2+aa3y+b4t2+a5+aa6ty+a82+aa9y2.(3)
The functionH:A×A→A of the variables τ=(t,s) and ξ=(x,y), defined by (44)Hτ,ξ=A0+A1τ+A2ξ+A3τ2+A4τξ+A5ξ2,is an H(A)-differentiable lifting of F, where (45)A0=a0,b0,A1=a1,b1,A2=a3,a2+aa3,A3=a4,b4,A4=a6,a5+aa6,A5=a9,a82+aa9.Proof.
Writingf(t,x,y)=A0+A1x+A2y-(1/2)aA3x2+A3xy+A4y2 yields (46)A0=a0+a1t+a4t2,A1=a2+a5t,A2=a3+a6t,A3=a8,A4=a9,and the map (x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable if and only if (47)gt,x,y=B0+A1+aA2y+12A3+aA4y2,where B0=b0+b1t+b4t2. Thus, statements (1) and (2) are equivalent.
The rest of the proof is similar to that of Theorem14.All the quadratic vector fields which are differentiable with respect to algebras with Type (III) products have the form(48)x˙=A0+A1x+A2x2,y˙=B0+B1y+B2y2,where A0,B0,A1,B1,A2,B2 are real constants; see Section 2.2. The algebrizability of (x,y)↦(f(t,x,y),g(t,x,y)) with respect to algebras with Type (III) products can be verified, by considering t 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.
LetA be an algebra with Type (III) product defined by the constant a and let f,g be the polynomials (35). The following statements are equivalent. (1)
The map(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable.(2)
The functionsf and g are given by (49)ft,x,y=a0+a1t+a2x+a4t2+a5tx+a7x2,gt,x,y=b0+b1t+b3y+b4t2+b6ty+b9y2.(3)
The functionH:A×A→A of the variables τ=(t,s) and ξ=(x,y), defined by (50)Hτ,ξ=A0+A1τ+A2ξ+A3τ2+A4τξ+A5ξ2,is an H(A)-differentiable lifting of F, where A0=(a0,b0), A1=(a1,b1), A2=(a2,b3), A3=(a4,b4), A4=(a5,b6), and A5=(a7,b9).Proof.
Function(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable if and only if (51)ft,x,y=a0+a1t+a2x+a4t2+a5tx+a7x2,gt,x,y=b0+b1t+b3y+b4t2+b6ty+b9y2.Thus, first and second statements are equivalents. The function H given by H(τ,ξ)=A0+A1τ+A2ξ+A3τ2+A4τξ+A5ξ2, where A0=(a0,b0), A1=(a1,b1), A2=(a2,b3), A3=(a4,b4), A4=(a5,b6), A5=(a7,b9), τ=(r,s), and ξ=(x,y), satisfies H(te,x,y)=(f(t,x,y),g(t,x,y)). Thus, second and third statements are equivalents.By Theorems14, 15, and 16, an algebrization of quadratic systems is a Riccati equation over an algebra dξ/dτ=P(τ)+Q(τ)ξ+R(τ)ξ2, where P(τ)=A0+A1τ+A3τ2, Q(τ)=A2+A4τ, and R(τ)=A5. 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 (52)ft,x,y=1+7t-x-3y+5t2-tx-3ty-6xy-6y2,gt,x,y=1+t+x+y+t2+tx+ty+x2+4xy+y2.The matrices (53)B1=102-1,B2=0130satisfy 〈Bi,∂(f,g)/∂(x,y)〉=0 for i=1,2. Thus, F:(x,y)↦(f(t,x,y),g(t,x,y)) is A-algebrizable for an algebra A with Type (I) product defined by the constants a=2 and b=3.
The conditions of Theorem14 are satisfied; then the H(A)-differentiable lifting H of F can be written as (54)Hτ,ξ=A0+A1τ+A2ξ+A3τ2+A4τξ+A5ξ2,where A0=(1,1), A1=(7,1), A2=(-1,1), A3=(5,1), A4=(-1,1), and A5=(0,1).



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