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

Abstract

A new technique for solving a certain class of systems of autonomous ordinary differential equations over 𝕂^𝑛 is introduced (𝕂 being the real or complex field). The technique is based on two observations: (1), if 𝕂^𝑛 has the structure of certain normed, associative, commutative, and with a unit, algebras 𝔸 over 𝕂, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations over ℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.

1. 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 𝑑𝑥(𝑥)𝑑𝑡=𝑓,𝑥∈Ω⊂𝕂^𝑛,𝑡∈ℝ, (1.1) with 𝑓∶Ω⊂𝕂^𝑛→𝕂^𝑛 having certain regularity conditions.

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 Schrödinger 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 𝑑𝜁=𝑔(𝜁)𝑑𝑡,𝜁∈Ω^′⊂𝔸,𝑡∈ℝ, (1.2) 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 𝕂^𝑛, that is, 𝔹=𝑅(𝔸), then in fact 𝑓 is 𝔸-differentiable on Ω. The problem of determining if a map 𝑓∶Ω⊂𝕂^𝑛→𝕂^𝑛 is 𝔸-differentiable for some algebra 𝔸 is treated indirectly in [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 𝑑𝜁=𝑔(𝜁)𝑑𝑡,𝜁∈Ω^′⧵𝒮⊂𝔸,𝑡∈ℝ, (1.3) 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 ℎ₁ and ℎ₂ 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.

2. Algebras

We introduce 𝕂-algebras (see, e.g., [24]).

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⁻¹𝑎=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.

2.1. Algebras and Their Fundamental Representations

We define the first fundamental representation.

If ℬ={𝛽₁,…,𝛽_𝑛} is an ordered basis of an algebra 𝔸, the product between the elements of ℬ is given by 𝛽_𝑖𝛽_𝑗=^𝑛_𝑘=1𝑐_𝑖𝑗𝑘𝛽_𝑘, (2.1) 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 𝑅𝑥₁𝛽₁+𝑥₂𝛽₂+⋯+𝑥_𝑛𝛽_𝑛=𝑥₁𝑅₁+𝑥₂𝑅₂+⋯+𝑥_𝑛𝑅_𝑛, (2.2) 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 (1)𝑐_𝑖𝑗𝑘=𝑐_𝑗𝑖𝑘, for all 𝑖,𝑗,𝑘∈{1,2…,𝑛} and (2)∑_𝑙𝑐_𝑖𝑗𝑙𝑐_𝑙𝑘𝑠=∑_𝑙𝑐_𝑗𝑘𝑙𝑐_𝑖𝑙𝑠, for all 𝑖,𝑗,𝑘,𝑠∈{1,2…,𝑛}, respectively.

Using 𝑅 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.

Example 2.2. Let 𝔸 be the linear space ℝ³ with the product between the elements of the standard basis ℬ={𝑒₁,𝑒₂,𝑒₃} given in the following equation: 𝑒₁𝑒₂𝑒₃𝑒₁𝑒₁𝑒₂𝑒₃𝑒₂𝑒₂−2𝑒₁+2𝑒₂+𝑒₃−𝑒₁+𝑒₂+𝑒₃𝑒₃𝑒₃−𝑒₁+𝑒₂+𝑒₃−𝑒₁+2𝑒₃, (2.3) that extends to the product in ℝ³ given by 𝑥,𝑦,𝑧(𝑢,𝑣,𝑤)=𝑒₁𝑥,𝑒₂𝑦,𝑒₃𝑧𝑒₁𝑢,𝑒₂𝑣,𝑒₃𝑤=𝑥𝑢𝑒₁𝑒₁+𝑥𝑣𝑒₁𝑒₂+𝑥𝑤𝑒₁𝑒₃+𝑦𝑢𝑒₁𝑒₂+𝑦𝑣𝑒₂𝑒₂+𝑦𝑤𝑒₂𝑒₃+𝑧𝑢𝑒₁𝑒₃+𝑧𝑣𝑒₂𝑒₃+𝑥𝑤𝑒₃𝑒₃=𝑥𝑢−2𝑦𝑣−𝑦𝑤−𝑧𝑣−𝑧𝑤𝑒₁+𝑥𝑣−𝑦𝑢−𝑧𝑦𝑣−𝑦𝑤+𝑧𝑣𝑒₂+𝑧𝑤+𝑧𝑢+𝑦𝑣+𝑦𝑤+𝑧𝑣+𝑧𝑤𝑒₃. (2.4)

The product between the elements of ℬ define the structure constants 𝑐_𝑖𝑗𝑘 for 𝑖,𝑗,𝑘∈{1,2,3}. So, ℝ³ with the product given is an algebra 𝔸.

2.2. Normal Algebras and Their Tensor Products

Let 𝐵₁,𝐵₂,…,𝐵_𝑙 be matrices 𝐵_𝑖∈𝑀(𝑘_𝑖,𝕂), each 𝐵_𝑖 of one the following four types: (𝑟),,,𝐼𝑠00⋯01𝑠0001𝑠0⋮⋱000⋯𝑠𝑎−𝑏𝑏𝑎𝐶00⋯0₂𝐶000𝐼₂𝐶0⋮⋱000⋯𝐶, (2.5) where 𝐶=𝑢−𝑣𝑣𝑢,𝐼₂=1001, (2.6)𝑟,𝑠,𝑎,𝑏,𝑢,𝑣∈𝕂 and 𝑘₁+⋯+𝑘_𝑙=𝑛. In this case we will say that the matrix 𝐵 given by 𝐵𝐵=¹0⋯00𝐵₂0⋮⋱00⋯𝐵_𝑙, (2.7) 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_𝑖𝐷_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_𝑖𝐷_𝑖+𝑏_𝑖𝐽_𝑖, where 𝐷_𝑖 is a diagonal matrix and 𝐽_𝑖 is a matrix with 𝐽²_𝑖=−𝐷_𝑖. The complex Jordan blocks may be written in the form 𝐵_𝑖=𝑎_𝑖𝐷_𝑖+𝑏_𝑖𝐽_𝑖+𝑁_𝑖, where 𝐷_𝑖 is the identity and 𝐽_𝑖 is a matrix with 𝐽²_𝑖=−𝐷_𝑖 and 𝑁_𝑖 is a nilpotent matrix of order 𝑘_𝑖/2.

We define the matrices {𝛽_𝑖,𝑗∶1≤𝑖≤𝑙,1≤𝑗≤𝑘_𝑖} whose entries are in the block 𝐵_𝑖 as follows.(i)If 𝐵_𝑖 is a real simple block, 𝛽_𝑖,1∶=𝜎_𝑖(1), in this case 𝑘_𝑖=1. (ii)If 𝐵_𝑖 is a real Jordan block, 𝛽_𝑖,1∶=𝜎_𝑖(𝐷_𝑖) and 𝛽_𝑖,𝑗∶=𝜎_𝑖(𝑁_𝑖^𝑗−1) for 𝑗=2,…,𝑘_𝑖. (iii)If 𝐵_𝑖 is a simple complex block, 𝛽_𝑖,1∶=𝜎_𝑖(𝐷_𝑖) and 𝛽_𝑖,2∶=𝜎_𝑖(𝐽_𝑖). (iv)If 𝐵_𝑖 is a complex Jordan block, 𝛽_𝑖,1∶=𝜎_𝑖(𝐷_𝑖), 𝛽_𝑖,2∶=𝜎_𝑖(𝐽_𝑖), 𝛽_{𝑖,2𝑗−1}∶=𝜎_𝑖(𝑁_𝑖^2𝑗−1), and 𝛽_𝑖,2𝑗∶=𝛽_𝑖,2𝜎_𝑖(𝑁_𝑖^2𝑗−1) for 𝑗=2,…,𝑘_𝑖/2.

Observe that the product of the matrices 𝛽_𝑖1,𝑗1 and 𝛽_𝑖2,𝑗2 is the zero matrix if 𝑖₁≠𝑖₂. The products of 𝛽_𝑖,𝑗1 and 𝛽_𝑖,𝑗2 for 𝑖=1,…,𝑙 are as follows.(i)If 𝐵_𝑖 is a real simple block, 𝛽_𝑖,1𝛽_𝑖,1=𝛽_𝑖,1. (ii)If 𝐵_𝑖 is a real Jordan block, then 𝛽_𝑖,1𝛽_𝑖,𝑗1𝛽_𝑖,1𝛽_𝑖,1𝛽_𝑖,𝑗1𝛽_𝑖,𝑗2𝛽_𝑖,𝑗1𝛽_{𝑖,𝑗1+𝑗2}, (2.8) for 2≤𝑗₁,𝑗₂≤𝑘_𝑖, where 𝛽_𝑖,𝑗1=0 when 𝑗₁≥𝑘_𝑖+1. (iii)If 𝐵_𝑖 is a complex simple block, then 𝛽_𝑖,1𝛽_𝑖,2𝛽_𝑖,1𝛽_𝑖,1𝛽_𝑖,2𝛽_𝑖,2𝛽_𝑖,2−𝛽_𝑖,1 (2.9)(iv)If 𝐵_𝑖 is a complex Jordan block, then 𝛽_𝑖,1𝛽_𝑖,2𝛽_{𝑖,2𝑗1−1}𝛽_𝑖,2𝑗1𝛽_𝑖,1𝛽_𝑖,1𝛽_𝑖,2𝛽_{𝑖,2𝑗1−1}𝛽_𝑖,2𝑗1𝛽_𝑖,2𝛽_𝑖,2−𝛽_𝑖,1𝛽_𝑖,2𝑗1−𝛽_{𝑖,2𝑗1−1}𝛽_{𝑖,2𝑗2−1}𝛽_{𝑖,2𝑗2−1}𝛽_𝑖,2𝑗2𝛽_{𝑖,2(𝑗1+𝑗2−1)−1}𝛽_{𝑖,2(𝑗1+𝑗2−1)}𝛽_𝑖,2𝑗2𝛽_𝑖,2𝑗2−𝛽_{𝑖,2𝑗2−1}𝛽_{𝑖,2(𝑗1+𝑗2−1)}−𝛽_{𝑖,2(𝑗1+𝑗2−1)−1}, (2.10) for 2≤𝑗₁,𝑗₂≤𝑘_𝑖/2, where 𝛽_𝑖,𝑗=0 when 𝑗≥2𝑘_𝑖+1.

The commutativity of the elements in the set ℬ=𝛽_1,1,…,𝛽_1,𝑘1,…,𝛽_𝑙,1,…,𝛽_{𝑙,𝑘𝑙}, (2.11) 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,𝑘1,…,𝛽_𝑙,1,…,𝛽_{𝑙,𝑘𝑙}} 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 𝔸₁ and 𝔸₂ in 𝑀(𝑛,𝕂) are linearly equivalent if there exists an invertible matrix 𝐵∈𝑀(𝑛,𝕂) such that 𝔸₁={𝐵𝐴𝐵⁻¹∶𝐴∈𝔸₂}.

For a proof of the following result, see [30].

Proposition 2.6. Let 𝔸 and 𝔹 be algebras. There exists a product in 𝔸⊗𝔹 satisfying 𝑎₁⊗𝑏₁𝑎₂⊗𝑏₂=𝑎₁𝑎₂⊗𝑏₁𝑏₂, (2.12) where 𝑎₁𝑎₂ and 𝑏₁𝑏₂ 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 𝑝- and 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 𝑃𝔸𝑃⁻¹⊗𝑄𝔹𝑄⁻¹=[𝑃⊗𝑄][𝔸⊗𝔹][𝑃⊗𝑄]⁻¹. (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 𝑝- and 𝑞-dimensional 𝕂-algebras, and 𝑅₁∶𝔸→𝑀(𝑝,𝕂) and 𝑅₂∶𝔹→𝑀(𝑞,𝕂) be first fundamental representations associated to the basis 𝒜={𝛼₁,…,𝛼_𝑝} and ℬ={𝛽₁,…,𝛽_𝑞}, respectively. Then, 𝑅∶𝔸⊗𝔹→𝑀(𝑝𝑞,𝕂) defined by 𝑅=𝑅₁⊗𝑅₂ is the first fundamental representation of 𝔸⊗𝔹 associated to the base {𝛼_𝑖⊗𝛽_𝑗∶1≤𝑗≤𝑝,1≤𝑗≤𝑞}.

Proof. Let {𝛼₁,…,𝛼_𝑝} be a base of 𝔸 and let {𝛽₁,…,𝛽_𝑞} be a base of 𝔹. We use the notations 𝐶_𝑖 and 𝐺_𝑗 for the matrices 𝑅₁(𝛼_𝑖) and 𝑅₂(𝛽_𝑗), respectively, for 1≤𝑖≤𝑝, 1≤𝑗≤𝑞. The set 𝛼_𝑖⊗𝛽_𝑗∶1≤𝑖≤𝑝,1≤𝑗≤𝑞 (2.14) is a base for 𝔸⊗𝔹 (see [30]). In order to find the structure constants of 𝔸⊗𝔹 we take the products 𝛼_𝑖⊗𝛽_𝑙𝛼_𝑗⊗𝛽_𝑠=𝛼_𝑖𝛼_𝑗⊗𝛽_𝑙𝛽_𝑠=^𝑝_𝑘=1𝑐_𝑖𝑗𝑘𝛼_𝑘⊗^𝑞_𝑡=1𝑑_𝑙𝑠𝑡𝛽_𝑡=^𝑝𝑞_𝑘=1𝑡=1𝑐_𝑖𝑗𝑘𝑑_𝑙𝑠𝑡𝛼_𝑘⊗𝛽_𝑡, (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_𝑖𝑗𝑘𝑑_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.
On the other hand we have that 𝑅₁(𝛼_𝑖)⊗𝑅₂(𝛽_𝑙)=𝐶_𝑖⊗𝐺_𝑙. The tensor product of 𝐶_𝑖 and 𝐺_𝑙 is given by 𝐶_𝑖⊗𝐺_𝑙=𝑐_𝑖11𝑑_𝑖11𝑑_𝑖21…𝑑_𝑖𝑞1⋮𝑑_𝑖1𝑞𝑑_𝑖2𝑞…𝑑_𝑖𝑞𝑞⋯𝑐_𝑖𝑝1𝑑_𝑖11𝑑_𝑖21…𝑑_𝑖𝑞1⋮𝑑_𝑖1𝑞𝑑_𝑖2𝑞…𝑑_𝑖𝑞𝑞⋮𝑐_𝑖1𝑝𝑑_𝑖11𝑑_𝑖21…𝑑_𝑖𝑞1⋮𝑑_𝑖1𝑞𝑑_𝑖2𝑞…𝑑_𝑖𝑞𝑞⋯𝑐_𝑖𝑝𝑝𝑑_𝑖11𝑑_𝑖21…𝑑_𝑖𝑞1⋮𝑑_𝑖1𝑞𝑑_𝑖2𝑞…𝑑_𝑖𝑞𝑞, (2.16) from which we see that in the position (𝑘𝑡,𝑗𝑠) the element 𝑐_𝑖𝑗𝑘𝑑_𝑙𝑠𝑡 appears. Thus, we have the equalities of matrices 𝐻_𝑖𝑙=𝐶_𝑖⊗𝐺_𝑙, for 1≤𝑖,𝑙≤𝑛. Therefore, 𝑅=𝑅₁⊗𝑅₂.

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=𝔹₁⊗⋯⊗𝔹_𝑚, 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.

3. 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 𝕂^𝑛. If 𝑓∶𝑈⊂𝕂^𝑛→𝕂^𝑛 is a differentiable map 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_ℬ𝑓(𝑥) for the Jacobian matrix at the point 𝑥 of 𝑓 with respect to a base ℬ.

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 𝑥₀∈𝔸 if there exists an element 𝑓^′(𝑥₀)∈𝔸, which one calls the 𝔸-derivative of 𝑓 at 𝑥₀, satisfying lim_ℎ→0𝑓𝑥₀+ℎ−𝑓𝑥₀−𝑓^′𝑥₀ℎ‖ℎ‖=0, (3.1) where 𝑓^′(𝑥₀)ℎ denotes the product in 𝔸 of 𝑓^′(𝑥₀) 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 𝑥₀ if and only if 𝐽_ℬ𝑓(𝑥₀)∈𝑅(𝔸) and is continuous as a function of 𝑥₀, where ℬ∶={𝑒₁,…,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, 𝐽_ℬ𝑓(𝑥) is the image of 𝑓^′(𝑥) under the map 𝑅, that is, if 𝑓 is 𝔸-differentiable, then 𝐽_ℬ𝑓(𝑥)=^𝑛_𝑖=1𝑢_𝑖(𝑥)𝑅_𝑖, (3.2) 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 𝑔𝑦=𝑅∘𝑓∘𝑅⁻¹𝑦, (3.3) then 𝑔 is 𝔹-differentiable and its differential at 𝑦 is given by 𝑔^′(𝑦)=∑^𝑛_𝑖=1𝑢_𝑖(𝑅⁻¹(𝑦))𝑅_𝑖, thus, the relation between the Jacobian matrix of 𝑓 and the 𝔹-differential of 𝑔 is 𝐽_ℬ𝑓(𝑥)=𝑔^′(𝑅(𝑥)).

The matrix equation (3.2) is equivalent to the 𝑛² equations 𝜕𝑓_𝑖𝜕𝑥_𝑗=^𝑛_𝑙=1𝑢_𝑙𝑐_𝑙𝑗𝑖, (3.4) for all 𝑖,𝑗∈{1,2,…,𝑛}. Using (3.4) and the associativity of the algebra, we can obtain ^𝑛_𝑖=1𝑐_𝑖𝑘𝑠𝜕𝑓_𝑖𝜕𝑥_𝑗=^𝑛_𝑖=1𝑐_𝑖𝑗𝑠𝜕𝑓_𝑖𝜕𝑥_𝑘, (3.5) appearing in [29, p. 646].

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, see [27, 28, 31–33].

Suppose we consider a basis 𝒜={𝛼₁,…,𝛼_𝑛} of 𝔸, where 𝛼_𝑖=∑^𝑛_𝑗=1𝑠_𝑗𝑖𝑒_𝑗 for 𝑖=1,…,𝑛, 𝑠_𝑗𝑖∈𝕂, it can be proved that 𝐽_𝒜𝑓=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=1𝑠_𝑗𝑖𝑅_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.

Theorem 3.4. Let 𝑓∶Ω⊂𝕂^𝑛→𝕂^𝑛 be a 𝐶¹ map defined in the open set Ω. 𝑓 is 𝔸-differentiable for an algebra 𝔸 if and only if the set of matrices 𝐽𝑓(𝑥)∶𝑥∈Ω (3.6) 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={𝛼₁,…,𝛼_𝑛} be a base for 𝕋 as given in Corollary 2.11. Then ℬ={𝐵𝛼₁𝐵⁻¹,…,𝐵𝛼_𝑛𝐵⁻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=1𝑢_𝑖(𝑥)𝛽_𝑖, where 𝑢_𝑖∶Ω→𝕂 are functions and 𝛽_𝑖=𝐵𝛼_𝑖𝐵⁻¹.
Now consider the base 𝒢={𝛾₁…,𝛾_𝑛} of 𝕂^𝑛 defined by 𝛾_𝑖=∑^𝑛_𝑗=1𝑏_𝑗𝑖𝑒_𝑗, where 𝐵=(𝑏_𝑖𝑗) and {𝑒₁,…,𝑒_𝑛} is the standard base of 𝕂^𝑛. Then, we have that 𝐽_𝒢𝑓=𝐵⁻¹(𝐽𝑓)𝐵. Thus, 𝐽_𝒢𝑓(𝑥)=𝐵^−1𝑛_𝑖=1𝑢_𝑖(𝑥)𝛽_𝑖𝐵=𝐵^−1𝑛_𝑖=1𝑢_𝑖(𝑥)𝐵𝛼₁𝐵⁻¹𝐵=^𝑛_𝑖=1𝑢_𝑖(𝑥)𝛼_𝑖, (3.7) in other words 𝐽_𝒢𝑓(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.

4. Reduction to a Variable in the Algebra

Sometimes 𝑓=(𝑓₁,…,𝑓_𝑛)∶Ω⊂𝕂^𝑛→𝕂^𝑛 can be expressed as a function of a variable 𝜁=∑^𝑛_𝑖=1𝑥_𝑖𝑒_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.

Proposition 4.1. With the same hypothesis as Theorem 3.4, there exists a basis 𝒟={𝛿₁,…,𝛿_𝑛} of invertible elements of the algebra 𝔸 such that the following diagram commutes Ω⊂𝕂−→^𝑛𝑓𝕂^𝑛Ω𝐴↓↓𝐴_′𝑔⊂𝔸−→𝔸, (4.1) where 𝐴 is the matrix associated to the change of basis ℰ→𝒟, where ℰ={𝑒₁,…,𝑒_𝑛} is the canonical basis of 𝕂^𝑛.

Proof. By a change of basis one has the following commutative diagram Ω⊂𝕂−→^𝑛𝑓𝕂^𝑛Ω𝐵↓↓𝐵_′′⊂𝔸→−^𝑔𝔸, (4.2) 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={𝛿₁,…,𝛿_𝑛} of regular elements of 𝔸, where without loss of generality 𝛿₁=𝑒. Then if 𝐴 is the matrix associated to the change of basis from ℰ to 𝒟, let 𝑔=𝐴∘𝑓∘𝐴⁻¹. Note that the domain of 𝑔 is Ω^′=𝐴(Ω).

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=1𝑦_𝑗𝛿_𝑗, for some regular basis 𝒟={𝛿₁,…,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={𝛿₁,…,𝛿_𝑛} of regular elements of the algebra 𝔸, with 𝛿₁=𝑒, without loss of generality we may assume that ||𝛿_𝑗||=1. Introducing the variable 𝜁 in the base 𝒟 as 𝜁=∑^𝑛_𝑖=1𝑦_𝑖𝛿_𝑖=∑^𝑛_𝑖=1𝑥_𝑖𝑒_𝑖, we then have 𝑓(𝑥₁,…,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 𝜁_𝑘=𝑦₁𝛿₁−𝑦_𝑘𝛿_𝑘+^𝑛_{𝑗=2𝑗≠𝑘}𝑦_𝑗𝛿_𝑗=−𝑦_𝑘𝛿_𝑘+^𝑛_{𝑗=1𝑗≠𝑘}𝑦_𝑗𝛿_𝑗=𝜁−2𝑦_𝑘𝛿_𝑘. (4.3) Then we have 𝑦_𝑘𝑒=𝜁−𝜁_𝑘2𝛿_𝑘⁻¹,for𝑘=2,…,𝑛, (4.4) and since 𝑦₁𝛿₁=𝜁−^𝑛_𝑗=2𝑦_𝑗𝛿_𝑗=𝜁−^𝑛_𝑗=2𝜁−𝜁_𝑗2, (4.5) then, 𝑦₁1𝑒=2(3−𝑛)𝜁+^𝑛_𝑗=2𝜁_𝑗𝛿₁⁻¹. (4.6) Furthermore one can define the differential operators 𝜕=1𝜕𝜁𝑛^𝑛_𝑗=1𝛿_𝑗⁻¹𝜕𝜕𝑦_𝑗,𝜕𝜕𝜁_𝑘=12𝛿₁⁻¹𝜕𝜕𝑦₁−𝛿_𝑘⁻¹𝜕𝜕𝑦_𝑘,for𝑘=2,…,𝑛, (4.7)(4.8) which satisfy 𝜕𝜁𝜕𝜕𝜁=1𝜁_𝑘=𝜕𝜁𝑛−2𝑛𝜕for𝑘=2,…,𝑛,𝜁_𝑘𝜕𝜁_𝑘=1𝜕𝜁𝜕𝜁_𝑘=𝜕𝜁_𝑗𝜕𝜁_𝑘=0for𝑘=2,…,𝑛,𝑘≠𝑗. (4.9)
If 𝑓 is 𝔸-differentiable, 𝑔 is also 𝔸-differentiable, hence 𝑔^′(𝜁)=𝑑𝑔𝑑𝑡𝜁+𝑡𝛿_𝑘|_𝑡=0=𝑑𝑔𝑑𝑡𝜁+𝑡𝛿_𝑗|_𝑡=0,∀𝑗,𝑘, (4.10) and since 𝑑𝑔𝑑𝑡𝜁+𝑡𝛿_𝑘|_𝑡=0=𝛿_𝑘⁻¹𝜕𝑔𝜕𝑦_𝑘(𝜁), (4.11) then for 𝑘=2,…,𝑛𝜕𝑔𝜕𝜁_𝑘(𝜁)=12^𝑛_𝑖=1𝛿₁⁻¹𝜕𝑔_𝑖𝜕𝑦₁(𝜁)−𝛿_𝑘⁻¹𝜕𝑔_𝑖𝜕𝑦_𝑘(𝜁)𝛿_𝑖=0, (4.12) so the right hand expression of 𝑔(𝜁)=∑^𝑛_𝑖=1𝑔_𝑖(𝜁)𝛿_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.
Moreover, by (4.7), (4.10), and (4.11) one has that 𝑔^′(𝜁)=𝜕𝑔𝜕𝑦₁(𝜁)=𝜕𝑔(𝜁).𝜕𝜁 (4.13)

In the following example we show how we can reduce the variables of a map by substituting for variables in an algebra.

Example 4.3. Let 𝑓(𝑥₁,𝑥₂,𝑥₃)=(𝑥²₁,2𝑥₁𝑥₂,𝑥²₂+2𝑥₁𝑥₃). Then 𝐽𝑓=2𝑥₁002𝑥₂2𝑥₁02𝑥₃2𝑥₂2𝑥₁, (4.14) the 𝐽𝑓 is in normal form, hence 𝑓 is 𝔸-differentiable. If 𝑁=000100010 (4.15) and 𝐼 is the identity, the multiplication in the associated algebra 𝑅(𝔸) is given by multiplication of matrices 𝐼, 𝑁, and 𝑁² representing, respectively, 𝑅(𝑒₁),𝑅(𝑒₂),and𝑅(𝑒₃), where ℰ={𝑒₁,𝑒₂,𝑒₃} is the canonical basis of ℝ³. It is easy to see that 𝑒₂ and 𝑒₃ do not have inverse in the algebra 𝔸. Consider the basis of regular elements 𝒟=𝛿₁=𝑒₁,𝛿₂=𝑒₁+𝑒₂,𝛿₃=𝑒₁+𝑒₃, (4.16) we see that the matrix associated to the change of basis from ℰ to 𝒟 is 𝐴=1−1−1010001, (4.17) so 𝑓 is transformed to 𝑔=𝐴∘𝑓∘𝐴⁻¹, 𝑔𝑦₁𝛿₁+𝑦₂𝛿₂+𝑦₃𝛿₃=𝑦²₁−2𝑦²₂−2𝑦₂𝑦₃−𝑦²₃𝛿₁+2𝑦₂𝑦₁+𝑦₂+𝑦₃𝛿₂+𝑦²₂+2𝑦₃𝑦₁+𝑦₂+𝑦₃𝛿₃. (4.18) Thus 𝜁=𝑦₁𝛿₁+𝑦₂𝛿₂+𝑦₃𝛿₃ (4.19) is the variable in the algebra 𝔸. Now consider the conjugates 𝜁₂=𝑦₁𝛿₁−𝑦₂𝛿₂+𝑦₃𝛿₃,𝜁₃=𝑦₁𝛿₁+𝑦₂𝛿₂−𝑦₃𝛿₃. (4.20) Then 𝑦₁1𝑒=2𝜁₂+𝜁₃𝛿₁⁻¹,𝑦₂1𝑒=2𝜁−𝜁₂𝛿₂⁻¹,𝑦₃1𝑒=2𝜁−𝜁₃𝛿₃⁻¹, (4.21) so by substituting this in 𝑔 and simplifying we obtain 𝑔(𝜁)=𝜁², (4.22) the function in the variable of the algebra.

5. 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 𝑓∶Ω⊂ℝ^𝑛→ℝ^𝑛,𝑔∶Ω^′⊂𝔸→𝔸, (5.1) 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 𝑔(𝜁)Φ(𝜁)=−Φ^′(𝜁),𝑓(𝑥)=−𝐽𝜙(𝑥)⁻¹𝜙(𝑥), (5.2) with 𝜙(𝑥)=(Φ∘𝐴)(𝑥).

Proof. We need to show that there exists Φ∶Ω^′⊂𝔸→𝔸 such that 𝑔(𝜁)Φ(𝜁)=−Φ^′(𝜁). (5.3) Noticing that log(Φ(𝜁))^′=Φ^′(𝜁)Φ(𝜁)1=−𝑔(𝜁), (5.4) and by Remark 3.3, it follows that Φ(𝜁)=exp−^𝜁𝑑𝑧𝑔(𝑧), (5.5) for 𝜁∈Ω^′⧵𝒮, where 𝒮=𝜁∈Ω^′∶𝑔(𝜁)∉𝔸^∗, (5.6) with 𝔸^∗ 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. On the other hand, if 𝑔(𝜁) is the polynomial function 𝑔(𝜁)=𝑐_𝑛𝜁^𝑛+⋯+𝑐₁𝜁+𝑐₀ with 𝑐_𝑛 singular, then 𝑔(𝜁) may be regular for all 𝜁∈𝔸, (see [26, p. 418]), in which case 𝒮=∅.

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].

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.

Remark 5.6. Further characterization and properties of 𝒮 will be studied elsewhere. In what follows we assume that 𝒮≠Ω^′.

Let 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛⧵{0}, then consider the projection onto 𝑆^𝑛−1(𝑥)=𝑥‖𝑥‖∈𝑆^𝑛−1. (5.7)

Corollary 5.7. 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, 𝑑𝑥(𝑥),𝑑𝑡=𝑓 (5.8) correspond to the level curves of the function ℎ₁(𝑥)=Π𝜙(𝑥), (5.9) moreover the real-valued function, ℎ₂(𝑥)=log𝜙(𝑥), (5.10) is a linear function of 𝑡 along the solutions 𝑥(𝑡) of (5.8).

Proof. By Theorem 3.4, Proposition 4.1, and Theorem 4.2, the solutions 𝑥(𝑡) of (5.8) are in correspondence with the solutions 𝜁(𝑡) of 𝑑𝜁=𝑔(𝜁).𝑑𝑡 (5.11) On the other hand, Corollary 5.1 shows that Φ(𝜁(𝑡))=Φ𝜁𝑡₀exp−𝑡−𝑡₀, (5.12) 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 𝑥₀∈Ω⧵𝐴⁻¹(𝒮) at time 𝑡₀∈ℝ, one needs only to plot the level curve {𝑥∈Ω∶ℎ₁(𝑥)=ℎ₁(𝑥₀)}. Moreover one can find explicitly the point 𝑥(𝑡₁), for 𝑡₁∈ℝ, as the intersection of the level curve ℎ₁(𝑥)=ℎ₁(𝑥₀) and the hypersurface ℎ₂(𝑥)=ℎ₂(𝑥₀)−𝑡₁+𝑡₀.

Example 5.9. Consider the function 𝑓𝑥,𝑦=𝑢𝑥,𝑦,𝑣𝑥,𝑦, (5.13) where 𝑢=1𝑥,𝑦62+6𝑥+−22+20𝑥+7𝑦4+5𝑥²+5+8𝑥−1+𝑦+𝑦−4+5𝑦2+4𝑥+5𝑦4+5𝑥²,𝑣=1+8𝑥1+𝑦+𝑦4+5𝑦𝑥,𝑦65−4+6𝑦−−4+5𝑥+4𝑦4+5𝑥²++8𝑥−1+𝑦+𝑦−4+5𝑦4−7𝑥−20𝑦4+5𝑥².+8𝑥1+𝑦+𝑦4+5𝑦 (5.14) Let 𝐴=1221, (5.15) and notice that 𝐴⋅𝐽𝑓𝐴⁻¹𝑥,𝑦⋅𝐴⁻¹=𝑎𝑏𝑎𝑥,𝑦𝑥,𝑦−𝑏𝑥,𝑦𝑥,𝑦, (5.16) with 𝑎2𝑥,𝑦=1−𝑥²+−2+𝑦²+𝑥−6+4𝑥+3𝑦𝑥²+−2+𝑦²²+𝑥4𝑥−32+𝑦𝑥²+2+𝑦²²−2𝑥²+2+𝑦²,𝑏𝑥,𝑦=4𝑦𝑥⁴+−4+𝑦²²+2𝑥²4+𝑦²²×48−96𝑥−24𝑥²−16𝑥³−9𝑥⁴+2𝑥⁵+2(−3+2𝑥)4+𝑥²𝑦²+(3+2𝑥)𝑦⁴. (5.17) Hence 𝐴⋅𝐽𝑓(𝐴⁻¹(𝑥,𝑦))⋅𝐴⁻¹ belongs to the normal algebra ℂ, so the previous results are available.
Thus 𝑔𝑥,𝑦=𝐴∘𝑓∘𝐴⁻¹𝑥,𝑦=𝑢₁𝑥,𝑦,𝑣₁𝑥,𝑦, (5.18) with 𝑢₁𝑥,𝑦=−1+𝑥+6−4𝑥−3𝑦2𝑥²+−2+𝑦²+6−4𝑥+3𝑦2𝑥²+2+𝑦²,𝑣₁=𝑥𝑥,𝑦−12+12𝑥+𝑥³𝑦+2−2+𝑥²𝑦³+𝑦⁵𝑥⁴+−4+𝑦²²+2𝑥²4+𝑦² (5.19) is just a (complex) analytic function, since 𝐽𝑔𝑥,𝑦=𝐴⋅𝐽𝑓𝐴⁻¹𝑥,𝑦⋅𝐴⁻¹∈𝑅(ℂ), (5.20) so by letting 𝑧=𝑥+𝑖𝑦, we have 𝑔(𝑧)=𝑢₁𝑧+𝑧2,𝑧−𝑧2𝑖+𝑖𝑣₁𝑧+𝑧2,𝑧−𝑧=2𝑖2−𝑧²+𝑧³4+𝑧². (5.21)
By Corollary 5.1 one has Φ(𝑧)=exp−^𝑧𝑑𝜁𝑔(𝜁), (5.22) so that Φ(𝑧)=𝑧−1−𝑖𝑧−1+𝑖^𝑖(𝑧+1)⁻¹=𝑒^{2arctan(1−𝑧)}.𝑧+1 (5.23) Note that in ℝ² 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 ℎ(𝑧)(Φ(𝑧𝑦=Arg))=−arctan+11+𝑥2log1−4𝑦2+(−2+𝑥)𝑥+𝑦2+𝑦 (5.24) is a constant of motion ℎ associated to 𝑔(𝑧) and the one associated to 𝑓(𝑥,𝑦) is ℎ𝑥,𝑦=(ℎ∘𝐴)𝑥,𝑦=−arctan2𝑥+𝑦+11+𝑥+2𝑦24log1−2𝑥+𝑦2+5𝑥².+𝑦−2+5𝑦+𝑥2+8𝑦 (5.25) So by Corollary 5.8 the trajectories associated to 𝑥^′(𝑡),𝑦^′(𝑡)=𝑓𝑥(𝑡),𝑦(𝑡), (5.26) are the level curves of ℎ(𝑥,𝑦). These are presented in 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 Table 1).

In order to parametrize the solution we need to calculate ℎ₂(𝑥,𝑦) which turns out to be ℎ₂𝑥,𝑦=−arctan−1+𝑥+2𝑦1−2𝑥−𝑦+arctan1−𝑥−2𝑦−11+2𝑥+𝑦2log(2𝑥+𝑦)²+1+𝑥+2𝑦². (5.27) According to Corollary 5.8 we proceeded to calculate the intersection the level curves ℎ(𝑥,𝑦)=−1.68195 and ℎ₂(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.

Table 1 

Coordinates of the intersection of the level curves ℎ(𝑥,𝑦)=−1.68195 and ℎ₂(𝑥,𝑦)=2.26841−𝑡 for 𝑡=0,1,2,3,4.

Acknowledgments

The first author wishes to thank Jesús Muciño-Raymundo for introducing him to the study of complex vector fields and for many interesting, insightful, and productive discussions. This work partially funded by UABC Grant no. 0196, and by CONACYT Grant CB-2010/150532.