Adv Syst Sci Appl 2020; 01:119–127
Published online at http://ijassa.ipu.ru/index.php/ijassa/article/view/866

Local Normal Forms of Autonomous Quasi-Linear Constrained
Differential Systems

Alexander M. Kotyukov1*, Stanislav O. Nikanorov1, Natalya G. Pavlova1,2

1V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
2Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Russia

Abstract: The paper presents a study of impasse (singular) points of autonomous quasi-linear
constrained differential systems, also called differential-algebraic equations. The interest in such
systems is motivated by their applications in various problems of pure and applied mathematics,
including control theory, biology, and electric engineering. Local normal forms of such systems
in a neighborhood of their impasse points are established.

Keywords: impasse point, direction field, normal form, diffeomorphism, symmetry

1. INTRODUCTION

We consider systems of differential equations in the form

A(ξ)ξ′ = b(ξ), ξ′ =
dξ

dt
, ξ = (ξ1, . . . , ξn)∗ ∈ Rn, (1.1)

where A = (aij) is an n× n matrix and b = (b1, . . . , bn)∗ is a vector function. The
components aij , bj are assumed to be C∞-smooth functions on ξ. There are many possible
names of systems (1.1): differential systems of Sobolev type, generalized vector fields,
descriptor systems, quasi-linear constrained† differential systems, etc.

The interest in systems (1.1) is motivated by their applications in various problems of pure
and applied mathematics (including control theory and electric engineering). For instance,
systems (1.1) describe the dynamic of electric circuit in nonlinear RLC-networks (networks
consisting of a resistor, an inductor, and a capacitor). See [1]– [14] and the references therein.
In a pioneer work [15], systems (1.1) appear as a tool for studying a systems of PDEs of the
mixed type, which describes the motion of a body filled with a viscous incompressible fluid.
In a recent series of papers [16]– [20], systems (1.1) in dimension n = 3 describe geodesic
lines in singular metrics.

There exists several different approaches for studying systems (1.1) and, more generally,
nonlinear systems of differential equations

F (t, ξ, ξ′) = 0, ξ′ =
dξ

dt
, ξ = (ξ1, . . . , ξn)∗ ∈ Rn, (1.2)

∗Corresponding author: amkotyukov@mail.ru
†The reason for the this term will soon become clear.



120 A.M. KOTYUKOV, S.O. NIKANOROV, N.G. PAVLOVA

which are often called implicit differential equations or differential-algebraic equations
(DAEs). Here F : R2n+1 → Rn is assumed to be a C∞-smooth mapping. There exists several
approaches to investigation of systems (1.1), (1.2).

An analytical approach is based on the decoupling procedure. The idea is to rearrange
terms within the given system so that it is decomposed in two subsystems of lower dimensions
(separated as far as possible), where the first subsystem is equivalent to a standard ODE of
of maximum possible dimension and the second one is a DAE having a special form. The
highest goal is so-called complete decoupling, where the both subsystems are completely
independent, that is, they do not have common unknowns. For linear DAEs with constant
coefficients this method is comparable with the Weierstrass–Kronecker normal form of
regular matrix pencils. This approach is presented in the book [21].

Another geometric approach goes back to H. Poincaré.‡ He used it for a single implicit
differential equation F (x, y, y′) = 0, that is, the case n = 1. Further development of this
approach is given, e.g., in [22]– [34]. In the modern terminology, the main idea is the
following.

By Jk denote the space of k-jets of vector functions ξ(t) : R→ Rn. In the case of
systems (1.2), consider a manifold Mn+1

F ⊂ J1, J1 ' R2n+1, defined by the equation F = 0.
Moreover, DAE (1.2) defines a direction field on Mn+1

F , whose integral curves are 1-graphs
(Legendrian lifts) of integral curves of (1.2). In other words, we pass from the multivalued
vector direction field given by DAE (1.2) in J0 (i.e., the (t, ξ)-space) to a single-valued
direction field on the manifold Mn+1

F . The natural projection π : J1 → J0 sends integral
curves of the field defined on Mn+1

F to integral curves of DAE (1.2). The restriction of
the projection π : Mn+1

F → J0 is a mapping of two (n+ 1)-dimensional manifolds. It has
singularities at those points of Mn+1

F where the matrix (∂F/∂ξ′) vanishes. Integral curves of
(1.2) have singularities (generically, cusps) at corresponding point of the (t, ξ)-space.

In the case of systems (1.1), the above construction is essentially simplified: no need to
consider the space J1. Indeed, writing system (1.1) in the pfaffian form

a11dξ1 + a12dξ2 + · · ·+ a1ndξn − b1dt = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
an1dξ1 + an2dξ2 + · · ·+ anndξn − bndt = 0,

(1.3)

one can see that system (1.1) defines a direction field in the (t, ξ)-space:

ξ̇1 = ∆1(ξ), . . . , ξ̇n = ∆n(ξ), ṫ = ∆(ξ), (1.4)

where ∆ is the determinant of the matrix A, and ∆i is the determinant of the matrix obtained
from A by replacing of its ith column with b. Here the dot over a symbol means the
differentiation by a new parameter playing the role of time.

A principal difference between system (1.1) and usual autonomous ODEs is that system
(1.1) (and consequently, field (1.4)) possesses so-called degenerate hypersurface

Γ = {ξ : ∆(ξ) = 0},

which is also called the criminant of the system (1.1). Generically, there are no integral curves
of (1.1) passing through Γ; see [29]. Therefore, Γ is also called impasse hypersurface of the
system, and points of Γ are called impasse points. It is worth observing that if system (1.1)
describes a model of a natural process in real time, the existence of impasse points in the
applicability domain often means that the model is not adequate.

‡Mémoire sur les courbes définies par les équations différentielles (1885).

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



LOCAL NORMAL FORMS OF DIFFERENTIAL SYSTEMS 121

The germs of two systems (1.1) are called Ck-equivalent, if there exists a local Ck-
diffeomorphism of the ξ-space that conjugates the germs of the corresponding direction fields
(1.3). Our goal is to simplify field (1.3) (and consequently, the corresponding system (1.1))
in a neighborhood of its impasse points using an appropriate local Ck-diffeomorphism of the
ξ-space. Here k ≥ 1 is integer or∞ or ω. Cω is the standard symbol for the class of analytic
maps.§

The germs of two systems (1.1) are called orbitally Ck-equivalent, if there exist a local
Ck-diffeomorphism of the ξ-space and Ck-mapping of the independent variable t 7→ τ(t, ξ)
that conjugate the germs of the corresponding direction fields (1.3).

The classification of systems (1.1) at all impasse points is not observable. According to
the general ideology of singularity theory, a number of classes of impasse points that admits
relatively simple normal forms is determined. For the Cω-equivalence, many results of this
sort appear in [28], where systems (1.1) are considered in real, complex and even infinite
dimensional (Banach) complex spaces. For the Ck-orbital equivalence, many results appear
in [32].

In this paper, we present some new results aboutCk-normal forms of the germs of systems
(1.1) at their impasse points of a special type.

2. MAIN RESULTS

There are several geometric objects naturally connected with system (1.1).
The firts object of thid sort is the criminant Γ. Other important object are the family of

linear operators
A(ξ) : TξRn → TξRn,

their images ImA(ξ) and kernels KerA(ξ). The criminant Γ is the locus of points ξ where
dim ImA(ξ) < n or, equivalently, dim KerA(ξ) > 0.

The simplest type of impasse points is so-called non-singular impasse points ξ ∈ Γ that
satisfy three following conditions:

1. d∆(ξ) 6= 0, that is, Γ is a regular hypersurface.
2. dim KerA(ξ) = 1 and the direction KerA(ξ) is transversal to Γ.
3. b(ξ) /∈ ImA(ξ), that is, ImA(ξ)⊕ 〈b(ξ)〉 = Rn.

Theorem 2.1:
The germ of system (1.1) at every its non-singular impasse point is C∞-equivalent (in the
analytic category, Cω-equivalent) to

ξ′1 = 0, . . . , ξ′n−1 = 0, ξn ξ
′
n = ±1.

Moreover, in the category of orbital equivalence, one can reduce ±1 to 1.

The proof of Theorem 2.1 can be found in [27, 28, 32].
Now consider a point ξ◦ ∈ Γ such that the 1st and 2nd conditions hold true, but the 3rd

condition fails. From dim KerA(ξ◦) = 1 (the 2nd condition) and the known identity

dim KerA(ξ) + dim ImA(ξ) = n

it follows that dim ImA(ξ◦) = n− 1, i.e., rank of the matrix A(ξ◦) = n− 1. Then there
exists i ∈ {1, . . . , n} such that rank of the matrix obtained from A(ξ◦) by eliminating the
ith column is n− 1, and the germs of ∆, ∆1, . . . ,∆n belong to the ideal I = 〈∆,∆i〉 in the
ring of smooth functions, and the set of singular points (equilibriums) of the corresponding

§In speaking about Cω-equivalence, we assume that the systems (1.1) are also analytic.

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



122 A.M. KOTYUKOV, S.O. NIKANOROV, N.G. PAVLOVA

field (1.4) is given by two equations: ∆ = ∆i = 0. (The detailed proof of this fact can be
found in [31].)

Therefore, singular point of the field (1.4) are not isolated, they fill a submanifold W ⊂ Γ
of codimension 2. At every point ξ ∈ W in a neighborhood of ξ◦, the spectrum of the
linear part of the field (1.4) is (λ1, λ2, 0, . . . , 0). Here λ1,2 are complex numbers, generically
Reλ1,2 6= 0 at almost all points of W .
Lemma 2.1:
Assume that Reλ1,2(ξ◦) 6= 0. Then in a neighborhood of ξ◦, the following statements hold
true:

(i) The set W is the center manifold of the field (1.4).
(ii) There exist i1, i2 ∈ {1, . . . , n} such that the ideal I = 〈∆i1 ,∆i2〉.

Proof
The first statement is trivial.

For the second statement, remark that it is possible to select as generators of I any two
elements v, w ∈ I such that dv, dw at the point ξ◦ are linearly independent. Consider the
Jacobi matrices

J0 =

∥∥∥∥∥∂(∆1, . . . ,∆n,∆)

∂(ξ1, . . . , ξn, t)

∥∥∥∥∥, J =

∥∥∥∥∥∂(∆1, . . . ,∆n)

∂(ξ1, . . . , ξn)

∥∥∥∥∥.
at the point ξ◦. Since the functions ∆1, . . . ,∆n,∆ do not depend on t, the spectrums of
matrices J0 and J have the form (λ1, λ2, 0, . . . , 0) with the same λ1,2. Therefore, among the
functions ∆1, . . . ,∆n there exist a couple v = ∆i1 , w = ∆i2 whose differentials at ξ◦ are
linearly independent. Therefore, v, w are generators of the ideal I .

Lemma 2.1 shows that using an appropriated renaming of the variables

(ξ1, . . . , ξn) 7→ (x, y, z), z = (z1, . . . , zn−2),

one can write the corresponding field (1.4) in the form

ẋ = v, ẏ = w, żi = αiv + βiw, i = 1, . . . , n− 2,

ṫ = av + bw,
(2.5)

Here a, b, α, β, v, w are C∞ (resp., Cω) functions on x, y, z such that v, w vanish at ξ◦. The
ideal I = 〈v, w〉, and the center manifold W consisting of singular points is given by the
equations v = w = 0.

Since all components of the field (2.5) do not depend on t, one can consider the projection
of (2.5) to the ξ-space:

ẋ = v, ẏ = w, żi = αiv + βiw, i = 1, . . . , n− 2. (2.6)

The classification of fields (2.6) much simpler than the classification of fields with isolated
singular points with the same spectrum, see [17, 18, 31]. It is explained by the fact that the
non-trivial dynamics is connected with the center manifold of the field. In the case of (2.6), the
center manifolds consists of singular points of the fields, therefore the non-trivial dynamics
is identically zero.
Lemma 2.2:
Assume that Reλ1,2(ξ◦) 6= 0 and there are no resonances

p1λ1(ξ◦) + p2λ2(ξ◦) = 0, p1,2 ∈ Z+, p1 + p2 ≥ 1. (2.7)

Then for any integer k ≥ 1, the germ of (2.6) is Ck-equivalent to the germ

ẋ = ṽ(x, y, z), ẏ = w̃(x, y, z), żi = 0, i = 1, . . . , n− 2, (2.8)

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



LOCAL NORMAL FORMS OF DIFFERENTIAL SYSTEMS 123

at the origin, with some ṽ, w̃ ∈ 〈x, y〉. Moreover, if Reλ1,2(ξ◦) have the same sign, the above
equivalence holds true with k =∞.

Without loss of generality, further we assume that |λ1(ξ◦)| ≤ |λ2(ξ◦)|. Put

λ◦ = λ1(ξ◦) : λ2(ξ◦), µ◦ = λ2(ξ◦) : λ1(ξ◦).

Lemma 2.3:
Assume that the conditions of Lemma 2.2 hold true. Then the germ of the field (2.6) is orbitally
Ck-equivalent to the germ (2.8), where ṽ, w̃ have the form

(a) ṽ = x, w̃ = λ(z)y, if λ1,2(ξ◦) are real and µ◦ /∈ N ∪Q−,
(b) ṽ = λ(z)x, w̃ = y + α(z)xµ◦ , if µ◦ ∈ N \ {1},
(c) ṽ = α(z)x+ β(z)y, w̃ = −β(z)x+ α(z)y, if λ1,2(ξ◦) are complex.

Note that λ(0) = λ◦, µ(0) = µ◦ in the cases (a), (b). In the case (c), α(0), β(0) are
respectively the real and imaginary parts of λ1,2(ξ◦).

Lemmas 2.2, 2.3 can be found in [17, 18, 31].
In the cases (a), (b), (c) enumerated in Lemma 2.3, consider three families of local

diffeomorphisms (Rn, 0)→ (Rn, 0) given by the following changes of the variables x, y:

(a) x 7→ xϕ, y 7→ yϕλ(z), (2.9)

(b) x 7→ xϕ,

y 7→ yϕµ(z) + α(z)µ(z)xµ◦
ϕµ(z) − ϕµ◦
µ(z)− µ◦

, z 6= 0,

y 7→ yϕµ(z) + α(z)µ(z)xµ◦ϕµ◦ lnϕ, z = 0,

(2.10)

with all smooth strictly positive functions ϕ = ϕ(x, y, z), and

(c)
x 7→ eα(z)ϕ(x cos β(z)ϕ+ y sin β(z)ϕ),

y 7→ eα(z)ϕ(y cos β(z)ϕ− x sin β(z)ϕ),
(2.11)

with all smooth functions ϕ = ϕ(x, y, z).
Lemma 2.4:
The families of diffeomorphisms (2.9), (2.10), (2.11) consist of symmetries of the direction
field (2.8) in the cases (a), (b), (c), respectively.

In other words, every diffeomorphism from the families (2.9), (2.10), (2.11) sends vector
field (2.8) into a parallel vector field¶ in the cases (a), (b), (c), respectively.

It is worth observing that families (2.9), (2.10), (2.11) are subgroups of the whole group
of symmetries of the direction field (2.8) in the cases (a), (b), (c), respectively, but non of
them coincide with the whole group. For instance, the family (2.9) does not contain the linear
changes

x 7→ c1x, y 7→ c2y

with arbitrary constants c1,2 6= 0.
The proof of Lemma 2.4 is by direct calculations and omitted.
Now we can get the main result of the paper.

Theorem 2.2:
Let ξ◦ ∈ Γ be an impasse point of the system (1.1) satisfying the above conditions. In addition,
in the cases (a), (b), we assume that the eigenvectors with λ1,2(ξ◦) are transversal to the

¶Two vector fields are parallel, if one of them is obtained from another by multiplication by a non-vanishing smooth function.
In other words, two vector fields are parallel, if the corresponding direction fields coincide.

Copyright © 2020 ASSA. Adv Syst Sci Appl (2020)



124 A.M. KOTYUKOV, S.O. NIKANOROV, N.G. PAVLOVA

criminant Γ. Then for any integer k ≥ 1, the germ of system (1.1) is Ck-equivalent to the
germ

(x+ y)ψx′ = ṽ, (x+ y)ψy′ = w̃, z′1 = 0, . . . , z′n−2 = 0, (2.12)

at the origin, where ṽ, w̃ are defined in Lemma 2.3 and ψ = ψ(x, y, z) is a C∞-smooth
function, ψ(0) 6= 0. Under the same conditions, the germ of system (1.1) is orbitally Ck-
equivalent to (2.12) with the same ṽ, w̃ and ψ ≡ 1. Moreover, if Reλ1,2(ξ◦) have the same
sign, the above equivalences hold true with k =∞.

Proof
By Lemma 2.3, there exist a Ck-smooth local diffeomorphism

f : (ξ1, . . . , ξn) 7→ (x, y, z), z = (z1, . . . , zn−2),

that sends the point ξ◦ to 0 and brings the germ of the direction field (2.6) at ξ◦ to the orbital
normal form (2.8). Then f transforms the corresponding direction field (2.5) into the form

ẋ = ṽ, ẏ = w̃, żi = 0, i = 1, . . . , n− 2,

ṫ = aṽ + bw̃,
(2.13)

where ṽ, w̃ are defined in 2.3 and a, b are smooth functions on the variables x, y, z.
The criminant of the system (1.1) corresponding to (2.13) is given by the equation

aṽ + bw̃. The condition that the eigenvectors with λ1,2(ξ◦) are transversal to the criminant
yields a(0) 6= 0, b(0) 6= 0. Let us show that there exists a diffeomorphism of the families
(2.9), (2.10), (2.11) in the cases (a), (b), (c), respectively, that convert the criminant Γ into a
hyperplane, for instance, the hyperplane x+ y = 0.

Consider that case (a). Then ṽ = x, w̃ = λ(z)y. By Lemma 2.4, any diffeomorphism of
the family (2.9) transforms the direction field (2.13) into

ẋ = x, ẏ = λ(z)y, żi = 0, i = 1, . . . , n− 2,

ṫ = axϕ+ bλ(z)yϕλ(z) = aϕ (x+ a−1bλ(z)ϕλ(z)−1y).
(2.14)

Without loss of generality, assume that a−1(0)b(0)λ(0) > 0. Otherwise one can make the
change variables y 7→ −y, which preserves the first n− 1 components of the field (2.14) and
brings its last component to the desirable form.

Thus, it suffices to establish the existence of a positive smooth function ϕ satisfying the
equation

a−1(xϕ, yϕλ(z), z) b(xϕ, yϕλ(z), z)λ(z)ϕλ(z)−1 = 1. (2.15)
The derivative of the left-hand side of (2.15) by ϕ is not equal to zero. By the implicit function
theorem, a function ϕ with the desirable properties exists in a neighborhood of the origin.

The diffeomorphism (2.9) with ϕ founded above preserves the first n− 1 components of
the direction field (2.14) and transforms its last component into

ṫ = ψ(x, y, z) (x+ y),

where ψ is a smooth function non-vanishing at the origin. The corresponding system (1.1)
can be written in the form

(x+ y)ψx′ = x, (x+ y)ψy′ = λ(z)y, z′1 = 0, . . . , z′n−2 = 0. (2.16)

Finally, if we deal with the orbital equivalence, one can make the change of the
independent variable: t 7→ τ , where the new and old variables are connected with the
differential equation

dt

dτ
= ψ(x, y, z).

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LOCAL NORMAL FORMS OF DIFFERENTIAL SYSTEMS 125

The geometric sense this equation is that we make a parameterization integral curves of
system (2.16), which depends on a point on every curve, so that the function ψ in (2.16)
becomes identically 1.

The case (b) is similar to (a) and omitted.
Consider that case (c). It is easy to see that (α− β)a− (α + β)b, (α + β)a+ (α− β)b

do not simultaneously vanish at the origin. Without loss of generality, we assume that the
latter functions does not vanish at the origin.

By Lemma 2.4, any diffeomorphism of the family (2.11) transforms the direction field
(2.13) with

ṽ = α(z)x+ β(z)y, w̃ = −β(z)x+ α(z)y

into the direction field whose first n− 1 components coincides with those in (2.13) and the
nth component is

ṫ = xeαϕ((αa− βb) cos βϕ− (αb+ βa) sin βϕ)+

yeαϕ((αa− βb) sin βϕ+ (αb+ βa) cos βϕ).

Thus, it suffices to establish the existence of a smooth function ϕ such that

(αa− βb) cos βϕ− (αb+ βa) sin βϕ = (αa− βb) sin βϕ+ (αb+ βa) cos βϕ, (2.17)

where α, β are given functions on z, while a, b depend on ϕ:

a = a(eαϕ(x cos βϕ+ y sin βϕ), eαϕ(−x sin βϕ+ y cos βϕ), z),

b = b(eαϕ(x cos βϕ+ y sin βϕ), eαϕ(−x sin βϕ+ y cos βϕ), z).

Equation (2.17) can be transformed to the equivalent form

tan(βϕ) =
(α− β)a− (α + β)b

(α + β)a+ (α− β)b
(2.18)

The existence of a function ϕ satisfying equation (2.18) follows from the implicit function
theorem. The remaining part of the proof in the case (c) repeats the reasonings carried out in
the case (a).

ACKNOWLEDGEMENTS

This paper was written with the financial support of RFBR (grants no. 19-01-00080, no.
20-01-00610). Theorem 2.2 was obtained by the first author with the financial support of the
Russian Science Foundation (project no. 20-11-20131).

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	Introduction
	Main results