Abstract
We study the classification of the hyperbolic singularities to 3-dimensional interval linear differential equations as an application of interval eigenvalues using the Constraint Interval Arithmetic (CIA). We also present the ideas to calculate the interval eigenvalues using the standard interval arithmetic.
Keywords: Interval eigenvalues, Dynamical systems, Classification of singularities
Introduction
Many applied problems have uncertainties or inaccuracies due to data measurement errors, lack of complete information, simplification assumption of physical models, variations of the system, and computational errors. An encoding of uncertainty as intervals instead of numbers when applicable is an efficient way to address the aforementioned challenges.
When studying an interval problem we need to first understand what is the context. In this presentation, we are concerned if there exists dependence, independence or both in the parameters involved. In accordance to this context we need to choose the appropriated arithmetic. Where we have the total independence or dependence, we can use interval arithmetic or single level arithmetic (SLA). If we are studying a problem where there is the independence as well as the dependence in parameters then constraint interval arithmetic (CIA) is a good choice.
We are interested in studying the interval eigenvalue problem associated with differential equations. This problem has many applications in the fields of mechanics and engineering. The first interval eigenvalues results were obtained by Deif [5], Deif and Rohn [15], Rohn [16]. Subsequently, approximation methods results were obtained by Qiu et al. [14], Leng et al. [10], Hladik [9] and Hladik et al. [6–8].
This presentation establishes conditions on the parameters of interval linear autonomous differential systems to classify the hyperbolic equilibrium point in 3-dimensions. Moreover a detailed study is given for an example using CIA along with a computational method for complex conjugate eigenvalues where we obtain the lower and upper bounds of the real eigenvalue.
Preliminaries
The following outlines the standard interval arithmetic WSMA (Warmus, Sunaga and Moore Arithmetic) and CIA (constraint interval arithmetic).
Let 
be such that 
and 
then for WSMA arithmetic we have the following operations:
Remark: Note that in WSMA arithmetic 
is never 0 unless 
is a real number (width zero) nor is 
.
Definition 1
[11] An interval 
CI (constraint interval) representation is the real single-valued function 
Constraint interval arithmetic (CIA) is 
, where 
and 
The set of 
interval matrices will be denoted by 
An interval matrix 
is interpreted as a set of real 
matrices
 |
Denote by 
where 
and 
are matrix whose entries are given by right and left sides of all intervals numbers 
respectively. In the CI context each element in 
is given by 
Definition 2
[5] Given 
an interval matrix in 
, the set of eigenvalues is given by:
 |
In addition, we denote by 
the midpoint and the radius of [A], respectively.
In what follows we use the notation 
to mean the dependence of the choice of the values for 
in the interval 
.
Definition 3
[12] Let be an interval matrix 
then the CI matrix is defined by
 |
where 
for 
and 
and the symbol 
denotes componentwise multiplication. Then, we say that 
is an eigenvalue of 
if 
i.e., 
where 
is the identity matrix of order n.
Remark: Here for each choice of matrix 
we have a deterministic problem to calculate eigenvalues. We can get the interval eigenvalues by minimizing and maximizing 
by varying all 
between 0 and 1.
To classify the equilibrium point in a 3-dimensional linear differential system, firstly we need to know how we can classify it in according the eigenvalues obtained of the matrix of the coefficients from linear differential system.
Consider a linear three-dimensional autonomous systems 
of the form
 |
1 |
where the 
are constants. Suppose that (1) satisfies the existence and uniqueness theorem. Given a matrix of order 
, we have the following possibilities for the real canonical forms:
 |
where the eigenvalue 
with 
for pure imaginary, 
for real case and both are different of zero for complex 
. If the singularities in matrix A are hyperbolic 
then we have the following possibilities:
Remark: We are not interested in the cases 
and/or the real eigenvalue equal to zero since we cannot classify the equilibrium point.
Interval 3-Dimensional Linear Differential System
Given system (1) with the initial conditions, we can to consider the Initial Value Problem with uncertainty, where the initial conditions and/or coefficients are uncertainty. The behavior of the solution trajectories are not changed if only its initial condition has a small perturbation. For example, consider the system 
where 
with the initial conditions 
then we have the following unstable trajectories (see Fig. 1).
Fig. 1.
The graph for x(t), y(t), z(t) com initial conditions 
and 
, respectively.
Remark: When only the initial conditions are intervals then the eigenvalues do not change and the stability or instability is kept. But, if in (1) the entries in the matrix of coefficient vary, then we need to evaluate what will happen with the equilibrium point in (1).
Then, we consider in (1), A as an interval matrix. According to Definition 3, we have the following problem:
 |
2 |
where 
for 
Observe that system (2):
Has an unique equilibrium point at the origin (0, 0, 0) if given the matrix
the 
for 
- For
, the eigenvalues are obtained from the equation:
where
3 
Theorem 1
If system (2) has a unique equilibrium point at (0, 0, 0), then there is a matrix 
such that the equilibrium point is classified according to Table 1.
Table 1.
Classification of the hyperbolic flows in dimension 3.
| Classification | Eigenvalues 
|
|---|---|
| Attractors (stable) |
 and  or 
|
| Saddle point |
 and  or  and 
|
| Repellors (unstable) |
 and  or 
|
Proof
Given the matrix 
in system (2), the nature of the equilibrium is defined according to the zeroes of the characteristic polynomial of 
 |
4 |
The roots of the polynomial of order 3 in (4) are defined according to the following discriminant [4]:
 |
5 |
where 
and 
Then, we have:
If
, (4) has 3 different real roots;If
, (4) has 1 real root and 2 complex (conjugate) roots;If
and 
then (4) has 3 real roots, where two of them are equal;If
, then (4) has 3 equal real roots.
The analysis depends of the entries in matrix 
for 
.
-
First case: if all entries of matrix
in system (2) are dependent, then 
and the eigenvalues are obtained from the equation: 
where 
Then, the classification can be obtained using the particular expression (5).
Second case: if the matrix
is symmetric, then 
and 
In this case, the eigenvalues are obtained from the equation: 
where 
Third case: if all elements of the matrix are independent and the eigenvalues are obtained from Eq. (3). For each
we have the characteristic polynomial of degree 3.
Note that if in all cases we have 
we have the deterministic case and for each 
we need to get the sign of the roots for the characteristic polynomial of degree 3 to study the stability in (2). Here we want to choose 
so that there is an unique equilibrium [12].
Remark 1
For the square matrix of order 3, it is difficult to find, explicitly, the regions of the hypercube of dimension 9 that give a complete classification of the singularities of the characteristic matrix equation of system (2), even in the case of symmetry or the case of total dependence. Considering these, we will first describe the method called Cylindrical Algebraic Decomposition (CAD) used to find the lower and upper bound for real interval eigenvalue (see [1–3]).
To this end, we deal with a semi-algebraic set 
which a finite union of sets defined by polynomial equations and inequalities with real coefficients. The CAD provides a partition of S into semi-algebraic pieces which are homeomorphic to 
for 
. Moreover, classification problems involving such a set S, reduces to the computing of a finite number of sample points in each connected component, and then facing a polynomial optimization question. The algorithm is implemented in RAGlib (Real Algebraic Geometry library) of the software Maple. For example, if 
the first step is a reduction to compute sample points in each component of S defined with non-strict inequalities. There is a connected component 
of 
and a suitable 
that can be found using notions of critical values and asymptotic critical values. The next step of the algorithm addresses an algebraic problem.
In the next example, we analyze a particular 3-dimensional interval differential system when all entries in the matrix are dependent and independent via CI. For the independent case, firstly we find conditions to get 3, 2 and 1 real eigenvalues and one real and a pair of complex as was described in the proof of the Theorem 1. Besides, in Proposition 1 and 2 we find the real interval eigenvalue by using techniques from real algebraic geometry. The same method cannot be used to find the complex interval eigenvalue, since the 
real ordering is not longer available. Finally, we compare the values obtained with the Deif’s method [5] and Rohn’s method [16].
Example 1
Consider the system of the differential equations 
where 
is an interval matrix written as
 |
6 |
Here to simplify the notation, 
in such way 
implies that the characteristic polynomial
 |
7 |
where 
The solution of (7), 
for 
subject to 
is obtained using the cube root formula
 |
8 |
Thus, the real interval eigenvalue is
 |
9 |
Firstly, consider in (7) that all interval entries in matrix are dependent, then 
and, we have the following equation for the eigenvalues:
.Thus, the eigenvalues are:
 |
and 
is
 |
.
For 
, and so on.
Analysing the graph of
 |
we can conclude that for 
between 0.29 and 0.39 we have three real eigenvalues, for 
there is a unique triple real root and in other cases we have one real and two complex eigenvalues.
Secondly, consider 
are independent, then can be proved that for the real case, the min/max of the eigenvalues are obtained at a corner point on the boundary of the space of the parameters in a hypercube of the 7-dimension. Note that to obtain conditions for the complex eigenvalues is not easy, because the complex set is not an ordered set and we have an optimization problem to find the conditions for the parameters 
For all 
, the 
matrix 
has at least one real eigenvalue, therefore for all 
one can define 
(resp.
as the minimal (resp. maximal) real eigenvalue of 
. Let us now consider the compact set 
.
We denote by 
the characteristic polynomial of 
. Then by considering the discriminant (5), where 
and 
we have the following analysis.
There are three different real eigenvalues for
. The left side is defined by parameters 
that define the characteristic equation 
and the right side by 
such that 
. We have that the equilibrium point in this case are saddle and repulsor, respectively.There is one real and a complex conjugate for
The left side is defined by parameters 
that define the characteristic equation 
and the right side by 
such that 
. Note that the equilibrium point in both cases are repulsing.- If
and 
, then (7) has 3 real roots, where two them are equal. Then,
and
. (7) has two equal roots if, and only if, (7) and its first derivative have the same roots. Then we have the condition 
Moreover, if 
we have the condition 
Then for example 
then 
with characteristic equation given by 
and 
respectively. If 
, that is, 
, (7) has one triple real root. The expression is
where
For example, if 
is the unique eigenvalue.
Equation (5) is equivalent to 
and to 
, but to find the roots of (7) we need to use the last one [4], so that
 |
10 |
Our first goal is to estimate the extreme values of the multiple roots in 
. We begin by dealing with the multiple roots of the derivative 
of the characteristic polynomial. We observe that the parameter 
is not present in this derivative. Let us denote 
. Let
 |
11 |
In the Eq. (11,) let us now consider the compact set 
. Then, 
has a double root in an interior point of 
if for
we have 
and in this case 
The real eigenvalues can be shown to be in 
in accordance to the Propositions 1 and 2, which follow.
Third, Deif [5] considers the interval matrix 
and found that 
and 
.
Fourth, by using the Rohn’s Method outlined in [12], we found 
and 
.
However, in the method using CI, the matrix 
and 
give us real eigenvalues in the interval 
In the complex case, we find numerically, 
for 
and 
respectively (Fig. 2).
Fig. 2.
Trajectories solutions for 
and 
respectively.
The Propositions 1 and 2 proof that the real interval eigenvalue is 
. Firstly, it is necessary to find a value 
such that for all 
does not intersects the discriminant of 
.
Proposition 1
is the upper bound for the real interval eigenvalue in (7) and is the greatest root of 
, obtained from the Eq. (7) for 
Proof
Let 
. First, we show that 
is non-empty. Note that for 
we have 
such that 
By taking 
sufficiently small, one gets 
with 
Then, the maximum is not attained at an interior set.
We consider on 
a function 
. Since 
is smooth on 
, 
.
The gradient 
of P with respect to 
is such that:
 |
12 |
Moreover, 
is the greatest root of 
which is a degree 3 polynomial with positive leading coefficient. Hence 
. It follows that the respective coordinates of 
and 
have opposite signs.
The transposed gradient 
is
 |
13 |
Let us consider a point 
and 
,then the signs of the coordinates for 
are 
The strategy to find the maximum eigenvalue is to choose one direction for the gradient, considering one constant, to find the directions where it is increasing, because we need to build the trajectory as a piecewise function in hypercube 
. Then we consider the smooth vector field 
defined on 
 |
14 |
Let 
and 
be the trajectory of 
such that 
. Along this trajectory 
strictly increases, hence 
remains in 
. Let 
. For all 
, we have 
and 
. In addition 
, 
and 
have exactly the same signs, excepting where the sign does not change, as well as 
and 
.
It follows that 
decreases, so that it cannot be the minimum on 
and there exists a minimal 
such that 
. Furthermore, if one puts 
, this proves that 
.
We decrease/increase iteratively each coordinate 
to 0 or 1 accordingly to the (constant, non-zero) corresponding gradient coordinate sign. We finally get a point 
(
) such that 
and 
. Observe that 
.
We now have to deal with the remaining free coordinates 
. So let us consider the polynomial
 |
with 
. The problem we are facing is exactly the same as the original one for P. For all 
, denote by 
the greatest root of 
. We introduce the non-empty semi-algebraic subset:
 |
For all 
, and all 
, the gradient 
equals 
where 
then 
, such that
.
.
.
We integrate 
from a point 
and get after at most two other iterations, a point 
such that 
, with 
. To 
corresponds the unique point 
such that 
. Moreover 
.
This results in 
and one can compute explicitly the value 
which is realized at 
.
Proposition 2
is the lower bound for the real interval eigenvalue in (7) and is the smallest root of 
, obtained from the Eq. (7) for 
.
Proof
The proof is similar to the proof of Proposition 1.
Remark 2
Therefore, in this example we showed:
- Given an equation
The discriminant
15 
of 
is a polynomial in the indeterminates 
with integer coefficients (explicitly computed as the determinant of a Sylvester matrix, see [13]). The set 
is exactly the set 
. Then, varying the coefficients in (15) we can get what type of roots it has. In particular the condition on the discriminant for 
defines the type of roots for the polynomial equation when 
in (7) are varying. In the real case, we have methods to find them, but in the complex case it is not easy to characterize them completely; There are other methods to find the bounds for interval eigenvalue for (7), but it is not easy(in general not possible) to define the matrix by choosing the entries in interval matrix to get the corresponding eigenvalues. That is, once has the max/min eigenvalues the matrix that generated these eigenvalues is impossible to find. This is not true with the CI approach. We always know the matrix that generated the eigenvalues;
CIA gives us the option to choose the parameters for each eigenvalue in the way we can find the matrix explicitly, but may be a NP-hard procedure;
Many authors consider the interval matrix
. By CIA, we get 
for 
and 
for 
.
is obtained from CIA taking 
, but for the element 
and by CIA 
, then 
if 
. This means that methods used by Deif [5], Rohn [16] and [9] are not equivalent. Note the elements 
neither 
.Mathematica and Maple were used as tools to analyze and get some results.
Conclusion
This research outlined a method, involving semi algebraic sets theory, for which the stability of interval linear differential equations can be analyzed via constraint intervals. As a by product, a method for obtaining conditions about parameters to get real or complex eigenvalues of interval matrices of order 
were developed.
Contributor Information
Marie-Jeanne Lesot, Email: marie-jeanne.lesot@lip6.fr.
Susana Vieira, Email: susana.vieira@tecnico.ulisboa.pt.
Marek Z. Reformat, Email: marek.reformat@ualberta.ca
João Paulo Carvalho, Email: joao.carvalho@inesc-id.pt.
Anna Wilbik, Email: a.m.wilbik@tue.nl.
Bernadette Bouchon-Meunier, Email: bernadette.bouchon-meunier@lip6.fr.
Ronald R. Yager, Email: yager@panix.com
Marina Tuyako Mizukoshi, Email: tuyako@ufg.br, http://www.ime.ufg.br.
Alain Jacquemard, Email: Alain.Jacquemard@u-bourgogne.fr, https://math.u-bourgogne.fr/.
Weldon Alexander Lodwick, Email: Weldon.Lodwick@ucdenver.edu, https://clas.ucdenver.edu/mathematical-and-statistical-sciences.
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