1. Introduction
Consider the ordinary differential equation
where
are real variables,
for
and
with
(
is the set of polynomials with real coefficients). We also assume that
. The derivative of
y with respect to
x will be denoted as
or
.
We study the solutions of Equation (
1) of the form
where
, not necessarily constant (otherwise it is trivial). The computation of solutions (either polynomial or rational) of a nonlinear differential equation has a remarkable role in understanding their dynamical properties and their whole set of solutions. In 1936 it was Rainville in [
1] the first one who determined all the Riccati differential equations
, with
and
being polynomials that have polynomial solutions. Not only this, he also provided an algorithm for computing these solutions. Campbell and Golomb in [
2] gave an algorithm for obtaining all polynomial solutions of the generalized Riccati equation of the form
where
for
. More recently, Behloul and Cheng in [
3] gave another algorithm to detect either polynomial or rational solutions of
where
are polynomials in
x for
. Equation (
1) was previously studied in [
4] where the authors provide an upper bound on the number of polynomial solutions of Equation (
1) in terms of
n and the degree of
. In particular when this degree is not zero the upper bound is very far from being optimal. Moreover, for some very particular classes and some particular values of
n they study properties of their polynomial solutions. In the present paper, with very different techniques, we first provide universal bounds (in the sense that they only depend on
n) on the polynomial solutions of Equation (
1) that are much more accurate and we also show that these upper bounds are optimal. Second we consider polynomial limit cycles and their multiplicity. We recall that besides fixed points (very well-understood), limit cycles (that is, isolated periodic orbits) are the asymptotic state of other solutions of a given system and thus of great importance for understanding their dynamical properties. Limit cycles are the subject of the so-called Hilbert’s 16th problem, which essentially asks for an upper bound, uniformly in terms of the degree n, for the maximum number of limit cycles of a planar polynomial vector field. This problem lasts for more than 100 years and all attempts to solve it (all failed) has produced significant progress mainly in the geometric theory of planar vector fields, but also in bifurcation theory, the study of normal forms, foliations and also in some topics in algebraic geometry. Up to now, it is unknown whether such a uniform upper bound exists, even when the degree is two. Solution for Hilbert’s 16th problem mostly consist on restricting the classes of vector fields to particular simpler ones (such as our Equation (
1)), see e.g., [
5] for an overview.
When the left-hand side of Equation (
1) is of the form
, there are more studies of either the number of polynomial solutions or of rational solutions. In this easier scenario, the polynomial solutions of these equations have been intensively investigated. In the case when the functions
are periodic we want to mention the papers [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15] and the references therein. A special mention is done to the papers [
16,
17,
18,
19]. In the first two, the authors study the number of polynomial solutions of either polynomial Riccati or polynomial Bernouilli equations, and Abel equations of certain type, and in the third one, the authors obtain the maximum number and the multiplicity of the polynomial limit cycles of
where
for
and
. We also want to mention that the study of the number of rational solutions of either polynomial Riccati or polynomial Bernouilli equations was done in [
20] and of Abel equations was done in [
21,
22].
The following theorem, which is our first main theorem, establishes for the differential Equation (
5), the maximum number of coprime polynomial solutions (i.e.,
satisfying
) and
periodic polynomial solutions (i.e., solutions of the form
where
is
and satisfy
). It also establishes the solutions that are the inverse of a polynomial. We recall that the period of a periodic polynomial solution is one but we could choose any period to define a periodic solution because by making an affine change in the variable
x one can considers just period one.
Theorem 1. The following statements hold for the polynomial ordinary differential Equation (1). - (a)
It has at most n solutions that are constant, and there are examples with exactly n constant solutions.
- (b)
If it has a polynomial solution which is not constant, then it has infinitely many polynomial solutions.
- (c)
If , the difference between two coprime polynomial solutions is a constant.
- (d)
If , there are examples with n polynomial periodic solutions.
- (e)
If , given n non-constant polynomials coprime among each other, there are equations with such n solutions.
- (f)
If , given non-constant polynomials, coprime among each other, there are no equations with such solutions.
- (g)
If , it has at most n polynomial solutions that are coprime among each other, and there are examples with such n polynomial solutions.
Theorem 1 is proved in
Section 2. In the context of the proof of Theorem 1 (see
Section 2) in a different but similar exploratory application and in a little broader context, one typical case leading to systems of equations is in multiple linear regressions by maximizing the likelihood under assumption of generalized Gauss-Laplace distribution error. As further exploratory cases one may also want to consider the cases involving characteristic polynomials (see [
23]).
From Theorem 1 we have readily the following result. Let
be the space of
functions
satisfying
endowed with the supremum topology. A periodic solution
of (
1) is said to be a
limit cycle if there exists in
a neighborhood of
without any other periodic solution.
Theorem 2. Any ordinary differential Equation (1) has at most n polynomial limit cycles coprime among each other and this bound is reached. We also study the multiplicity of the differential Equation (
1). In order to define the multiplicity we will consider the so-called
translation operator. Let
be the solution of Equation (
1) defined for
such that
. The function
satisfying
is called the
translation operator associated with (
1). Following Lloyd [
24,
25], the multiplicity of
as a zero of
is the
multiplicity of a limit cycle of (
1) associated with the isolated zero
of the translation operator.
Theorem 3. Let be a periodic solution of Equation (1) and assume that with . Then φ is: - (i)
a limit cycle of multiplicity one if and only if ;
- (ii)
a limit cycle of multiplicity two if and only if and ;
- (iii)
a limit cycle of multiplicity three if and only if and ;
- (iv)
a limit cycle of multiplicity greater than or equal to four, or it belongs to a continuum of periodic orbits if .
The proof of Theorem 3 is given in
Section 3.
2. Proof of Theorem 1
Let
be a polynomial solution of Equation (
1) which is constant. Then the polynomial in
y,
must be divisible by
and since, its degree in
y is
n, it has at most
n different constant roots. Therefore, there are at most
n different constant solutions of Equation (
1). Please note that
is of the form (
1) and has
n constant solutions. Statement (a) is proved.
For statement (b), let
be a polynomial solution of Equation (
1) which is not a constant, i.e.,
Since
p divides the left-hand side of Equation (
4), it divides the right-hand side too, and so,
=
for some
. Hence, Equation (
1) becomes
If
, then
for
and Equation (
5) becomes
and so
which proves statement (b).
Now assume
and let
be another solution of (
1) coprime with
. Then we must have that
for some
. In short, since
and
are coprime we must have
with
. We consider the change
and we transform Equation (
5) into the equation
where each
is a polynomial depending on
and
for
and some powers of
, for
. Let
be another polynomial solution of Equation (
6) different from
, i.e.,
Then since
divides the right-hand side of Equation (
7), it divides the left-hand side, too. Therefore
which implies that
is a constant. Therefore, any polynomial solution of Equation (
6) must be a constant. In short the difference of two coprime polynomial solutions of (
5) must be a constant. This proves statement (c).
Since any solution which is constant is also periodic, statement (d) is a direct consequence of statement (c).
To prove statement (e), assume that Equation (
5) has
n non-constant polynomial solutions
for
that are coprime among each other. In view of statement (b) we can write
=
with
for
and
. We have
Since
divides the left-hand side of Equation (
8), it divides
for any
,
j =
. Hence,
where
. Therefore, we have the equation
Since two solutions differ by a constant, we have that
So we have the following system of
n equations with the unknowns
where
M is the matrix
We claim that the solutions
of Equation (
9) are polynomials.
To prove the claim, we will first show that
is a constant. Please note that
M is the Vandermonde matrix
with
Again, taking into account that
M is the Vandermonde matrix, its inverse
is
Therefore, in view of (
9), we have a polynomial solution in
for any polynomial
. This proves statement (e).
Now we prove statement (f). In this case we will show that there is exactly one set of polynomials
satisfying
where
and that for this unique set of polynomials we have
(and so
) which is not possible. Let
Then we can write
where we have dropped the dependence in
x to obtain a lighter notation. We claim that
We prove the claim (
12) by induction over the dimension
of the matrix
. Assume
. Then
becomes the
matrix
Now we assume it is true for
and we will prove it for
(obtaining an
-matrix
). Let
be the
ith row. We subtract to each row
for
, the first row
and we get that
Let
be the
i-th column. We multiply the column
by
and we add it to the column
for
. Then
is equal to
Therefore, by the Laplace theorem for determinants we obtain that
where
is the
matrix
Clearly,
where
where it is again a
matrix
(starting in
instead of
). By the induction hypotheses we have that
and so
which proves (
12).
Now we want to compute the inverse of
. We compute the first row of the matrix
=
, and we will show that
is equal to
Indeed, note that in view of the fact that the determinant of (
10) is the same as the determinant of (
12) and by the forms of the matrices
M and
, it is clear that
is equal to
.
It follows from (
13) that
We prove the by induction over the number of elements
n. If
it is clear. Now assume it is true for
and we will show it for
n. We write the left-hand side of (
14) as
Each of the sums in (
15) now has the form of the original sum in (
14), except on
elements, and the values turn out nicely by induction. Indeed,
and
and both are zero by induction hypotheses.
Hence, in view of (
14) we have that
. However, then
, which is not possible and the proof of statement (f) follows.
Statement (g) is an immediate consequence of statements (e), (f) and (d). The proof of Theorem 1 is complete.
3. Proof of Theorem 3
Let
and assume that
is a periodic solution. We make the change of variables
and we obtain
Please note that
and so
is always well defined.
With the change of variables
, the periodic solution
becomes the solution
(is constant and so it is periodic). Please note that
The right-hand side of Equation (
16) is 0 when
. Set
to be the solution of Equation (
16) with initial condition
. Please note that
. We want to study the solutions of Equation (
16) around
. We consider the translation operator
and we study it near
. If
is identically zero then the periodic solution belongs to a continuum of periodic solutions. If
is not identically zero then the multiplicity of
as a limit cycle of the differential equation is the multiplicity of
as a zero of
which, by the change of variables, is the multiplicity of
for the initial Equation (
5).
We expand
in Taylor series around
and we get
where
are differentiable functions for
and
while
, because
. We have that
satisfies the equality
where
was introduced in (
2). Now we expand this equation in Taylor series around
. Since this identity must hold for any value of
in the neighborhood around
, the coefficients of the same powers of
must be equal. Hence, we obtain the following system of differential equations for the functions
for
(we dropped the dependency of
in
x for simplicity of the notation):
The solution of the system in (
17) satisfies
where the functions
for
are the ones provided in the statement of the theorem. We observe that
for any
x and so, the former expressions are well defined. These expressions yield that
if and only if
. In addition, for fixed
, we have
and
for all
i such that
if and only if,
for all
i such that
.
The translation operator reads as
and so
is a limit cycle of multiplicity
if and only if
for all
i such that
and
.
Finally, if , then either is a limit cycle of multiplicity greater than or equal to 4, or it belongs to a continuum of periodic solutions. This concludes the proof of the theorem.