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ON THE REDUCIBILITY OF SYSTEMS WITH A QUASIPERIODIC MATRIX
A. E. GEL’MAN
In the monograph [1] N. P. Erugin raised the question of the reducibility of systems with almost periodic coefficients.
In 1957, in [2], a method* was proposed for studying systems with a quasiperiodic matrix, making it possible to single out a class of reducible systems of this kind.
The exposition (without proofs) was carried out for the case of two-dimensional systems. Applying this method, L. Ya. Andrianova, in [3], singled out a similar class for systems of order \(n\).
In this article complete proofs are given of the main results announced in [2].
1. SOME PROPERTIES OF BOHR FUNCTIONS
By Bohr functions (they are also often called quasiperiodic or conditionally periodic functions) one understands the class of uniformly almost periodic functions [6] for which the set of frequencies \(\{\Lambda_k\}\) has a finite basis, i.e., there exist \(n\) real numbers \(\omega_1, \omega_2, \ldots, \omega_n\)** such that
\[ \Lambda_k=\sum_{i=1}^{n} C_i^{(k)}\omega_i, \]
where \(C_i^{(k)}\) are integers.
It is known that a Bohr function can also be defined as the limit of a uniformly convergent sequence of trigonometric polynomials, the frequencies of whose individual terms are integral linear combinations of the given numbers \(\omega_1, \omega_2, \ldots, \omega_n\).
Definition 1. Let
\[
P(t)=\sum_{|k_1|+|k_2|+\cdots+|k_n|\le H}
a_{k_1 k_2 \ldots k_n}
e^{it(k_1\omega_1+k_2\omega_2+\cdots+k_n\omega_n)}.
\]
We shall call \(H=H(P)\) the height of the polynomial \(P(t)\).
Theorem 1. Every Bohr function \(f(t)\) can be represented in the form of a uniformly convergent series of trigonometric polynomials
\[
f(t)=\sum_{k=0}^{\infty} P_k(t)
\]
such that \(H(P_k)\le k\).
* This method differs from the methods of Siegel and Kolmogorov, usually applied in problems with “small divisors” (see, on this subject, the works of V. I. Arnold [4, 5]).
** Here and below it is assumed that \(\omega_1,\omega_2,\ldots,\omega_n\) are linearly independent, i.e., there do not exist integers \(C_1,C_2,\ldots,C_n\), not all zero, such that
\[
\sum_{k=1}^{n} C_k\omega_k=0.
\]
Proof. From the definition of the function \(f(t)\) it follows that it can be represented in the form of a uniformly convergent series of trigonometric polynomials
\[ f(t)=\sum_{k=0}^{\infty} Q_k(t). \]
To each \(Q_k(t)\) we assign a positive number \(r_k\) in the following way:
\[ r_0=H(Q_0), \]
\[ r_k=\max [H(Q_k),\ r_{k-1}+1]. \]
We now define a sequence of trigonometric polynomials \(P_j(t)\) as follows:
\[ P_j(t)= \begin{cases} Q_k(t), & \text{if } j=r_k,\\ 0, & \text{if } j\ne r_k. \end{cases} \]
Consider the series \(\sum_{j=0}^{\infty} P_j(t)\). It is obvious that this series differs from the series \(\sum_{k=0}^{\infty} Q_k(t)\) only by terms equal to zero, and therefore it also converges uniformly. On the other hand, by construction the terms of this series satisfy the condition \(H(P_j)\le j\).
The theorem is proved.
Definition 2. We shall call a uniformly convergent series of trigonometric polynomials \(\sum_{k=0}^{\infty} P_k(t)\) a regular series of the Bohr function \(f(t)\) if
\[ f(t)=\sum_{k=0}^{\infty} P_k(t) \]
and
\[ H(P_k)\le k. \]
It is quite obvious that every Bohr function has an infinite set of regular series. Indeed, if, for example, \(\sum_{k=0}^{\infty} P_k(t)\) is a regular series of the function \(f(t)\), then the series \(\sum_{k=k_0}^{\infty} Q_k(t)\), where
\[ Q_k(t)= \begin{cases} \displaystyle \sum_{j=0}^{k_0} P_j(t), & \text{if } k=k_0,\\[6pt] P_k(t), & \text{if } k>k_0, \end{cases} \]
will also be a regular series.
Definition 3. We shall call the radius of convergence of a regular series of the Bohr function \(f(t)\) the radius of convergence of the power series
\[ \bar f(\lambda)=\sum_{k=0}^{\infty}\lambda^k \sup_{-\infty<t<+\infty}|P_k(t)|. \]
From the definition of a regular series it is clear that the radius of convergence of a regular series is not less than one; for any trigonometric polynomial it is equal to \(\infty\).
In what follows we shall be mainly interested only in those Bohr functions which have a regular series with radius of convergence greater than one. We shall call them \(H\)-functions.
Functions of this kind, as will be shown by the theorems of the next paragraph, possess a number of interesting properties.
2. \(H\)-FUNCTIONS
Let \(B\) be the set of all \(H\)-functions with frequency basis \(\omega_1,\omega_2,\ldots,\omega_n\).
Theorem 1. The set \(B\) is an algebra.*
Proof. For the proof it is enough to show that if
\[ f_1(t)\in B \quad \text{and} \quad f_2(t)\in B, \]
then
\[ f_1(t)+f_2(t)\in B \quad \text{and} \quad f_1(t)\cdot f_2(t)\in B. \]
Let \(f_1(t)\in B\) and \(f_2(t)\in B\), and let their representations in the form of regular series be
\[ f_1(t)=\sum_{k=0}^{\infty} P_k^{(1)}(t), \]
\[ f_2(t)=\sum_{k=0}^{\infty} P_k^{(2)}(t). \]
Introduce the power series under consideration
\[ f_1(t,\lambda)=\sum_{k=0}^{\infty} \lambda^k P_k^{(1)}(t), \]
\[ f_2(t,\lambda)=\sum_{k=0}^{\infty} \lambda^k P_k^{(2)}(t), \]
and construct their sum and product in accordance with the rules for operations on power series:
\[ F_1(t,\lambda)=f_1(t,\lambda)+f_2(t,\lambda)= \]
\[ =\sum_{k=0}^{\infty}\lambda^k\left[P_k^{(1)}(t)+P_k^{(2)}(t)\right] =\sum_{k=0}^{\infty}\lambda^k Q_k^{(1)}(t), \]
\[ F_2(t,\lambda)=f_1(t,\lambda)\cdot f_2(t,\lambda) =\sum_{k=0}^{\infty}\lambda^k \sum_{k_1+k_2=k} P_{k_1}^{(1)}(t)\cdot P_{k_2}^{(2)}(t)= \]
\[ =\sum_{k=0}^{\infty}\lambda^k Q_k^{(2)}(t). \]
* The operations in this algebra are addition and multiplication, understood in the arithmetic sense.
It is easy to see that for the polynomials \(Q_k^{(1)}\) and \(Q_k^{(2)}\) the inequalities hold:
\[ H\bigl(Q_k^{(1)}\bigr) \leq k, \]
\[ H\bigl(Q_k^{(2)}\bigr) \leq k. \]
The first of these inequalities is obvious, while the second follows from the fact that the height of the product of two trigonometric polynomials does not exceed the sum of their heights.
The series \(F_1(t,\lambda)\) and \(F_2(t,\lambda)\) are majorized by the series \(\overline{f_1}(\lambda)+\overline{f_2}(\lambda)\) and \(\overline{f_1}(\lambda)\cdot \overline{f_2}(\lambda)\), respectively, and therefore their radii of convergence exceed one. Consequently, the series
\[ \sum_{k=0}^{\infty} Q_k^{(1)}(t) \quad \text{and} \quad \sum_{k=0}^{\infty} Q_k^{(2)}(t) \]
are regular series of the functions \(f_1(t)+f_2(t)\) and \(f_1(t)\cdot f_2(t)\), having radius of convergence greater than one, as was required to prove.
With regard to an \(H\)-function, it is easy to establish a theorem that resolves the question of the boundedness of its antiderivatives.
Theorem 2. Let \(R(R>1)\) be the radius of convergence of some regular series of an \(H\)-function with zero mean value. If
\[ \overline{\lim}_{H=|C_1|+|C_2|+\cdots+|C_n|} \left|C_1\omega_1+C_2\omega_2+\cdots+C_n\omega_n\right|^{-\frac{1}{H}} < R, \]
then an antiderivative of such a function is also an \(H\)-function.
Proof. To prove this theorem it suffices to show that the series obtained as a result of formal term-by-term integration of some regular series of the given \(H\)-function will be majorized by a numerical series with radius of convergence greater than one.
As is known [6], the Fourier series of the Bohr function \(f(t)\) has the form
\[ \sum a_{m_1m_2\ldots m_n} e^{it(\omega_1m_1+\omega_2m_2+\cdots+m_n\omega_n)}, \]
where
\[ a_{m_1m_2\ldots m_n} = \lim_{T\to\infty} \frac{1}{T} \int_{0}^{T} f(t)e^{-it(\omega_1m_1+\omega_2m_2+\cdots+\omega_n m_n)}\,dt . \]
Let us estimate \(a_{m_1m_2\ldots m_n}\), where
\[ |m_1|+|m_2|+\cdots+|m_n|=H. \]
Let
\[ f(t)=\sum_{k=0}^{\infty} P_k(t) \]
be the representation of the Bohr function by its regular series.
Then, since \(P_1(t), P_2(t), \ldots, P_{H-1}(t)\) do not contain terms of the form
\[ e^{it(m_1\omega_1+m_2\omega_2+\cdots+m_n\omega_n)}, \]
where \(|m_1|+|m_2|+\cdots+|m_n|=H\), we obtain
\[ a_{m_1m_2\ldots m_n} = \lim_{T\to\infty} \frac{1}{T} \int_{0}^{T} e^{-it(m_1\omega_1+m_2\omega_2+\cdots+m_n\omega_n)} f(t)\,dt = \]
\[ = \lim_{T \to \infty} \frac{1}{T} \int_0^T e^{-it(m_1\omega_1+m_2\omega_2+\ldots+m_n\omega_n)} \left[\sum_{k=H}^{\infty} P_k(t)\right] dt . \]
It follows from this that, if we introduce the notation
\[ s_k=\sup_{-\infty<t<+\infty}|P_k(t)|, \quad \text{then} \quad |a_{m_1m_2\ldots m_n}| \leq \sum_{k=H}^{\infty} s_k . \]
The corresponding coefficient of the Fourier series of the primitive \(f(t)\) will be estimated as follows:
\[ \left| \frac{a_{m_1m_2\ldots m_n}} {m_1\omega_1+\ldots+m_n\omega_n} \right| \leq \frac{\displaystyle \sum_{k=H}^{\infty} s_k} {|m_1\omega_1+\ldots+m_n\omega_n|} \leq \]
\[ \leq \frac{\displaystyle \sum_{k=H}^{\infty} s_k} {\displaystyle \min_{|m_1|+|m_2|+\ldots+|m_n|=H} |m_1\omega_1+m_2\omega_2+\ldots+m_n\omega_n|} . \]
Such an estimate, obviously, holds for any coefficient of the Fourier series of the primitive function for which \(|m_1|+|m_2|+\ldots+|m_n|=H\). The number of such coefficients \(N\), obviously, is equal to the number of representations of the number \(H\) as a sum of \(n\) nonnegative integers, multiplied by \(2^n\) (since each of the \(m_k\) in any such representation may be positive or negative). Consequently,
\[ N=2^n\frac{(n+H-1)!}{(n-1)!H!} =2^n\frac{(H+1)(H+2)\ldots(H+n-1)}{(n-1)!} = \]
\[ =2^n\cdot H^{n-1} \frac{\left(1+\frac{1}{H}\right)\left(1+\frac{2}{H}\right)\ldots\left(1+\frac{n-1}{H}\right)} {(n-1)!} \leq 2^n n H^{\,n-1}. \]
Therefore, in order that the primitive function be an \(H\)-function, it is sufficient that the series
\[ \sum_{H=1}^{\infty} u_H, \]
where
\[ u_H= \frac{2^n n H^{n-1}} {\displaystyle \min_{|m_1|+|m_2|+\ldots+|m_n|=H} |m_1\omega_1+m_2\omega_2+\ldots+m_n\omega_n|} \sum_{k=H}^{\infty}s_k, \]
have radius of convergence greater than one, i.e., that the inequality
\[ \varlimsup_{H\to\infty}\sqrt[H]{u_H}<1 \]
hold.
Since the radius of convergence of the series \(\displaystyle \sum_{k=0}^{\infty}\lambda^k s_k\) is \(R\), it follows, as is known,
the same will be the radius of convergence of the series \(\sum_{H=1}^{\infty} \lambda^H \sum_{k=H}^{\infty} s_k\), i.e., the equality
\[ \varlimsup_{H\to\infty} \sqrt[H]{\sum_{k=H}^{\infty} s_k}=\frac{1}{R}. \]
Hence it follows that
\[ \varlimsup_{H\to\infty} \sqrt[H]{u_H} = \frac{ \varlimsup_{H\to\infty} |m_1\omega_1+m_2\omega_2+\cdots+m_n\omega_n|^{-1/H} }{R} \]
and, consequently, by virtue of the condition of the theorem,
\[ \varlimsup_{H\to\infty} \sqrt[H]{u_H}<1. \]
The theorem is proved.
For the convenience of the subsequent exposition we introduce some notation. Let
\[ F(z,t)=\sum_{k=0}^{\infty} z^k f_k(t), \]
where all \(f_k(t)\) are Bohr functions.
Represent each of the functions \(f_k(t)\) in the form of the sum of its regular series:*
\[ f_k(t)=\sum_{i=1}^{\infty} P_i^{(k)}(t). \]
Consider the functions
\[ f_k(t,\lambda)=\sum_{i=1}^{\infty}\lambda^i P_i^{(k)}(t),\qquad \bar f_k(\lambda)=\sum_{i=1}^{\infty}\lambda^i \sup_{-\infty<t<+\infty}|P_i^{(k)}(t)| \]
and introduce the notation
\[ \bar F(z,t,\lambda)=\sum_{k=0}^{\infty} z^k f_k(t,\lambda) \quad\text{and}\quad \bar F(z,\lambda)=\sum_{k=0}^{\infty} z^k \bar f_k(\lambda). \]
In what follows we shall assume that the function \(F(z,t)\) is such that the series \(\bar F(z,1)\) has a nonzero radius of convergence; this requirement is certainly satisfied, for example, in the case when \(F(z,t)\) is a polynomial with respect to \(z\); we shall also assume that \(F(z,t)\) is nonlinear in \(z\).
The following holds.
Theorem 3. In order that the differential equation
\[ \dot z+Az=F(z,t)\qquad (A^2>0) \]
have a Bohr function as its solution, it is sufficient that the equation
\[ |A|z=\bar F(z,1) \tag{i} \]
have a nonnegative root.
* The regular series here are constructed in a special way: for all \(k\) the equality \(P_0^{(k)}=0\) holds. The possibility of choosing them in this way follows from the remark to Definition 2.
If, however, equation (i) has a simple nonnegative root \(\bar z\), and the point \((\bar z,1)\) does not lie on the boundary of the domain of convergence of the series \(\bar F(z,\lambda)\), then this solution is an \(H\)-function.
Proof. We introduce for consideration two auxiliary equations with parameter \(\lambda\):
\[ \dot z+Az=\widetilde F(z,t,\lambda) \tag{ii} \]
and
\[ |A|z=\bar F(z,\lambda). \tag{iii} \]
We shall solve equation (ii) by the method of a small parameter; namely, we shall seek its solution (a Bohr function) in the form of a series
\[ z(t,\lambda)=\sum_{k=0}^{\infty}\lambda^k z_k(t). \]
Equation (ii) then decomposes into an infinite set of linear equations of the form
\[ \dot z_k+Az_k=W_k(t). \]
By induction it is easy to prove that each of these equations has a unique solution \(z_k\) bounded on \((-\infty,+\infty)\), that this solution is a trigonometric polynomial, and that the height of this polynomial does not exceed \(k\):
\[ H(z_k)\leqslant k. \]
Indeed, for \(k=0\) we have
\[ \dot z_0+Az_0=0 \quad (\text{since } \widetilde F(z,t,0)\equiv 0). \]
This equation obviously has the unique solution bounded on \((-\infty,+\infty)\)
\[ z_0=0. \]
This solution is a trigonometric polynomial, and \(H(z_0)\leqslant 0\).
Suppose now that \(z_k\) \((k\leqslant n)\) possess the properties described above, and prove that then \(z_{n+1}\) will also possess these same properties.
As is not difficult to see, \(W_{n+1}(t)\) is the coefficient of \(\lambda^{n+1}\) in the function
\[ \widetilde F\left(\sum_{k=1}^{n} z_k\lambda^k,\ t,\ \lambda\right), \]
and therefore is a finite sum of various products of the trigonometric polynomials \(z_k\) and \(P_i\), the sum of the indices in such products being exactly \(n+1\). (For the indices of the polynomials are equal to the exponents of the powers of \(\lambda\), and the sum of such exponents in the coefficient of \(\lambda^{n+1}\) is equal to \(n+1\).)
Since, by the induction hypothesis, the height of each of these polynomials does not exceed its index, then, taking into account that the height of a product of polynomials does not exceed the sum of their heights, it becomes clear that the height of the polynomial \(W_{n+1}(t)\) does not exceed \(n+1\).
The equation
\[ \dot z_{n+1}+Az_{n+1}=W_{n+1}(t) \]
will then have a unique solution bounded on \((-\infty,+\infty)\)—a trigonometric polynomial of the same height as \(W_{n+1}(t)\) (which is easy to verify by solving, for example, this equation by the method of undetermined coefficients).
Thus, we have proved that the formal series satisfying equation (ii) has the form
\[ z(\lambda)=\sum_{k=1}^{\infty}\lambda^k z_k(t), \]
where \(z_k(t)\) are trigonometric polynomials of height not exceeding \(k\).
With the help of equally simple considerations we establish that equation (iii) also has a formal series solution
\[ \overline z(\lambda)=\sum_{k=1}^{\infty}\overline z_k\lambda^k, \]
where \(\overline z_k\) are nonnegative numbers.
Indeed, as is easy to see, \(\overline z_0=0\), and \(\overline z_{n+1}\) will satisfy the equation
\[ |A|\overline z_{n+1}=\overline W_{n+1}, \]
where \(\overline W_{n+1}\) is the coefficient of \(\lambda^{n+1}\) in the power series
\[ \overline F\left(\sum_{k=1}^{n}\lambda^k\overline z_k,\lambda\right), \]
and, consequently, if one makes the inductive assumption of the nonnegativity of all \(\overline z_k\) \((k\le n)\), then from this will follow the nonnegativity of \(\overline z_{n+1}\).
We now prove that the series \(\overline z(\lambda)\) is a majorant series for \(z(\lambda)\). Since \(z_0=\overline z_0=0\), the inequality \(|z_0|\le \overline z_0\) is valid. Suppose that the inequality \(|z_k|\le \overline z_k\) is valid for all \(k\) \((k\le n)\), and prove it for \(k=n+1\).
Since each coefficient of the polynomial
\[ \sum_{k=0}^{n}z_k\lambda^k \]
does not exceed in modulus the corresponding coefficient of the polynomial
\[ \sum_{k=0}^{n}\overline z_k\lambda^k, \]
and the series \(\overline F(z,\lambda)\) is a majorant series for the series \(\widetilde F(z,t,\lambda)\), the series
\[ \overline F\left(\sum_{k=0}^{n}\overline z_k\lambda^k,\lambda\right) \]
will be a majorant for the series
\[ \widetilde F\left(\sum_{k=0}^{n}z_k\lambda^k,t,\lambda\right), \]
whence it follows that
\[ |W_{n+1}(t)|\le \overline W_{n+1}. \]
Let us now estimate \(z_{n+1}(t)\) and find \(\overline z_{n+1}\). Using the known inequality* for a solution bounded on \((-\infty,+\infty)\) of the equation
\[ \dot z+Az=\varphi(t) \]
\[ |z|\le \frac{1}{|A|}\sup_{-\infty<t<+\infty}|\varphi(t)|, \]
we obtain
\[ |z_{n+1}|\le \frac{1}{|A|}\sup_{-\infty<t<+\infty}|W_{n+1}(t)|. \]
* See, for example, the monograph [7].
$\overline z_{n+1}$ is found as the solution of the equation $|A|\overline z_{n+1}=\overline W_{n+1}$:
\[ \overline z_{n+1}=\frac{\overline W_{n+1}}{|A|}. \]
Hence, if we take into account the inequality obtained above, $|W_{n+1}(t)|\leqslant \overline W_{n+1}$, it follows that $|z_{n+1}|\leqslant \overline z_{n+1}$, as was required to prove.
It is now clear that, in order to complete the proof of the theorem, it remains only to investigate the domain of convergence of the series $\overline z(\lambda)$—the formal solution of equation (iii).
From a lemma in [8] it follows that the radius of convergence of the series $\overline z(\lambda)$ is given by the formula
\[ \Lambda=\sup_{[z,\lambda(z)]\in D}\lambda(z), \]
where $\lambda(z)$ is understood to mean the function*, satisfying the equation
\[ z=\frac{1}{|A|}\,\overline F(z,\lambda), \]
and $D$ means the part of the domain of convergence of the function $\overline F(z,\lambda)$ situated in the positive quadrant $(z\geqslant 0,\lambda\geqslant 0)$. At the same time it is also known that, if $\overline F(z,\lambda)$ is a nonlinear function of $z$, then the series $\overline z(\lambda)$ converges at the point $\lambda=\Lambda$. From the definition of $\Lambda$ it is clear that if $\lambda>\Lambda$, then equation (iii) cannot have a nonnegative solution.
Therefore, if it is known that equation (iii) has a positive solution for $\lambda_0>0$, then it follows that $\lambda_0\leqslant \Lambda$ and, consequently, the series $\overline z(\lambda_0)$ is convergent.
Hence follows the first assertion of Theorem 3: if equation (i) has a nonnegative root, then, in view of what was set forth above, the series
\[ \overline z(1)=\sum_{k=1}^{\infty}\overline z_k \]
with positive terms is a convergent series. Since it was proved earlier that it is a majorant series for the series-solution of the differential equation under study, this latter series will be a uniformly convergent series of trigonometric polynomials. Hence it is clear that the equation has as its solution a Bohr function, as was required to prove.
Let us prove the second assertion of Theorem 3.
From the definition of $\Lambda$ it is clear that the function $\overline z(\lambda)$ has a singularity at the point $\Lambda$. From equation (iii) it is seen that
\[ \overline z'(\lambda)= \frac{\dfrac{1}{|A|}\,\overline F'_{\lambda}[\overline z(\lambda),\lambda]} {1-\dfrac{1}{|A|}\,\overline F'_{z}[\overline z(\lambda),\lambda]} \]
and, consequently, if the point $[\overline z(\Lambda),\Lambda]$ does not lie on the boundary of the domain $D$, then
\[ 1-\frac{1}{|A|}\,\overline F'_z[\overline z(\Lambda),\Lambda]=0. \]
In other words, in this case, for $\lambda=\Lambda$ equation (iii) has a multiple root. Consequently, if it is known that equation (iii)
* The fact that such a unique function exists follows trivially from the monotonicity of the function $\overline F(z,\lambda)$ with respect to $\lambda$ (for $\lambda\geqslant 0,\ z\geqslant 0$).
for \(\lambda=\lambda_0\) has a nonmultiple nonnegative root \(z_0\) and the point \((z_0,\lambda_0)\) does not lie on the boundary of the domain \(D\), then the inequality
\[
\Lambda>\lambda_0
\]
holds.
Hence, putting \(\lambda_0=1\), we obtain the second assertion of Theorem 3. Indeed, the series \(z(1)\) will be a majorant series for \(z(1)\), and the series \(z(\lambda)\) has radius of convergence greater than one. Consequently, the series \(z(1)\) is a proper series of a solution of the original equation, and its radius of convergence exceeds one.
Corollary 1. If \(F(z,t)\) is a polynomial with respect to \(z\), whose coefficients are \(H\)-functions, and equation (i) has a simple nonnegative root, then the original differential equation also has a solution \(z(t)\) which is an \(H\)-function. This assertion is obvious, for in this case the domain \(D\) is the strip
\[
0\le z<+\infty,\quad 0\le \lambda\le R\ (R>1).
\]
Therefore the point \((1,z_0)\) cannot lie on the boundary of the domain \(D\).
Let us note a particularly simple special case, important for what follows. Let \(F(z,t)\) be a quadratic trinomial
\[
F(z,t)=z^2a(t)+zb(t)+c(t).
\]
From elementary considerations it is clear that, in order that the equation
\[
|A|z=z^2\overline{a}(1)+z\overline{b}(1)+\overline{c}(1)=\overline{F}(z,1)
\]
have a simple positive root, it is sufficient that
\[
\overline{b}(1)+2\sqrt{\overline{a}(1)\cdot\overline{c}(1)}<|A|.
\]
Consequently, if \(a(t)\), \(b(t)\), and \(c(t)\) are \(H\)-functions and
\[
\overline{b}(1)+2\sqrt{\overline{a}(1)\cdot\overline{c}(1)}<|A|,
\]
then the Riccati equation
\[
\dot z+Az=F(z,t)
\]
has as its solution an \(H\)-function.
It is easy to see that the radius of convergence of the proper series of this solution is the positive root of the equation
\[
\overline{b}(\lambda)+2\sqrt{\overline{a}(\lambda)\overline{c}(\lambda)}=A. \tag{a}
\]
3. On the reducibility of systems with a Bohr matrix
Consider the system
\[
\dot z=P(t)z, \tag{1}
\]
where
\[
P(t)=
\begin{Vmatrix}
p_{11}(t) & p_{12}(t)\\
p_{21}(t) & p_{22}(t)
\end{Vmatrix},
\]
and \(p_{ij}(t)\) are Bohr functions.
In the monograph [1] N. P. Erugin proved that if the Riccati equation
\[
\dot\tau=p_{12}+(p_{22}-p_{11})\tau-p_{21}\tau^2 \tag{2}
\]
has a solution \(\tau(t)\) bounded on \((-\infty,+\infty)\), then the question of the reducibility of this system is connected with the possibility of the equalities
\[
\int_0^t(-\tau p_{21}-p_{11})\,dt=b_1t+\Phi_1(t),
\]
\[
\int_0^t(\tau p_{21}-p_{22})\,dt=b_2(t)+\Phi_2(t), \tag{3}
\]
where \(b_1\) and \(b_2\) are constants, and \(\Phi_1(t)\) and \(\Phi_2(t)\) are bounded functions. Moreover, if \(b_1 \ne b_2\), then these equalities are a necessary and sufficient condition for the reducibility of the system under consideration.
Suppose that in system (1) all \(p_{ij}(t)\) are \(H\)-functions and, if we introduce the notation
\[ A=\lim_{T\to\infty}\frac{1}{T}\int_0^T\bigl(p_{22}(t)-p_{11}(t)\bigr)\,dt, \]
\[ b(t)=p_{22}(t)-p_{11}(t)-A, \]
the inequality
\[ \overline{b}(1)+2\sqrt{\overline{p}_{12}(1)\cdot \overline{p}_{21}(1)}<|A| \]
is satisfied.
Then, by the corollary to Theorem 3, the Riccati equation (2) has an \(H\)-function as its solution. Let \(r\) be the radius of convergence of its regular series. Let \(r_1\) and \(r_2\) be the radii of convergence of the regular series of the functions \(p_{11}(t)\) and \(p_{22}(t)\).
Denote \(R=\min(r,r_1,r_2)\). Obviously, \(R>1\).
True is
Theorem 4. If all the assumptions made above are satisfied and
\[ \varlimsup_{H\to\infty}|m_1\omega_1+m_2\omega_2+\ldots+m_n\omega_n|^{-\frac{1}{H}}<R, \]
then system (1) is reducible.
Proof. The radius of convergence of the regular series of the function \(\tau p_{21}(t)\) is equal to the smaller of the radii of convergence of the regular series of the factors. From equation \((\alpha)\) it is evident, however, that \(r\) does not exceed the radius of convergence of the regular series \(p_{21}(t)\). Therefore the radius of convergence of each of the integrand functions in equalities (3) is not less than \(R\). Hence, if the condition of Theorem 4 is satisfied, then by Theorem 2 the equalities (3) are fulfilled, and \(\Phi_1(t)\) and \(\Phi_2(t)\) will be \(H\)-functions.
Thus, for the proof of the theorem it remains only to show that \(b_1\ne b_2\), or, what is the same thing, that
\[ \lim_{T\to\infty}\frac{1}{T}\int_0^T(2\tau p_{21}+p_{11}-p_{22})\,dt\ne 0, \]
i.e. that
\[ \lim_{T\to\infty}\frac{1}{T}\int_0^T 2\tau p_{21}\,dt\ne A. \]
This inequality will be proved if we show that
\[ 2\sup|\tau|\cdot \sup|p_{21}|<|A|. \]
Since it was proved earlier that \(z(t,\lambda)\) (the series-solution of the equation \(\dot z+Az=F(z,t,\lambda)\)) is majorized by \(\overline{z}(\lambda)\) (the series-root of the equation \(|A|z=\overline{F}(z,\lambda)\)), it is obvious that
\[ |z(t,1)|\le \overline{z}(1). \]
It follows from this that
\[ |\tau|\le \overline{z}(1), \]
where \(\bar z(1)\) is a root of the quadratic equation \(|A|z=\bar p_{21}(1)z^2+\bar b(1)z+\bar p_{12}(1)\):
\[ \bar z(1)=\frac{1}{2\bar p_{21}(1)} \left[|A|-\bar b(1)-\sqrt{(|A|-\bar b(1))^2-4\bar p_{21}(1)\bar p_{12}(1)}\right]. \]
From the inequality
\[ \bar b(1)+2\sqrt{\bar p_{12}(1)\cdot \bar p_{21}(1)}<|A|, \]
it obviously follows that
\[ \bar z(1)<\frac{|A|}{2\bar p_{21}(1)} \]
and, consequently,
\[ 2|\tau|\,|p_{21}|<2\bar z(1)\bar p_{21}(1)<|A|, \]
as was required to prove.
References
- Erugin N. P. Reducible systems. Proceedings of the Steklov Institute of Mathematics, vol. XIII, 1946.
- Gelman A. E. Doklady Akademii Nauk SSSR, 116, No. 4, 1957.
- Andrianova L. Ya. Vestnik LGU, No. 7, mathematical series, issue 2, 1962.
- Arnold V. I. Problems of the motion of artificial celestial bodies (Small denominators and the problem of stability in classical and celestial mechanics). Academy of Sciences, Moscow, 1963.
- Arnold V. I. Uspekhi Matematicheskikh Nauk, 18, issue 6 (114), 1963.
- Levitan B. M. Almost Periodic Functions. GITTL, Moscow, 1953.
- Malkin I. G. Some Problems in the Theory of Nonlinear Oscillations. GITTL, Moscow, 1956.
- Gelman A. E. The small-parameter method for operator equations. Doklady Akademii Nauk SSSR, 118, No. 6, 1958.
Received by the editors
December 10, 1964
Leningrad Electrotechnical Institute
named after V. I. Ulyanov-Lenin