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UDC 517.919
ON THE ANALYTIC FORM OF SOLUTIONS OF LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH QUASIPERIODIC COEFFICIENTS
A. I. ZAITSEV
§ 1. BASIC DEFINITIONS AND STATEMENT OF THE PROBLEM
Consider the system of differential equations
\[ \frac{dx}{dt}=P(t)x, \tag{1} \]
where \(x=(x_1,\ldots,x_n)\) is a vector, and \(P(t)=\|p_{sk}(t)\|_1^n\) is a quasiperiodic matrix [1] of order \(n\) with frequency basis \(\beta=(\beta_1,\beta_2,\ldots,\beta_m)\).
Our problem consists in clarifying the structure of the solutions of equation (1). As shown in [2], the solutions of system (1) are obtained from the solutions of the system
\[ Dx\equiv \frac{\partial x}{\partial u_1}+\cdots+\frac{\partial x}{\partial u_m} =F(u_1,\ldots,u_m)x \tag{2} \]
on the diagonal \(u_1=u_2=\cdots=u_m=t\), where
\(F(u_1,\ldots,u_m)=\|f_{sk}(u_1,\ldots,u_m)\|_1^n\) is a periodic matrix with periods
\(\omega_1=\dfrac{2\pi}{\beta_1},\ldots,\omega_m=\dfrac{2\pi}{\beta_m}\) in \(u_1,\ldots,u_m\), respectively, and
\(f_{sk}(t,\ldots,t)=p_{sk}(t)\) \((s,k=1,2,\ldots,n)\).
In [2] the concept of a fundamental matrix of solutions of equation (2) was introduced, and it was shown that this fundamental matrix of solutions \(X\) satisfies the following finite-difference equation:
\[ X(u_1+\omega_1,\ldots,u_m+\omega_m) = X(u_1,\ldots,u_m)\, C(u_2-u_1,\ldots,u_m-u_1). \tag{3} \]
Everywhere below, by \(\overline{\Gamma}\) we shall denote
\(\Gamma(u_1+\omega_1,\ldots,u_m+\omega_m)\). Then our relation (3) is rewritten in the form
\[ \overline{X}=XC,\qquad C=C(u_2-u_1,\ldots,u_m-u_1). \tag{3_1} \]
Following [2], we shall call the matrix \(C\) a matrix constant on the diagonal. It is evident that \(DC=\Theta\), where \(\Theta\) is the zero matrix.
In clarifying the structure of the solutions of system (2), in [2] the form of the functions satisfying relation (3) is determined.
We shall approach the solution of this question from another point of view. Namely, we shall transform system (2) into a form convenient for investigation, in such a way that the coefficients of the transformed system remain periodic. To this end we introduce some notions needed below.
Consider two sets \(K\) and \(M\), whose elements are continuous and differentiable functions of a finite number of arguments. On the set \(K\) we define a linear differential operator \(D\).
Definition 1. We shall call the set \(K\) a producing set for the set \(M\) if any element of the set \(K\) is transformed by the operator \(D\) into one of the elements of the set \(M\), i.e., if \(r \in K\), then \(Dr \in M\).
Consider the set of square matrices with elements from \(M\) and \(K\). If the elements of some matrix \(T\) are contained in \(M\) (in \(K\)), then we shall say that the matrix \(T\) is contained in \(M\) (in \(K\)).
Definition 2. a) A matrix having the form
\[ \left\| \begin{array}{cccccc} a_1 & b_1 & 0 & \ldots & 0 & 0\\ 0 & a_1 & b_1 & \ldots & 0 & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \ldots & a_1 & b_1\\ 0 & 0 & 0 & \ldots & 0 & a_1 \end{array} \right\|, \]
will be called a generalized Jordan block and will be denoted by \(J_{q_1}(a_1,b_1)\), where \(q_1\) is the order of the block;
b) if \(b_1 \equiv 1\), then such a matrix \(J_{q_1}(a_1,1)\) will be called a normal Jordan block;
c) if \(b_1 \equiv 0\), then such a matrix \(J_{q_1}(a_1,0)\) will be called diagonal;
d) if generalized Jordan blocks stand on the main diagonal of a matrix, and the remaining elements are equal to zero, then we shall say that the matrix has generalized Jordan form, and denote it by \([J_{q_1}(a_1,b_1),\ldots, J_{q_k}(a_k,b_k)]\);
e) if blocks of type b) stand on the main diagonal of a matrix, and the remaining elements are equal to zero, then such a matrix \([J_{q_1}(a_1,1),\ldots, J_{q_k}(a_k,1)]\) will be called a matrix in normal Jordan form;
f) if blocks of type c) stand on the main diagonal of a matrix, and the remaining elements are equal to zero, then such a matrix \([J_{q_1}(a_1,0),\ldots, J_{q_k}(a_k,0)]\) will be called diagonal in the broad sense.
§ 2. THEOREM ON REDUCIBILITY IN \(M\)
In the monograph of A. M. Lyapunov [3] the concept of reducible systems of linear differential equations was introduced. Reducible systems, as Lyapunov showed, play an important role in the investigation of the stability of solutions of nonlinear systems. However, in his investigations Lyapunov did not express any general considerations about reducibility and, apart from the example of systems with periodic coefficients, gave no other examples.
The theory of reducible systems was in fact developed in the investigations of N. P. Erugin [4, 5] (p. 119). The study of this section of our paper is carried out on the basis of N. P. Erugin’s ideas.
Consider the equation
\[ Dx = P(u_1,\ldots,u_m)x, \tag{4} \]
where \(x\) is an \(n\)-dimensional vector, and \(P\) is a square matrix of order \(n\) from \(M\).
Let some matrix \(B(u_1,\ldots,u_m)\) be bounded together with \(DB\) and \(B^{-1}\) for all \(u_1,\ldots,u_m > 0\) and lying on the diagonal. We shall call this matrix a Lyapunov matrix.
Definition 3. Equation (4) will be called reducible in \(M\) if, by means of the transformation
\[ x = B(u_1,\ldots,u_m)y, \tag{5} \]
where \(B\) is a Lyapunov matrix, is reduced to the form
\[ Dy=Q(u_1,\ldots,u_m)y, \tag{6} \]
where \(Q\subset M\) and has one of the three forms c), d), or e), while the matrix \(B(u_1,\ldots,u_m)\) in the general case may fail to belong to \(M\).
We shall call the indicated reducibility, in contrast to the reducibility introduced by Lyapunov, reducibility in \(M\).
Lemma. Let equation (6) be given,
\[ Dy=Q(u_1,\ldots,u_m)y, \]
where \(Q\subset M\) and has the structure of one of the three forms c), d), or e). Let \(A\subset K\) and have the same form as \(Q\), and let \(DA=Q\). Then a solution of equation (6) has the form
\[ Y=\exp A(u_1,\ldots,u_m). \]
Indeed, since in the present case \(A\) and \(DA\) commute, we have
\[ DY=D(\exp A)=(\exp A)(DA)=(DA)(\exp A)=QY. \]
Theorem 1. In order that equation (4)
\[ Dx=P(u_1,\ldots,u_m)x \]
be reducible in \(M\), it is necessary and sufficient that a fundamental system of solutions can be represented in the form
\[ X(u_1,\ldots,u_m)=B(u_1,\ldots,u_m)\exp A(u_1,\ldots,u_m), \tag{7} \]
where \(B\) is a Lyapunov matrix, \(A\subset K\) and has one of the three forms c), d), or e).
Indeed, suppose equation (4), by means of the transformation (5), has been reduced to the form
\[ Dy=Q(u_1,\ldots,u_m)y. \]
Then, according to the lemma just established, \(Y=\exp A\), where \(A\subset K\) and has the form analogous to \(Q\). Hence \(X=B\exp A\), and necessity is proved.
Let us establish sufficiency. Suppose equation (4) has a solution of the form (7). Put
\[ x=B(u_1,\ldots,u_m)y, \]
where \(B=\exp(-A)\). Applying the operator \(D\) to both sides of (5), we obtain
\[ Dx=(DB)y+B(Dy)= \]
\[ =(DX)Ny-XN(DA)y+XN(Dy)=Px, \]
where, for brevity of notation, \(\exp -A\) is denoted by \(N\). Since \(Px=PXNy\), it follows that
\[ \underline{(DX)}Ny-XN(DA)y+XN(Dy)=\underline{PXN}y. \tag{8} \]
Since \(X\) is a solution of equation (4), the underlined terms in (8) cancel, and consequently \(Dy=(DA)y\), but \(DA=Q\). Thus
\[ Dy=Q(u_1,\ldots,u_m)y. \]
Corollary 1. If the set of real numbers is considered as \(M\), then in (7) the matrix \(A\) must have the form
\[ A=\sum_1^m \frac{\alpha_i u_i}{\omega}\,V, \tag{9} \]
where \(V\) is an absolutely constant matrix [2], \(\omega=\sum_{1}^{m}\alpha_j\) and \(\alpha_j=\mathrm{const}\).
Corollary 2. If in (9) we put \(u_1=\cdots=u_m=t\), then we obtain Erugin’s theorem [4], since then \(A=tV\) and \(D=\dfrac{d}{dt}\).
§ 3. Reducibility in the Case of Periodic Matrices
In what follows we shall assume that the set \(M\) consists of periodic functions of \(m\) independent variables \(u_1,\ldots,u_m\) with periods \(\omega_1,\ldots,\omega_m\) in \(u_1,\ldots,u_m\), respectively, and that \(K\) is the differential ring generated by \(M\).
Remark 1. If some matrix \(\Gamma \subset K\), then the relation
\[
\overline{\Gamma}=\Gamma+R,
\tag{*}
\]
holds, where \(R\) is some matrix constant on the diagonal.
Remark 2. For a given matrix \(R\) constant on the diagonal, one can find a matrix \(\Gamma\) satisfying relation (*) and having the same structure as \(R\).
Consider equation (2)
\[
Dx=F(u_1,\ldots,u_m)x.
\]
Theorem 2. If equation (2) is such that in equation (3) the matrix \(C\) has the form
\[
C=\exp R(u_2-u_1,\ldots,u_m-u_1),
\]
where \(R\) is a matrix constant on the diagonal and having one of the three forms d), e), or f), then equation (2) is reducible in \(M\) by means of a periodic matrix.
Indeed, since \(R\) is constant on the diagonal, it follows, by Remarks 1 and 2, that there exists in \(K\) a matrix \(A\) such that \(\overline{A}=A+R\), and \(A\) has the same structure as \(R\). It is not difficult to see that one of the solutions of equation (3), when \(C=\exp R\) and \(R\) is the matrix specified in the hypothesis of Theorem 2, is a matrix of the form
\[
X_1=\exp A(u_1,\ldots,u_m).
\]
If \(Z\) is some solution of the finite-difference equation (3), then, according to [2], the matrix \(Z_1=\Phi Z\), where \(\Phi\) is a periodic matrix, will also be a solution of equation (3). Therefore the fundamental matrix of solutions of the differential equation (2) in the case under consideration has the form
\[
X(u_1,\ldots,u_m)=B(u_1,\ldots,u_m)\exp A(u_1,\ldots,u_m),
\tag{10}
\]
where \(B(u_1,\ldots,u_m)\) is a periodic matrix with periods \(\omega_1,\ldots,\omega_m\) in \(u_1,\ldots,u_m\), respectively. Thus, on the basis of Theorem 1 and formula (10), we conclude that equation (2) is reducible by the transformation \(x=By\) (with the periodic Lyapunov matrix \(B\)) to the form
\[
Dy=Q(u_1,\ldots,u_m)y,
\]
where \(Q=DA\). Since \(A\subset K\), equation (2) is reducible in \(M\).
Theorem 3. If equation (2) is reducible in \(M\) by means of a periodic matrix, then the matrix \(C\) in relation (3) has the form \(C=\exp R\), where \(R\) is a matrix constant on the diagonal of generalized Jordan form.
Indeed, suppose that by means of the transformation \(x=By\), where \(B\) is a periodic Lyapunov matrix, equation (2) has been reduced to the form
\[ Dy=Q(u_1,\ldots,u_m)y, \]
where \(Q\) is a periodic matrix of generalized Jordan form. Thus, on the basis of established Lemma 1, we have
\[ Y=\exp A(u_1,\ldots,u_m), \]
and, consequently, \(X=B\exp A\), where \(DA=Q\). Then
\[ \overline{X}=\overline{B}\exp \overline{A}. \tag{11} \]
On the basis of Remarks 1 and 2 and the conditions of Theorem (3), equality (11) is rewritten in the form
\[ \overline{X}=B\exp(A+R)=B(\exp A)(\exp R)=X\exp R=XC. \tag{12} \]
From Theorems 2 and 3 we obtain the following theorem.
Theorem 4. In order that equation (2) be reducible to \(M\) by means of a periodic matrix, it is necessary and sufficient that there exist a fundamental matrix of solutions of equation (2) for which the matrix \(C\) in relation (3) has the form
\[ C=\exp R, \]
where \(R\) is a constant matrix whose diagonal is of one of the three forms c), d), or e).
In what follows, in order to clarify the structure of the solutions of equation (2), we shall solve in \(M\) the problem for the reduced equation (6). The functions standing on the main diagonal of the matrix \(Q(u_1,\ldots,u_m)\) will be called characteristic functions.
Remark. Condition (a) of [6] will be called the Harasakhala condition. In what follows we shall assume that the Harasakhala condition for system (2) is not satisfied. This means that in equation (6) the elements of the matrix \(Q\) depend on \(u_1,\ldots,u_m\) in a decisive way [6].
§ 4. STRUCTURE OF SOLUTIONS OF SYSTEMS REDUCIBLE TO \(M\)
Consider the following three cases.
\(1^\circ.\ Q=[J_{q_1}(\lambda_1,0),\ldots,J_{q_k}(\lambda_k,0)]\).
In this case system (6) splits into independently integrable groups of equations. It is not difficult to see that the fundamental system of solutions corresponding to the group of equations \((\nu_i=q_1+q_2+\cdots+q_{i-1},\ \nu_1=0)\)
\[ D(y_{\nu_i+1},\ldots,y_{\nu_{i+1}})=J_{q_i}(\lambda_i,0)(y_{\nu_i+1},\ldots,y_{\nu_{i+1}}), \tag{13} \]
is formed by functions of the form \(y_{jk}=\exp r_i\), where \(Dr_i=\lambda_i\). The original system (2) will have a fundamental matrix of solutions composed of functions of the form \((\nu_k=q_1+q_2+\cdots+q_{k-1},\ 1+\nu_k\le l_k\le \nu_{k+1})\)
\[ x_{j l_k}(u_1,\ldots,u_m)=\Phi_{j l_k}(u_1,\ldots,u_m)\exp r_k(u_1,\ldots,u_m), \tag{14} \]
where \(\Phi_{j l_k}\) is a periodic function, and \(r_k\) is such that \(Dr_k=\lambda_k\).
Corollary. If this system of solutions is considered on the diagonal, then for system (1) we obtain the following fundamental system of solutions:
\[ x_{j l_k}(t,\ldots,t)=\varphi_{j l_k}(t)\exp \rho_k(t), \]
where \(\varphi_{jl_k}(t)\) is a quasiperiodic function, and \(\rho_k(t)\) has a quasiperiodic derivative.
\(2^\circ.\ Q=[J_{q_1}(\lambda_1,1),\ldots,J_{q_k}(\lambda_k,1)]\).
In this case the system decomposes into independently integrable groups of equations of the form
\[ D(y_{\nu_i+1},\ldots,y_{\nu_{i+1}})=J_{q_i}(\lambda_i,1)(y_{\nu_i+1},\ldots,y_{\nu_{i+1}}). \tag{15} \]
It is not difficult to show that the fundamental matrix of solutions of equation (15) has the form
\[ \left\| \begin{array}{cccc} \exp r_i & 0 & \cdots & 0\\ P_1\exp r_i & \exp r_i & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ P_{q_i-1}\exp r_i & P_{q_i-2}\exp r_i & \cdots & \exp r_i \end{array} \right\|. \]
Then the original system of differential equations (2) will have a fundamental matrix of solutions composed of functions of the form
\[ x_{sl_k}(u_1,\ldots,u_m)=G_{sl_k}(u_1,\ldots,u_m)\exp r_k(u_1,\ldots,u_m) \tag{16} \]
with
\[
G_{sl_k}=\Phi_{sl_k}+\Phi_{sl_k-1}P_{s1}+\cdots+\Phi_{s1}P_{sl_k-1},
\]
where \(\Phi_{sj}\) are periodic functions with periods \(\omega_i\) in \(u_i\), \(P_{sj}\) is a form in the variables \(u_1,u_2,\ldots,u_m\) of \(j\)-th order, \(Dr_k=\lambda_k\), \(\lambda_k\) is a periodic function, and \(1+\nu_k\le l_k\le\nu_{k+1}\), \(s=1,2,\ldots,n\).
Corollary. On the diagonal, system (16) gives the following fundamental system of solutions of equation (1)
\[ x_{sl_k}(t,\ldots,t)=g_{sl_k}(t)\exp\rho_k(t) \]
with
\[
g_{sl_k}(t)=\varphi_{sl_k}(t)+\varphi_{sl_k-1}(t)t_{s1}(t)+\cdots+\varphi_{s1}(t)t_{sl_k-1},
\]
where \(\varphi_{sl_k}(t)\) are quasiperiodic functions, \(t_{sj}(t)\) is a form in the variable \(t\) of \(j\)-th order, and \(\rho_k(t)\) has a quasiperiodic derivative.
\(3^\circ.\ Q=[J_{q_1}(\lambda_1,b_1),\ldots,J_{q_k}(\lambda_k,b_k)]\).
From the form of the matrix \(Q\) it follows that system (6) decomposes into independently integrable groups of equations of the form
\[ D(y_{\nu_i+1},\ldots,y_{\nu_{i+1}})=J_{q_i}(\lambda_i,b_i)(y_{\nu_i+1},\ldots,y_{\nu_{i+1}}). \tag{17} \]
It is not difficult to show that the fundamental matrix of solutions of system (17) has the following form
\[ \left\| \begin{array}{cccc} e^{r_i} & 0 & \cdots & 0\\ N_1e^{r_i} & e^{r_i} & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ N_{q_i-1}e^{r_i} & N_{q_i-2}e^{r_i} & \cdots & e^{r_i} \end{array} \right\|, \]
where the function \(r_i\) is such that \(Dr_i\) is a periodic function. The functions \(N_j\) are chosen according to the following rule: \(DN_1\) is a periodic function,
\[ DN_2=b_iN_1,\ldots,\quad DN_{q_i-1}=b_iN_{q_i-2}. \]
The fundamental matrix of solutions of system (2) consists of functions of the form
\[ x_{s l_k}(u_1,\ldots,u_m)=G_{s l_k}(u_1,\ldots,u_m)e^{r_k(u_1,\ldots,u_m)}, \tag{18} \]
where \(G_{s l_k}=\Phi_{s l_k}+\Phi_{s l_k-1}N_{s1}+\cdots+\Phi_{s1}N_{s l_k-1}\). The functions \(\Phi_{sj}\) are periodic with periods \(\omega_i\) in \(u_i\); \(D r_i=\lambda_i\), where \(\lambda_i(u_1,\ldots,u_m)\) is a periodic function. The functions \(N_{sj}\) satisfy the following conditions: \(N_{s1}=b_k\), where \(b_k\) is a periodic function, \(D N_{s2}=b_k N_{s1}, \ldots, D N_{s l_k}=b_k N_{s l_k-1}\), where \(N_{s l_k-1}\) is a function of the preceding type.
Consequently. If we consider solution (18) on the diagonal, then for equation (1) we obtain the following fundamental system of solutions
\[ x_{s l_k}(t,\ldots,t)=g_{s l_k}(t)\exp \rho_k(t), \]
where \(g_{s l_k}(t)=G_{s l_k}(t,\ldots,t)\), and \(\rho_k(t)=r_k(t,\ldots,t)\), \(\nu_k+1\le l_k\le \nu_{k+1}\).
Let us call the diagonal of a set \(M\) the set composed of the functions belonging to \(M\) for which \(u_1=u_2=\cdots=u_m=t\), and denote it by \(Md\).
Theorem 5. If the system of differential equations (2) is reducible in \(M\) by means of a periodic matrix, then the system of differential equations (1) will be reducible on the diagonal \(Md\) by means of a quasi-periodic matrix.
The theorem is obtained as a consequence of Theorem 2 on the diagonal.
Remark 1. As the above investigations show, equation (1), reducible in the sense of Lyapunov, is also reducible in \(Md\), although the converse is not true in the general case.
Indeed, if we assume that the Harasahal condition for equation (2) is violated, then equation (1) is not reducible in the sense of Lyapunov, although it remains reducible in our sense.
Remark 2. In [2] several cases are considered of the structure of the matrix \(C\) in relation (3), for which the Harasahal condition is not fulfilled, and the types of solutions of equation (2) (respectively (1)) are indicated for these cases. As the above investigations show, equation (2) (respectively (1)) in all these cases is reducible in \(M\) (in \(Md\)), although it is not reducible in the sense of [2].
Remark 3. As the clarified structure of the solutions of equation (1) shows, in the case when it is reducible in \(Md\), it is regular in the sense of Lyapunov.
Indeed, if the characteristic numbers of system (1) (the reduced system) are denoted by \(p_k\), and the characteristic numbers of the adjoint system (also reduced) by \(q_k\), then it is clear that \(p_k+q_k=0\).
Theorem 6. If system (1) is reducible in \(Md\) and if the mean values [1] of its characteristic functions are strictly negative, then the zero solution of system (1) is asymptotically stable.
The proof follows from the structure of the solutions established by us and from Lyapunov’s theorem [3].
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Erugin N. P. Linear systems of ordinary differential equations with periodic and quasi-periodic coefficients. Minsk, Publishing House of the Academy of Sciences of the BSSR, 1963.
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Received by the editors
June 21, 1965
Kazakh State
University