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UDC 917.941.92
ON THE REDUCIBILITY OF LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH QUASI-PERIODIC COEFFICIENTS
V. I. ROMANOV, V. Kh. KHARASAKHAL
In the monograph [1] (p. 138) the question is posed of the reducibility (in the sense of Lyapunov–Erugin) of linear systems of differential equations with quasi-periodic coefficients.
Let us first note that, in the general case, linear systems with quasi-periodic coefficients are not reducible. But if some system turns out to be reducible, then the coefficients of the transformation need not necessarily be quasi-periodic functions.
Consider, for example, the system [2]
\[ \frac{dx}{dt}=P(t)x,\qquad P(t)= \left\| \begin{array}{cc} 0 & f(t)\\ -f(t) & 0 \end{array} \right\|, \]
where \(f(t)\) is a quasi-periodic function with zero mean value and with an unbounded integral, and \(x(x_1,x_2)\) is a vector. This system is reduced to a system with constant (equal to zero) coefficients by means of the transformation
\[ x=A(t)y,\qquad A(t)= \left\| \begin{array}{cc} \cos\int f(t)\,dt & \sin\int f(t)\,dt\\ -\sin\int f(t)\,dt & \cos\int f(t)\,dt \end{array} \right\|, \]
whose coefficients are not quasi-periodic functions.
Thus, it is of interest to single out, from the class of reducible systems with quasi-periodic coefficients, those systems which are reduced by means of a quasi-periodic transformation.
In the present paper a method of investigation is proposed by means of which this problem is solved.
- Consider the system
\[ \frac{dx}{dt}=P(t)x, \tag{1} \]
where \(P(t)=\|P_{sk}(t)\|_1^n\) is a quasi-periodic matrix with frequency basis \(\beta\{\beta_1,\ldots,\beta_m\}\), and \(x(x_1,\ldots,x_n)\) is a vector. It is known [3] (p. 118) that the quasi-periodic function \(P_{sk}(t)\) is the diagonal of a periodic function \(F_{sk}(u_1,\ldots,u_m)\) with periods \(\omega_k=\dfrac{2\pi}{\beta_k}\) in the variables \(u_k\), i.e.,
\[ P_{sk}(t)=F_{sk}(t,\ldots,t). \]
We form the system of partial differential equations
\[ Dx=F(u_1,\ldots,u_m)x, \tag{2} \]
where
\[ Dx=\frac{\partial x}{\partial u_1}+\frac{\partial x}{\partial u_2}+\cdots+\frac{\partial x}{\partial u_m}, \qquad F(t,\ldots,t)=P(t). \]
The system (2) with periodic matrix \(F(u_1,\ldots,u_m)\) constructed in this way gives rise, on the diagonal \(u_1=u_2=\cdots=u_m=t\), to the system (1) with quasiperiodic matrix \(P(t)\).
Let us note some facts needed here from the general theory of systems (2), developed in [4, 5].
Let the vectors
\[ x_1(x_{11},\ldots,x_{n1}),\ldots,x_n(x_{1n},\ldots,x_{nn})\bigl(x_{sk}=x_{sk}(u_1,\ldots,u_m)\bigr) \tag{3} \]
be \(n\) particular solutions of system (2).
Definition. A system of solutions (3) will be called fundamental if it satisfies the following condition: let \(x(x_1,\ldots,x_n)\) \(\bigl(x_k=x_k(u_1,\ldots,u_m)\bigr)\) be any preassigned solution of system (2); then there exist differentiable functions \(A_k(u_2-u_1,\ldots,u_m-u_1)\) \((k=1,\ldots,n)\) such that
\[ x=A_1x_1+\cdots+A_nx_n . \]
Let \(x(u_1,\ldots,u_m)\) be the matrix composed of the functions (3), and let \(|x(u_1,\ldots,u_m)|\) be its determinant. Then it has been proved that if \(|x(u_1,\ldots,u_m)|\) is nonzero for all values of \(u_k\), then the system (3) is fundamental; moreover, the system (2) with fundamental matrix \(x\) has the determinant
\[ \left| \begin{array}{ccccc} Dx_s & Dx_{s1} & \cdot & \cdot & Dx_{sn}\\ x_1 & x_{11} & \cdot & \cdot & x_{1n}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ x_n & x_{n1} & \cdot & \cdot & x_{nn} \end{array} \right|=0 \quad (s=1,\ldots,n). \tag{4} \]
Let \(C=C(u_2-u_1,\ldots,u_m-u_1)\) be a matrix depending on the differences \(u_k-u_1\). In what follows such matrices, as well as functions of a similar kind, will be called constant on the diagonal.
- We construct a completely integrable system
\[ \frac{\partial x}{\partial u_j}=f_j(u_1,\ldots,u_m)x \quad (j=1,\ldots,m), \tag{5} \]
such that
\[ \frac{\partial x}{\partial u_1}+\frac{\partial x}{\partial u_2}+\cdots+\frac{\partial x}{\partial u_m} =Dx=\sum_{j=1}^{m} f_j(u_1,\ldots,u_m)x=F(u_1,\ldots,u_m)x. \]
System (2) will be called a consequence of system (5). A fundamental system of solutions for system (5), and certain questions connected with it, are introduced in the same way as in the theory of ordinary differential equations. In particular, to a given fundamental matrix \(x\) there corresponds the completely integrable Pfaffian form
\[ \left| \begin{array}{ccccc} \dfrac{\partial x_s}{\partial u_j} & \dfrac{\partial x_{s1}}{\partial u_j} & \cdot & \cdot & \dfrac{\partial x_{sn}}{\partial u_j}\\ x_1 & x_{11} & \cdot & \cdot & x_{1n}\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ x_n & x_{n1} & \cdot & \cdot & x_{nn} \end{array} \right|=0 \quad (s=1,\ldots,n;\quad j=1,\ldots,m), \tag{6} \]
and the sum of these determinants gives system (4), i.e., system (4) is a consequence of system (6).
For different fundamental matrices of system (2) one can in this way construct a set of compatible systems (5), of which system (2) is a consequence. Thus the following is true.
Theorem 1. There exists an infinite set of compatible systems of type (5), of which one system (2) is a consequence.
Let us note that, for system (5), in principle it is not difficult to obtain conditions of complete integrability, and one can construct whole classes of systems of this kind that are known in advance not to be completely integrable.
It is not difficult to see that if \(x_1(u_1,\ldots,u_m)\) is a fundamental matrix of solutions of system (5), then the matrix \(x=x_1 C\), where \(C\) is a constant everywhere matrix [4], will also be a fundamental matrix of solutions, and all fundamental matrices of solutions of system (5) are contained in this formula.
Obviously, every solution of system (5) will be a solution of system (2). A fundamental matrix of solutions of system (5) will also be a fundamental matrix of solutions for system (2). The converse, of course, does not hold in the general case.
If \(x_1(u_1,\ldots,u_m),\ldots,x_n(u_1,\ldots,u_m)\) is some solution of system (5), and consequently also of system (2), then the diagonal functions \(x_1(t,\ldots,t),\ldots,\) \(\ldots,x_n(t,\ldots,t)\) constitute a solution of system (1) [5]. Thus, each solution of system (5) generates on the diagonal a solution of system (1).
In particular, a solution of system (5) periodic in all variables \(u_k\) generates on the diagonal \(u_1=u_2=\cdots=u_m=t\) a quasiperiodic solution of system (1).
- If system (2) is subjected to the transformation
\[ x=B(u_1,\ldots,u_m)y, \tag{7} \]
where \(B\) is a nonsingular matrix, then we obtain the equation
\[ Dy=(B^{-1}FB-B^{-1}DB)y. \]
Let the nonsingular matrix \(B\), together with the matrix \(DB\), be bounded for all positive values \(u_1,u_2,\ldots,u_m\) lying on the diagonal. A matrix \(B\) having this property will be called a Lyapunov matrix.
Definition [4]. We shall call system (2) reducible if, by means of transformation (7) with a Lyapunov matrix \(B\), it is transformed into a system with a matrix that is constant on the diagonal or constant everywhere.
Let us note that if \(x(u_1,\ldots,u_m)\) is a fundamental matrix of solutions of system (2), then, by virtue of the periodicity of the matrix \(F\), \(x(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m)\) is also a solution of system (2), and, consequently,
\[ x(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m) = x(u_1,\ldots,u_m)C, \tag{8} \]
where \(n_1,\ldots,n_m\) are arbitrary integers, and \(C\) is, in the general case, a matrix constant on the diagonal.
Theorem 2. In order that system (2) with periodic matrix \(F\) be reducible to a system with constant matrix by means of a periodic matrix, it is necessary and sufficient that there exist a completely integrable system (5) with periodic coefficients, of which system (2) is a consequence.
Proof. Suppose system (2) is reducible to a system with constant matrix by means of a periodic matrix. Then it is known [5] that there exists such a fundamental matrix of solutions \(x(u_1,\ldots,u_m)\)
of system (2), the matrix \(C\) in relation (8) will be constant everywhere. Let us now construct a completely integrable Pfaffian system with this fundamental matrix \(x\), of which system (2) is a consequence. If the coefficients \(f_{jk}^{(s)}(u_1,\ldots,u_m)\) of this constructed system are not periodic functions, then \(x(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m)\) will not be its solution and, consequently, relation (8) with a constant matrix \(C\) everywhere is not satisfied; and this proves necessity.
Let us establish sufficiency. Suppose there exists a completely integrable Pfaffian system of type (5) with periodic coefficients, of which system (2) is a consequence. Then, if \(x(u_1,\ldots,u_m)\) is a fundamental matrix of solutions for system (5), then \(x(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m)\) is also a solution of this system, and therefore
\[ x(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m)=x(u_1,\ldots,u_m)C, \]
where \(C\) is a constant matrix everywhere. But all solutions of system (5) are solutions of system (2), which proves theorem [5].
Thus:
I) All systems (2) with periodic matrices that are consequences of compatible systems (5) with periodic coefficients are reducible to systems with constant matrices by means of periodic matrices.
II) If there does not exist even one completely integrable system (5) with periodic coefficients, of which the given system (2) is a consequence, then system (2) is not reducible to a system with a constant matrix by means of a periodic matrix.
If the results obtained are considered on the diagonal \(u_1=\cdots=u_n=t\), then we obtain the theorem.
Theorem 3. Every system (1) with a quasiperiodic matrix, generated on the diagonal by a system (2) that is a consequence of a completely integrable Pfaffian system (5) with periodic coefficients, is reducible to a system with a constant matrix by means of a quasiperiodic matrix.
Example. Given the system
\[ \begin{aligned} \frac{dx_1}{dt} &=x_1(1+\cos t)\cos^2\sqrt{2}\,t +x_2\bigl(\cos\sqrt{2}\,t\,\sin\sqrt{2}\,t \\ &\qquad\qquad +\cos\sqrt{2}\,t\,\sin\sqrt{2}\,t\cos t-\sqrt{2}\bigr), \end{aligned} \tag{9} \]
\[ \begin{aligned} \frac{dx_2}{dt} &=x_1\bigl(\sin\sqrt{2}\,t\,\cos\sqrt{2}\,t +\cos t\,\sin\sqrt{2}\,t\,\cos\sqrt{2}\,t+\sqrt{2}\bigr)\\ &\qquad\qquad +x_2(1+\cos t)\sin^2\sqrt{2}\,t . \end{aligned} \]
System (9) is generated by the system
\[ \begin{aligned} Dx_1 &=x_1(1+\cos u_1)\cos^2\sqrt{2}\,u_2 +x_2\bigl(\cos\sqrt{2}\,u_2\,\sin\sqrt{2}\,u_2\\ &\qquad\qquad+\cos\sqrt{2}\,u_2\,\sin\sqrt{2}\,u_2\cos u_1-\sqrt{2}\bigr), \end{aligned} \]
\[ \begin{aligned} Dx_2 &=x_1\bigl(\sin\sqrt{2}\,u_2\,\cos\sqrt{2}\,u_2 +\cos u_1\,\sin\sqrt{2}\,u_2\,\cos\sqrt{2}\,u_2+\sqrt{2}\bigr)\\ &\qquad\qquad +x_2(1+\cos u_1)\sin^2\sqrt{2}\,u_2, \end{aligned} \]
which is a consequence of the completely integrable Pfaffian system
\[ \frac{\partial x_1}{\partial u_1} =x_1(1+\cos u_1)\cos^2\sqrt{2}\,u_2 +x_2(1+\cos u_1)\cos\sqrt{2}\,u_2\,\sin\sqrt{2}\,u_2, \]
\[ \frac{\partial x_1}{\partial u_2}=-\sqrt{2}\,x_2, \]
\[ \frac{\partial x_2}{\partial u_1} = x_1(1+\cos u_1)\sin\sqrt{2}\,u_2\cos\sqrt{2}\,u_2 + x_2(1+\cos u_1)\sin^2\sqrt{2}\,u_2, \]
\[ \frac{\partial x_2}{\partial u_2}=\sqrt{2}\,x_1 \]
with periodic coefficients. Consequently, system (9) is reducible by means of a nonsingular transformation with quasiperiodic coefficients.
References
- N. P. Erugin. Linear systems of ordinary differential equations with periodic and quasiperiodic coefficients. Publishing House of the Academy of Sciences of the BSSR, Minsk, 1963.
- Yu. G. Zolotarev. On the regularity of one class of systems of differential equations with almost periodic coefficients. Report at the 4th All-Union Mathematical Congress, July 5, 1961.
- B. M. Levitan. Almost-periodic functions. Gostekhizdat, 1953.
- V. Kh. Kharasakhal. DAN SSSR, 146, No. 6, 1290—1293, 1962.
- V. Kh. Kharasakhal. PMM, 27, issue 4, 672—682, 1963.
Received by the editors
June 19, 1965
Kazakh State University