UDC 517.917

ON QUASIPERIODIC SOLUTIONS OF NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS CONTAINING A SMALL PARAMETER

A. I. ZAITSEV, K. K. KAPISHEV, V. Kh. KHARASAKHAL

The paper investigates ordinary differential equations whose right-hand sides depend on a small parameter.

The case in which the right-hand sides of a system of differential equations depend continuously on the parameter has been studied rather well. In particular, the case of periodic solutions under continuous dependence of the right-hand sides on the parameter has been studied quite fully. The case of quasiperiodic solutions under continuous dependence of the right-hand sides on the parameter was studied in [1].

However, the case of quasiperiodic solutions in which the right-hand sides are discontinuous functions of the parameter has been little studied. Nevertheless, these problems must be studied for various kinds of applications in mathematical practice.

§ 1. ON ONE SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS

Consider the equation

\[ Dx \equiv \frac{\partial x}{\partial u_1}+\frac{\partial x}{\partial u_2}+\cdots+\frac{\partial x}{\partial u_m} = F(u_1,\ldots,u_m)x, \tag{1.1} \]

where \(x(x_1,\ldots,x_n)\) and \(F(u_1,\ldots,u_m)\) are vectors, and the functions \(F_s(u_1,\ldots,u_m,x)\) in the domain

\[ |u_j|<\infty,\quad |x_k|<\infty \quad (j=1,\ldots,m;\ k=1,\ldots,n) \]

will be assumed continuous with continuous partial derivatives.

It will be seen from what follows that the study of equation (1.1) is not an end in itself for us. We bring (1.1) into the investigation because, if it is considered under the condition

\[ u_1=u_2=\cdots=u_m=t, \tag{1.2} \]

then we obtain the equation in ordinary derivatives

\[ \frac{dx}{dt}=F(t,\ldots,t,x). \tag{1.3} \]

However, the study of equation (1.1) apparently is also of independent interest.

Every solution of equation (1.3) is obtained from some solution of equation (1.1) under condition (1.2). Therefore, first of all it is necessary to make sure

that the solutions of equation (1.1) exist. To this end, note that if one seeks a solution of equation (1.1) in the form \(\varphi(\varphi_1,\ldots,\varphi_n)\), \(\varphi_k(u_1,\ldots,u_m,x)=0\) \((k=1,\ldots,n)\), under the condition that the Jacobian
\[ \frac{\partial(\varphi_1,\ldots,\varphi_n)}{\partial(x_1,\ldots,x_n)} \]
is everywhere different from zero, then equation (1.1) is equivalent to the scalar linear equation
\[ Dz+\left(\frac{\partial z}{\partial x}\cdot F\right)=0, \tag{1.4} \]
where \(\left(\frac{\partial z}{\partial x}\cdot F\right)\) is the scalar product of the vectors \(F\) and \(\dfrac{\partial z}{\partial x}=\left(\dfrac{\partial z}{\partial x_1},\ldots,\dfrac{\partial z}{\partial x_n}\right)\).

The integration of (1.4) is reduced, as is known, to the integration of the system of ordinary differential equations
\[ \frac{du_2}{du_1}=1,\quad \frac{du_3}{du_1}=1,\ \ldots,\ \frac{du_m}{du_1}=1, \]
\[ \frac{dx}{du_1}=F(u_1,\ldots,u_m,x). \tag{1.5} \]

Thus, solutions of equation (1.1) exist if first integrals of system (1.5) exist.

Let us now pose the question of the existence and uniqueness of a solution of equation (1.1) subject to the condition
\[ x(u_1,u_2,\ldots,u_m)\big|_{u_1=u_1^0} =\pi(u_2,\ldots,u_m), \tag{1.6} \]
where \(\pi(\pi_1,\ldots,\pi_n)\) is a vector, and the \(\pi_s(u_2,\ldots,u_m)\) are differentiable functions of their arguments.

In what follows let \(u_1^0=0\). Integrating the first group of equations (1.5), we obtain
\[ u_j=u_1+u_j^0\quad (j=2,\ldots,m), \tag{1.7} \]
\[ \frac{dx}{du_1}=F(u_1,u_1+u_2^0,\ldots,u_1+u_m^0,x). \tag{1.8} \]

Let \(x=\theta(u_1,u_2^0,\ldots,u_m^0,x^0)\) be the solution of equation (1.8) passing through the point \((0,x^0)\), i.e.
\[ \theta(0,u_2^0,\ldots,u_m^0,x^0)=x^0. \]
We choose the values of the quantity \(x^0\) from the condition
\[ x^0=\pi(u_2^0,\ldots,u_m^0). \]
Then the functions
\[ u_j=u_1+u_j^0\quad (j=2,\ldots,m), \]
\[ x=\theta\bigl(u_1,u_2^0,\ldots,u_m^0,\pi(u_2^0,\ldots,u_m^0)\bigr) \tag{1.9} \]
will indeed be a unique solution of the system of equations (1.5) passing through the point
\[ (0,u_2^0,\ldots,u_m^0,\pi(u_2^0,\ldots,u_m^0)), \]
where \(u_2^0,\ldots,u_m^0\) are arbitrarily chosen quantities taking any finite values. Moreover, this solution exists for any finite values of \(u_1\). If from (1.7) the values of the quantities \(u_2^0,\ldots,u_m^0\) are substituted into (1.9), then we obtain a solution of equation (1.1)
\[ x=\theta\bigl(u_1,u_2-u_1,\ldots,u_m-u_1,\pi(u_2-u_1,\ldots,u_m-u_1)\bigr), \tag{1.10} \]

satisfying condition (1.6) and which will exist for all finite values of the variables \(u_1,\ldots,u_m\).

Obviously, the set of all solutions of (1.1) is contained in (1.10) for all possible different choices of the vector \(\pi\) satisfying the indicated conditions. Thus the general solution of equation (1.1) has been constructed.

From the manner in which the set of all solutions of equation (1.1) is formed, it follows that in this set there are no two distinct solutions corresponding, for \(u_1=0\), to the same initial conditions.

We note that all solutions of equation (1.3) are contained in (1.10) when condition (1.2) is fulfilled.

Let us now consider the linear equation

\[ Dx=\Phi(u_1,\ldots,u_m)x, \tag{1.11} \]

where \(\Phi(u_1,\ldots,u_m)=\|\Phi_{sk}(u_1,\ldots,u_m)\|_1^n\) is a square matrix, and \(x(x_1,\ldots,x_n)\) is a vector. Systems (1.11) were studied in [1, 2]. Here we shall briefly dwell on some properties of these systems.

A function \(f(u_2-u_1,\ldots,u_m-u_1)\), depending on the differences \(u_k-u_1\), as well as matrices with elements of this kind, will be called constant on the diagonals (1.2).

Let

\[ X(u_1,\ldots,u_m)=\|x_{sk}(u_1,\ldots,u_m)\|_1^n \tag{1.12} \]

be \(n\) solutions of equation (1.11), where the first index denotes the number of the function and the second the number of the solution.

The system of solutions (1.12) will be called fundamental if any solution \(\psi(x_1,\ldots,x_n)\) of equation (1.11) can be represented in the form

\[ \psi=X\cdot A, \]

where \(A(A_1,\ldots,A_n)\) is a vector whose components \(A_k(u_2-u_1,\ldots,u_m-u_1)\) are differentiable functions of their arguments that are constant on the diagonals (in particular, constant everywhere).

We note that if \(X_1(u_1,\ldots,u_m)\) is a fundamental matrix of solutions of equation (1.11), then the matrix

\[ X(u_1,\ldots,u_m)=X_1(u_1,\ldots,u_m)\cdot C, \tag{1.13} \]

where \(C\) is a nonsingular matrix constant on the diagonals, will also be a fundamental matrix of solutions, and all fundamental matrices of solutions of equation (1.11) are contained in formula (1.13).

Let now, in equation (1.11), the matrix \(\Phi(u_1,\ldots,u_m)\) be periodic with periods \(\omega_j\) in the variables \(u_j\), respectively, i.e., for any values of \(u_1,\ldots,u_m\) the relation

\[ \Phi(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m)=\Phi(u_1,\ldots,u_m), \]

holds, where \(n_1,\ldots,n_m\) are arbitrary integers. Then, under condition (1.2), system (1.11) generates the system

\[ \frac{dx}{dt}=\varphi(t)x \tag{1.14} \]

with quasiperiodic matrix \(\varphi(t)=\Phi(t,\ldots,t)\) with frequency basis
\(\beta\{\beta_1,\ldots,\beta_m\}\), \(\left(\beta_j=\dfrac{2\pi}{\omega_j}\right)\).

If \(X(u_1,\ldots,u_m)\) is a fundamental matrix of solutions of equation (1.11), then, by virtue of the periodicity of the matrix \(\Phi\), \(X(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m)\) is also a solution of equation (1.11), and on the basis of (1.13)

\[ X(u_1+n_1\omega_1,\ldots,u_m+n_m\omega_m) = X(u_1,\ldots,u_m)\cdot C, \tag{1.15} \]

where \(C\) is a nonsingular matrix constant on the diagonal or constant everywhere.

Relation (1.15) makes it possible to establish the structure (analytic form) of the solutions of equation (1.11), and hence also of equation (1.14) with a quasiperiodic matrix.

It has been shown [1, 2] that, in order that equation (1.11) be reducible to an equation with an everywhere constant matrix by means of the transformation \(x=B(u_1,\ldots,u_m)y\) with periodic matrix \(B\), it is necessary and sufficient that there exist such a fundamental matrix of solutions \(X(u_1,\ldots,u_m)\) for which the matrix \(C\) in relation (1.15) is everywhere constant.

In this case

\[ X(u_1,\ldots,u_m)=B(u_1,\ldots,u_m)\times \]

\[ {}\times \exp\left[ \frac{\alpha_1u_1+\cdots+\alpha_m u_m}{\alpha_1+\cdots+\alpha_m}\,A \right] \qquad (\alpha_j-\text{const}), \]

where \(B\) is periodic and \(A\) is an everywhere constant matrix.

For equation (1.14) this gives

\[ X(t,\ldots,t)=B(t,\ldots,t)\exp(tA). \]

Here \(B(t,\ldots,t)\) is a quasiperiodic matrix.

The equation \(|C-\lambda E|=0\), where \(E\) is the identity matrix, will in this case be called the characteristic equation of equation (1.11), and the quantities

\[ \alpha_j=\frac{m}{\omega_1+\cdots+\omega_m}\ln \lambda_j \]

the characteristic exponents of equation (1.14).

§ 2. Formulation of the problem

Consider the differential equation

\[ \frac{dx}{dt}=f(t,x,\varepsilon), \tag{2.1} \]

where \(f(f_1,\ldots,f_n)\) and \(x(x_1,\ldots,x_n)\) are vectors; moreover, the functions \(f_s\) are quasiperiodic with respect to \(t\), with common frequency basis \(\beta\{\beta_1,\ldots,\beta_m\}\), independent of the small parameter \(\varepsilon\), and for all real values of \(t\) are analytic functions of the variables \(x_1,\ldots,x_n\) and meromorphic functions of the parameter \(\varepsilon\) in the domain

\[ |x_s|\leq R,\qquad |\varepsilon|\leq \rho. \tag{G} \]

Speaking of any interval of variation of the parameter, we shall always assume that the value \(\varepsilon=0\) is one of the points of this interval.

By a solution of equation (2.1) for \(\varepsilon=0\) we shall mean the limit (if such exists), as \(\varepsilon\to0\), of some solution of this equation for \(\varepsilon\ne0\).

Let \(\varepsilon^{k_s} f_s^0(t,x_1^0,\ldots,x_n^0)\) be the term of lowest order with respect to \(\varepsilon\) in the expansion of the function \(f_s(t,x_1,\ldots,x_n,\varepsilon)\) in a series in integral powers of \(\varepsilon\). Let \(\gamma=\min(k_1,\ldots,k_n)\). We form the equation

\[ \frac{dx^0}{dt}=\varepsilon^\gamma f^0(t,x^0), \tag{2.2} \]

where \(f^0(f_1^0,\ldots,f_n^0)\) and \(x^0(x_1^0,\ldots,x_n^0)\) are vectors.

Following [3], we shall call equation (2.2) the simplified equation. We pose the problem. Suppose that the simplified equation admits a quasiperiodic solution

\[ x^0=x^0(t,\varepsilon)\qquad \bigl(x^0(x_1^0,\ldots,x_n^0)\bigr) \tag{2.3} \]

with frequency basis \(\beta\), which for all real values of \(t\) remains inside the domain \(G\). It is required to establish under what conditions equation (2.1), independently of the form of the terms discarded in comparison with the simplified equation, admits a quasiperiodic solution for arbitrary values of the parameter \(\varepsilon\) in some interval, tending, as \(\varepsilon=0\), to \(x^0(t,0)\).

If \(k_1=k_2=\cdots=k_n=0\), then the right-hand sides of (2.1) are analytic functions of the parameter, and the simplified equation is represented by equation (2.1) for \(\varepsilon=0\). Consequently, in this case the problem posed reduces to the following. Suppose that equation (2.1), whose right-hand side depends analytically on \(\varepsilon\), for \(\varepsilon=0\) admits a quasiperiodic solution \(x^0=x^0(t)\) with frequency basis \(\beta\), remaining for all real values of \(t\) inside the domain \(G\). It is required to find conditions under which equation (2.1) admits a quasiperiodic solution also for all numerically sufficiently small values of the parameter, tending to \(x^0(t)\) as \(\varepsilon=0\). This particular problem is similar to the well-known problem of Poincaré [4], and in works [1, 2] this case was considered in detail and it was shown that, if one passes to the systems of partial differential equations considered in § 1, then this problem, as in the periodic case, can be solved by Poincaré’s method.

Here we shall likewise solve the problem posed by passing to equations in partial derivatives. Therefore we consider the equation

\[ Dx=F(u_1,\ldots,u_m,x,\varepsilon), \tag{2.4} \]

where \(F(F_1,\ldots,F_n)\) and \(x(x_1,\ldots,x_n)\) are vectors, and \(F_s(u_1,\ldots,u_m,x_1,\ldots,x_n,\varepsilon)\) are periodic functions in the variables \(u_j\), respectively with periods \(\omega_j=\dfrac{2\pi}{\beta_j}\), and analytic functions of the variables \(x_1,\ldots,x_n\) and meromorphic functions of the parameter \(\varepsilon\) in the domain \(G\) for arbitrary real values of \(u_1,\ldots,u_m\).

Let \(\varepsilon^{k_s}F_s^0(u_1,\ldots,u_m,x_1,\ldots,x_n)\) be the term of lowest order with respect to \(\varepsilon\) in the expansion of the function \(F_s(u_1,\ldots,u_m,x_1,\ldots,x_n,\varepsilon)\) \((s=1,\ldots,n)\) in a series in integral powers of \(\varepsilon\). Let \(\gamma=\min(k_1,\ldots,k_n)\), and form the simplified equation

\[ Dx^0=\varepsilon^\gamma F^0(u_1,\ldots,u_m,x^0), \tag{2.5} \]

where \(F^0(F_1^0,\ldots,F_n^0)\) and \(x^0(x_1^0,\ldots,x_n^0)\) are vectors.

The problem is now posed as follows. Suppose that the simplified equation (2.5) admits a periodic solution

\[ x^0=x^0(u_1,\ldots,u_m,\varepsilon) \tag{2.6} \]

with periods \(\omega_j=\dfrac{2\pi}{\beta_j}\), which, for all real values \(u_1,\ldots,u_m\), remains inside the domain \(G\). It is required to establish under what conditions equation (2.4), independently of the form of the discarded terms, admits a periodic solution for arbitrary values of the parameter \(\varepsilon\), on some interval tending, as \(\varepsilon=0\), to \(x_s^0(u_1,\ldots,u_m,0)\).

Let us form the linear equation

\[ Dy=\varepsilon^\gamma py, \tag{2.7} \]

where \(p=\|p_{sj}\|_1^n\) is a matrix; \(y(y_1,\ldots,y_n)\) is a vector, and

\[ p_{sj}(u_1,\ldots,u_m)= \left( \frac{\partial F_s^0(u_1,\ldots,u_m,x_1^0,\ldots,x_n^0)} {\partial x_j^0} \right)_{x_i^0=x_i^0(u_1,\ldots,u_m,0)} \]

are periodic functions of the variables \(u_i\), with periods \(\omega_j=\dfrac{2\pi}{\beta_j}\).

In what follows, the variational equation (2.7) will be called determining [3], since the solution of the posed problem is predetermined by certain properties of this system.

In the periodic case, a problem similar to that formulated for equations (2.1) and (2.2) was posed in [3]. The quasiperiodic case is qualitatively different from the periodic one; for example, a linear system with quasiperiodic coefficients is, in general, not reducible in the sense of Lyapunov—Erugin [5]. However, by passing to partial differential equations, we shall show that the method proposed in [3] for the periodic case can be used without substantial changes also in the quasiperiodic case.

If \(X(u_1,\ldots,u_m)\) is a fundamental matrix of solutions of equation (2.7), then relation (1.15) is valid.

We shall consider here the cases: A) \(C\) is constant everywhere, B) \(C=\exp R\), where \(R\) is a matrix constant on the diagonal and having one of the following forms:

\[ R=[I_{q_1}(a_1,0),\ldots,I_{q_p}(a_p,0)], \]

\[ R=[I_{q_1}(a_1,1),\ldots,I_{q_p}(a_p,1)], \]

\[ R=[I_{q_1}(a_1,b_1),\ldots,I_{q_p}(a_p,b_p)], \]

\[ q_1+q_2+\cdots+q_p=n, \]

\[ I_{q_k}(a_k,b_k)= \left\| \begin{array}{ccccc} a_k & b_k & 0 & \cdots & 0\\ 0 & a_k & b_k & \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \cdots & a_k \end{array} \right\|. \]

Thus, the matrix \(R\) consists of the indicated blocks of order \(q_k\), and all its remaining elements are equal to zero. The functions \(a_j\) and \(b_j\) are constant on the diagonal, i.e., depend on the differences \(u_i-u_1\) \((i=2,\ldots,m)\).

§ 3. CASE A)

By means of the substitution

\[ v_j=\varepsilon^\gamma u_j\quad (j=1,\ldots,m) \tag{3.1} \]

we transform equation (2.7) to the form

\[ D_v y=\frac{\partial y}{\partial v_1}+\frac{\partial y}{\partial v_2}+\ldots+\frac{\partial y}{\partial v_m}=Qy, \tag{3.2} \]

where \(Q=\|q_{sj}\|_1^n\) is a matrix; \(y(y_1,\ldots,y_n)\) is a vector, and the coefficients \(q_{sj}(v_1,\ldots,\allowbreak v_m)\) are periodic functions with periods \(\Omega_j=\omega_j\varepsilon^\gamma\) in the variables \(v_j\), respectively, and, for sufficiently small \(|\varepsilon|\), are continuous.

In the case now under consideration, equation (3.2), and consequently also (2.7), is reducible by a transformation with a periodic matrix. The characteristic exponents \(\alpha_1,\ldots,\alpha_n\) of equation (2.7) and the characteristic exponents \(l_1,\ldots,l_n\) of equation (3.2) are related by

\[ \alpha_k=\varepsilon^\gamma l_k \quad (k=1,\ldots,n). \tag{3.3} \]

Let \(y=\alpha(v_1,\ldots,v_m)z\) be a transformation reducing equation (3.2) to the canonical form

\[ D_v z=Az, \tag{3.4} \]

where

\[ A= \left\| \begin{array}{ccccccc} l_1 & 0 & 0 & . & . & . & 0\\ \sigma_1 & l_2 & 0 & . & . & . & 0\\ 0 & \sigma_2 & l_3 & . & . & . & 0\\ . & . & . & . & . & . & .\\ 0 & 0 & 0 & . & . & . & l_n \end{array} \right\| \qquad (\sigma,l-\mathrm{const}) \]

and all \(\sigma_i=0\) if among the quantities \(l_1,l_2,\ldots,l_n\) there are no multiple ones, and \(\sigma_i\) is equal either to one or to zero in the opposite case.

Since equation (3.2) is reducible, such a transformation with continuous coefficients \(\alpha_{sk}(v_1,\ldots,v_m)\) exists, and it admits the inverse transformation \(z=\beta(v_1,\ldots,v_m)y\), with coefficients also possessing the indicated property. Moreover, both the one and the other coefficients are periodic with periods \(\Omega_j\) with respect to \(v_j\).

Consider the differential equation

\[ D_u y=\varepsilon^\gamma\bigl(py+\Phi(\varepsilon,u_1,\ldots,u_m)\bigr), \tag{3.5} \]

where \(\varepsilon^\gamma p\) is from the determining equation (2.7), and \(\Phi(\Phi_1,\ldots,\Phi_n)\) is a vector periodic with respect to \(u_1,\ldots,u_m\) with periods \(\omega_1,\omega_2,\ldots,\omega_m\).

Using the substitution (3.1), we transform equation (3.5) to the form

\[ D_v y=Qy+\Phi. \tag{3.6} \]

Applying the transformation that reduces equation (3.2) to the form (3.4), for equation (3.6) we obtain

\[ D_v z=Az+\beta\Phi. \]

Taking (3.3) into account, we obtain

\[ D_v z=Bz+\beta\Phi, \]

where the matrix \(B\) is obtained from the matrix \(A\) if in the latter \(l_j\) is replaced by \(\alpha_j\varepsilon^{-\gamma}\).

Since \(D_u z=\varepsilon^\gamma D_v z\), after the indicated transformations of equation (3.5) we obtain

\[ D_u z=Kz+\varepsilon^\gamma\beta\Phi, \]

where the matrix \(K\) is obtained from the matrix \(A\) if in it \(l_j\) is replaced by \(\alpha_j\), and \(\sigma_j\) by \(\sigma_j \varepsilon^\gamma\).

Assuming henceforth that among the quantities \(\alpha_j\) there are none equal to zero, and putting

\[ \rho_j = |\alpha_j|,\qquad b_{sj}=\frac{1}{\rho_s}\beta_{sj}\varepsilon^\gamma,\qquad A_{j-1}=\frac{1}{\rho_j}\varepsilon^\gamma\sigma_{j-1}, \]

we reduce equation (3.5) to the form (written, for convenience, in scalar form):

\[ Dz_1=\alpha_1 z_1+\rho_1\sum_{j=1}^{n} b_{1j}\Phi_j, \]

\[ Dz_s=\alpha_s z_s+\rho_s\left(A_{s-1}z_{s-1}+\sum_{j=1}^{n} b_{sj}\Phi_j\right)\quad (s=2,\ldots,n). \tag{3.7} \]

We shall say that equation (2.7) is reducible with respect to the parameter [3] if, for such an equation, all \(b_{sj}\) and \(A_{j-1}\) are continuous for all real values of \(u_1,\ldots,u_m\) and all \(\varepsilon\) in some interval.

Suppose that, in some interval \(a\leq \varepsilon \leq b\), the determining equation (2.7) is reducible with respect to the parameter and that among the characteristic exponents there is not a single one with real part equal to zero. Let, moreover, the functions \(\Phi_s(\varepsilon,u_1,\ldots,u_m)\) \((s=1,\ldots,n)\) be continuous in some interval \(h\subset [a,b]\) for all real values of \(u_1,\ldots,u_m\).

Theorem 3.1. Under the stated conditions, equation (3.5) admits a periodic solution with periods \(\omega_1,\ldots,\omega_m\), whatever the functions \(\Phi_s\) may be, and in this case one can specify a certain finite domain beyond whose boundary this periodic solution does not go, whatever the real values of \(u_1,\ldots,u_m\) and the values of \(\varepsilon\) in the interval \(h\) may be.

Proof. Let us first consider one scalar equation

\[ Dx=ax+f(u_1,\ldots,u_m), \tag{3.8} \]

where \(a=\delta+\mu\sqrt{-1}=\mathrm{const}\), \(f(u_1,\ldots,u_m)\) is a periodic function of period \(\omega_j\) with respect to \(u_j\), respectively. It is not difficult to see that equation (3.8) has, moreover, a unique periodic solution

\[ x(u_1,\ldots,u_m)=\exp(au_1)\int_{\infty}^{u_1} f(\tau,u_2-u_1+\tau,\ldots, \]

\[ \ldots,u_m-u_1+\tau)\exp(-a\tau)\,d\tau\quad (\delta>0), \tag{3.9} \]

\[ x(u_1,\ldots,u_m)=\exp(au_1)\times \]

\[ \times\int_{-\infty}^{u_1} f(\tau,u_2-u_1+\tau,\ldots,u_m-u_1+\tau)\exp(-a\tau)\,d\tau \quad (\delta<0). \tag{3.10} \]

Let equation (3.5) satisfy the stated conditions. It was shown above that (3.5), by means of nonsingular transformations, can be reduced to the form (3.7). Thus, a periodic solution of this system exists for all \(\varepsilon\) in the interval \(h\) for arbitrary \(\Phi_s(\varepsilon,u_1,\ldots,u_m)\) and is determined by the formulas

\[ y_s=\sum_{j=1}^{n} a_{sj}z_j\quad (s=1,\ldots,n), \]

where, according to (3.9) or (3.10),

\[ z_1=\exp(\alpha_1 u_1)\int_{u_1}^{\infty}\rho_1\sum_{j=1}^{n} b_{1j}\Phi_j(\varepsilon,\tau,u_2-u_1+\tau,\ldots, \]

\[ \ldots,u_m-u_1+\tau)\exp(-\alpha_1\tau)\,d\tau, \]

\[ z_k=\exp(\alpha_k u_1)\int_{u_1}^{\infty}\rho_k\left(A_{k-1}z_{k-1}+\sum_{j=1}^{n} b_{kj}\Phi_j\right)\times \]

\[ \times \exp(-\alpha_k\tau)\,d\tau \qquad (k=2,\ldots,n). \]

It is not difficult to see that the expressions

\[ \left|\exp(\alpha_i u_1)\int_{u_1}^{\infty}\left|\rho_i\exp(-\alpha_i\tau)\right|\,d\tau\right| \qquad (i=1,\ldots,n) \]

remain bounded for all \(\varepsilon\in h\) by certain numbers \(M_i\).

Let \(G_i,\ R_{k-1}\), and \(A_{ir}\) be quantities not less than the maximum values in \(h\) of the moduli, respectively, of the quantities

\[ \sum_{j=1}^{n} b_{ij}\Phi_j,\quad A_{k-1},\quad \alpha_{ir}, \]

and let \(a_1,a_2,\ldots,a_n\) be the quantities determined by the equalities

\[ a_1=M_1G_1,\quad a_k=M_k(A_{k-1}a_{k-1}+G_k)\qquad (k=2,\ldots,n), \]

then for all real values \(u_1,\ldots,u_m\) and values of \(\varepsilon\) in the interval \(h\) the conditions \(|z_j|\leq a_j\) hold, and consequently also the conditions

\[ |y_s|\leq \sum_{k=1}^{n} A_{sk}a_k \qquad (s=1,\ldots,n). \]

This proves

Theorem 3.2. Suppose the reduced differential equation (2.5) admits a periodic solution \(x^0(\varepsilon,u_1,\ldots,u_m)\), to which, on some interval \(g\) of variation of \(\varepsilon\), there corresponds a determining equation reducible with respect to the parameter. If, moreover, among the characteristic exponents of the determining equation there is not a single one with real part equal to zero, then within this interval there is contained some interval at all points of which equation (2.4) admits a periodic solution tending to \(x^0(0,u_1,\ldots,u_m)\) as \(\varepsilon=0\). This holds independently of the form of the terms discarded in comparison with the reduced system.

Proof. Suppose that the reduced differential equation admits some periodic solution \(x^0(\varepsilon,u_1,\ldots,u_m)\), \(x^0(x_1^0,\ldots,x_n^0)\). In what follows, speaking of this solution, we shall assume that it satisfies the condition of the problem and, in addition, the condition that for each nonzero quantity \(x_s^0(\varepsilon,u_1,\ldots,u_m)\) one can choose such a nonnegative integer \(m_s\) that the latter can be represented in the form \(\varepsilon^{m_s}x_s^*(\varepsilon,u_1,\ldots,u_m)\), where \(x_s^*(0,u_1,\ldots,u_m)\) is different from zero and finite. The quantity

\[ y_s^0=\varepsilon^{m_s}x_s^*(0,u_1,\ldots,u_m) \]

will be called the principal part of the quantity \(x_s^0(\varepsilon,u_1,\ldots,u_m)\). In the case when \(x_s^0(\varepsilon,u_1,\ldots,u_m)\equiv 0\), we shall assume that its principal part is also equal to zero.

We shall prove the existence of periodic series formally satisfying equation (2.4), and shall indicate a method for constructing such series under the assumption that the general solution of the determining equation (2.7) is known.

If we make the substitution

\[ x = y + y^0 \quad \bigl(y(y_1,\ldots,y_n),\; y^0(y_1,\ldots,y_n^0)\bigr), \]

then equation (2.4) is reduced to the form

\[ Dy=\varepsilon^\gamma(py+\varepsilon\varphi+\Phi), \tag{3.11} \]

where \(p=\|p_{sj}\|_1^n\) is the matrix of the determining equation (2.7), while \(\varphi(\varphi_1,\ldots,\varphi_n)\) and \(\Phi(\Phi_1,\ldots,\Phi_n)\) are vectors, and here the \(\varphi_s\) are constants independent of \(\varepsilon\) or periodic functions of the variables \(u_1,\ldots,u_m\), while the expansions of the quantities \(\Phi_1,\ldots,\Phi_n\) in series in integral positive powers of \(\varepsilon,y_1,\ldots,y_n\) contain no terms below the second dimension. We shall try formally to satisfy equation (3.11) by series of the form

\[ y=\varepsilon y^{(1)}+\varepsilon^2 y^{(2)}+\cdots \quad \bigl(y^{(k)}(y_1^{(k)},\ldots,y_n^{(k)})\bigr), \tag{3.12} \]

in which the quantities \(y_s^{(1)}, y_s^{(2)},\ldots\) would be periodic functions of the variables \(u_1,\ldots,u_m\), depending or not depending on \(\varepsilon\).

After the formal substitution in equation (3.11) of the quantities \(y_1,y_2,\ldots,y_n\) by the corresponding series (3.12), and after the subsequent grouping of terms with identical powers of \(\varepsilon\), we obtain

\[ \sum_{i=1}^{\infty} Dy^{(i)}\varepsilon^i = \varepsilon^\gamma \sum_{i=1}^{\infty}\bigl(py^{(i)}+\Phi^{(i)}\bigr)\varepsilon^i, \]

where \(\Phi^{(i)}(\Phi_1^{(i)},\ldots,\Phi_n^{(i)})\) and \(\Phi_s^{(i)}\) are known integral rational functions with periodic, with respect to \(u_1,\ldots,u_m\), coefficients only of those \(y_\alpha^{(1)}, y_\beta^{(2)},\ldots\) whose upper indices are smaller than the number \(i\), in particular \(\Phi_s^{(1)}=\varphi_s\).

Equation (3.11) will formally be satisfied if the coefficients of the series (3.12) are determined from the differential equations

\[ Dy^{(i)}=\varepsilon^\gamma\bigl(py^{(i)}+\Phi^{(i)}\bigr) \quad (i=1,2,\ldots). \tag{3.13} \]

Taking into account the structure of the quantities \(\Phi_s^{(i)}\), we conclude that from the differential equations (3.13) the \(y^{(1)}, y^{(2)},\ldots\) can be determined successively, and moreover, according to Theorem 3.1, as vector functions periodic with respect to \(u_1,\ldots,u_m\) and finite at \(\varepsilon=0\). It follows that, under the adopted conditions, equation (2.4) can always be formally satisfied by periodic series constructed in the indicated way,

\[ x=y^0+\varepsilon y^{(1)}+\varepsilon^2 y^{(2)}+\cdots, \]

which turn into \(x^0(0,u_1,\ldots,u_m)\) when \(\varepsilon=0\).

It remains to prove that inside the interval \(g\) there is contained an interval at all points of which the series (3.12) converge absolutely and uniformly for all real values of \(u_1,\ldots,u_m\). This, as well as some of the preceding arguments, can be done almost verbatim by repeating the exposition of [3].

If the proved Theorem 3.2 is considered under condition (1.2), then we obtain a solution of the problem posed for equation (2.1).

Theorem 3.3. Suppose that the simplified differential equation (2.2) admits a quasiperiodic solution \(x^0(\varepsilon,t)\), to which, in some interval \(g\) of variation of \(\varepsilon\), there corresponds a linear ordinary differential equation generated (under (1.2)) by a reducible, with respect to the parameter, determ-

dividing equation, for which the matrix \(C\) in relation (1.15) is everywhere constant.

If, in addition, among the characteristic exponents of the determining equation there is not a single one with real part equal to zero, then inside the interval \(g\) there is contained some interval at all points of which equation (2.1) admits a quasiperiodic solution tending to \(x^0(0,t)\) as \(\varepsilon = 0\). This holds independently of the form of the terms discarded in comparison with the simplified equation.

§ 4. CASE B)

If in equation (2.1) the functions \(f_s(t,x,\varepsilon)\) are periodic with respect to \(t\) (of common period), then such a case cannot arise at all, since then in relation (1.15) the matrix \(C\) is everywhere constant. Consequently, the case now under consideration is inherent only in functions \(f_s\) that are quasiperiodic with respect to \(t\).

Let \(y=\gamma(u_1,\ldots,u_m)z\), where \(\gamma(u_1,\ldots,u_m)\) is a periodic matrix with periods \(\omega_k\) in the variables \(u_k\), be a transformation reducing equation (2.7) to the form

\[ Dz=\Lambda z,\quad \Lambda = \begin{Vmatrix} \lambda_1 & 0 & 0 & \cdots & 0\\ r_1 & \lambda_2 & 0 & \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \cdots & \lambda_n \end{Vmatrix}, \]

where \(\lambda_i(u_1,\ldots,u_m)\) and \(r_i(u_1,\ldots,u_m)\) are periodic functions with periods \(\omega_k\) in the variables \(u_k\), respectively. In [6] it is shown that, if condition B) is satisfied, then such a transformation is always possible, and it admits an inverse transformation.

If equation (2.7) is considered under condition (1.2), then we obtain the equation

\[ \frac{dy}{dt}=\varepsilon^\nu Q(t)y, \tag{4.1} \]

where \(Q(t)=p(t,\ldots,t)\) is a quasiperiodic matrix with frequency basis \(\beta\{\beta_1,\ldots,\beta_m\}\).

Thus there always exists a transformation \(y=\alpha(t)z\) with a quasiperiodic matrix \(\alpha(t)\) of frequency basis \(\beta\), which reduces equation (4.1) to the form

\[ \frac{dz}{dt}=\eta z,\quad \eta = \begin{Vmatrix} \alpha_1 & 0 & 0 & \cdots & 0\\ \sigma_1 & \alpha_2 & 0 & \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot\\ 0 & 0 & 0 & \cdots & \alpha_n \end{Vmatrix}, \tag{4.2} \]

where \(\alpha_j\) \((j=1,\ldots,n)\) and \(\sigma_{j-1}\) are quasiperiodic functions of frequency basis \(\beta\).

We shall call equation (4.1) the determining equation.

Consider the equation

\[ \frac{dy}{dt}=\varepsilon^\nu\bigl(Q(t)y+\Phi(\varepsilon,t)\bigr), \tag{4.3} \]

where \(\varepsilon^\gamma Q(t)\) is from the defining equation (4.1), and \(\Phi(\Phi_1,\ldots,\Phi_n)\) is a vector in which the \(\Phi_s(\varepsilon,t)\) are quasiperiodic functions with respect to \(t\) with frequency basis \(\beta\). Using the indicated transformation, we bring equation (4.3) to the following form (written, for convenience, in scalar form):

\[ \frac{dz_1}{dt}=\alpha_1 z_1+\sum_{j=1}^{n}\beta_{1j}\Phi_j\varepsilon^\gamma,\qquad \frac{dz_k}{dt}=\alpha_k z_k+\sigma_{k-1}z_{k-1}+\sum_{j=1}^{n}\beta_{kj}\Phi_j\varepsilon^\gamma, \]

where \(\beta_{sk}(t)\) are the elements of the matrix \(a^{-1}(t)\).

Assuming henceforth that the functions \(\alpha_j\) are strictly different from zero and putting

\[ \rho_j=|\alpha_j|,\qquad b_{sj}=\frac{1}{\rho_s}\,\beta_{sj}\varepsilon^\gamma,\qquad A_{j-1}=\frac{\sigma_{j-1}}{\rho_j}, \]

we bring equation (4.3) to the form

\[ \frac{dz_1}{dt}=\alpha_1 z_1+\rho_1\sum_{j=1}^{n} b_{1j}\Phi_j, \]

\[ \frac{dz_s}{dt}=\alpha_s z_s+\rho_s\left(A_{s-1}z_{s-1}+\sum_{j=1}^{n} b_{sj}\Phi_j\right) \qquad (s=2,\ldots,n). \tag{4.4} \]

Again we shall say that equation (4.1) is reducible with respect to the parameter if all \(b_{sj}\) and \(A_{j-1}\) are continuous for all real values of \(t\) and all \(\varepsilon\) in some interval.

We shall call the functions \(\alpha_1,\ldots,\alpha_n\) the characteristic functions of equation (4.1).

Suppose that in some interval \(a\leq \varepsilon\leq b\) the defining equation (4.1) is reducible with respect to the parameter and that, in addition, the functions \(\Phi_s(\varepsilon,t)\) \((s=1,\ldots,n)\) are continuous in some interval \(h\subset [a,b]\) for all real values of \(t\).

Theorem 4.1. Under the stated conditions, equation (4.3) admits a quasiperiodic solution with frequency basis \(\beta\), whatever the functions \(\Phi_s\) may be; moreover, one can indicate a certain finite domain beyond whose boundary this quasiperiodic solution does not go, whatever the real values of \(t\) and the values of \(\varepsilon\) in the interval \(h\).

If one scalar equation is given,

\[ \frac{dx}{dt}=\alpha x+f(t), \tag{4.5} \]

where \(\alpha\) and \(f\) are quasiperiodic functions, then

\[ x(t)=\left(\exp\int_{0}^{t} a\,d\tau\right) \int_{t}^{\infty} f(\tau)\exp\left(-\int_{0}^{\tau} a\,dv\right)\,d\tau \qquad (a>\delta>0), \]

\[ x(t)=\left(\exp\int_{0}^{t} a\,d\tau\right) \int_{-\infty}^{t} f(\tau)\exp\left(-\int_{0}^{\tau} a\,dv\right)\,d\tau \qquad (a>-\delta) \]

will be, moreover, the unique quasiperiodic solution of equation (4.5).

Thus the proof of Theorem 4.1 is carried out in the same way as the proof of Theorem 3.1.

Consequently, by the same device one can obtain

Theorem 4.2. Suppose that the simplified differential equation (2.2) admits a quasiperiodic solution \(x^0(\varepsilon,t)\), to which, on some interval \(g\), as \(\varepsilon\) varies, there corresponds a reducible linear ordinary differential equation, generated (under (1.2)) by an equation in partial derivatives, for which the matrix \(C\) in relation (1.15) satisfies condition B. Then inside the interval \(g\) there is contained a certain interval at all points of which equation (2.1) admits a quasiperiodic solution which becomes \(x^0(0,t)\) when \(\varepsilon=0\). This holds independently of the form of the terms discarded in comparison with the simplified equation.

In conclusion, the authors express their gratitude to Professor Ya. V. Bykov for a number of suggestions made in the course of this work.

References

1. Kharasakhal V. Kh. Doctoral dissertation, Kiev, 1964.
2. Kharasakhal V. Kh. PMM, XXVII, no. 4, 672, 1963.
3. Volk I. M. PMM, X, nos. 5–6, 561, 1946.
4. Malkin I. G. Some Problems in the Theory of Nonlinear Oscillations. Gostekhizdat, 1956.
5. Erugin N. P. Reducible systems. Proceedings of the V. A. Steklov Mathematical Institute, XIII, 1946, pp. 5–18.
6. Zaitsev A. I. Differential Equations, 3, no. 2, 219–225, 1967.

Received by the editors
March 15, 1966

Kazakh State
University