ON THE REDUCTION OF AN ANALYTIC SYSTEM OF DIFFERENTIAL EQUATIONS TO LINEAR FORM

V. A. PLISS

Consider a system of differential equations of the form

\[ \frac{dx_i}{dt}=\lambda_i x_i+f_i(x_1,\ldots,x_n)\quad (i=1,\ldots,n), \tag{0.1} \]

where the functions \(f_i\) are power series in \(x_1,\ldots,x_n\), beginning with terms of order not lower than the second and converging in a sufficiently small neighborhood of the origin; \(\lambda_i\) are arbitrary complex numbers.

We shall study the question of the possibility of reducing system (0.1) to the form

\[ \frac{dy_i}{dt}=\lambda_i y_i \quad (i=1,\ldots,n) \tag{0.2} \]

by means of an analytic transformation

\[ x_i=y_i+\psi_i(y_1,\ldots,y_n)\quad (i=1,\ldots,n), \tag{0.3} \]

where the \(\psi_i\) are power series in \(y_1,\ldots,y_n\), beginning with terms of order not lower than the second and converging for sufficiently small \(|y_s|\).

Very interesting results in this direction were obtained comparatively recently by Siegel [1, 2]. Siegel assumed that for all nonnegative integers \(k_j\) for which

\[ \sum_{j=1}^{n} k_j>1, \]

the inequalities

\[ \sum_{j=1}^{n} k_j\lambda_j-\lambda_i\ne 0\quad (i=1,\ldots,n) \tag{0.4} \]

hold.

We shall not retain this assumption and shall consider the case when the inequalities (0.4) do not hold for all \(k_j\) for which \(k_1+k_2+\ldots+k_n>1\).

In this case formal series of the form (0.3) reducing system (0.1) to the form (0.2) by no means always exist.

In what follows we shall assume that the functions \(f_i(x_1,\ldots,x_n)\) are such that formal series (0.3) can be constructed, and we shall be interested in questions of convergence of such series.

1. The expansion of the functions \(f_i(x_1,\ldots,x_n)\) into series has the form

\[ f_i=\sum_{k_1+\ldots+k_n=m}^{\infty} M_i^{(k_1,\ldots,k_n)}x_1^{k_1}\ldots x_n^{k_n}, \tag{1.1} \]

where \(k_1,\ldots,k_n\) are nonnegative integers, and \(m\geqslant 2\) is the least order of the terms of the expansions of the functions \(f_i\) into the series (1.1). Let \(\varphi_i(y_1,\ldots,y_n)\) be functions of the form

\[ \varphi_i=\sum_{k_1+\cdots+k_n=m}^{2m-2} L_i^{(k_1,\ldots,k_n)}y_1^{k_1}\cdots y_n^{k_n} \quad (i=1,\ldots,n), \tag{1.2} \]

satisfying the equations

\[ \sum_{j=1}^{n}\lambda_j y_j\frac{\partial\varphi_i}{\partial y_j} -\lambda_i\varphi_i = \sum_{k_1+\cdots+k_n=m}^{2m-2} M_i^{(k_1,\ldots,k_n)}y_1^{k_1}\cdots y_n^{k_n}. \tag{1.3} \]

Then the change of variables

\[ x_i=y_i+\varphi_i(y_1,\ldots,y_n)\quad (i=1,\ldots,n) \tag{1.4} \]

will bring system (0.1) to the form

\[ \frac{dy_i}{dt}=\lambda_i y_i+g_i(y_1,\ldots,y_n)\quad (i=1,\ldots,n). \tag{1.5} \]

Let us study the character of the functions \(g_i\). Performing in system (0.1) the substitution (1.4), we obtain

\[ \lambda_i y_i+g_i+\sum_{j=1}^{n}\frac{\partial\varphi_i}{\partial y_j} (\lambda_j y_j+g_j) = \lambda_i y_i+\lambda_i\varphi_i+ \]

\[ +f_i(y_1+\varphi_1,\ldots,y_n+\varphi_n) \quad (i=1,\ldots,n), \]

whence, in view of (1.3), we find

\[ g_i+\sum_{j=1}^{n}\frac{\partial\varphi_i}{\partial y_j}g_j = f_i(y_1+\varphi_1,\ldots,y_n+\varphi_n)- \]

\[ - \sum_{k_1+\cdots+k_n=m}^{2m-2} M_i^{(k_1,\ldots,k_n)}y_1^{k_1}\cdots y_n^{k_n} \quad (i=1,\ldots,n). \tag{1.6} \]

Since the series representing the functions \(\varphi_i\) and \(f_i\) begin with terms of order not lower than \(m\), it is clear that the right-hand sides of equations (1.6) are series in powers of \(y_1,\ldots,y_n\), beginning with terms of dimension not lower than \(2m-1\).

It then follows from equations (1.6) that the functions \(g_i\) are series in powers of \(y_1,\ldots,y_n\), beginning with terms of dimension not lower than \(2m-1\).

1. The functions \(\psi_i(y_1,\ldots,y_n)\), by means of which system (0.1) is brought to the form (0.2), will be constructed by the method of successive approximations*.

Put \(y_i=x_{i1}\), \(g_i(x_{11},\ldots,x_{n1})=f_{i1}(x_{11},\ldots,x_{n1})\), \(\varphi_i(x_{11},\ldots,x_{n1})=\varphi_{i1}(x_{11},\ldots,x_{n1})\).

* We note that the method of successive approximations, used below (Newton’s method), was applied to the study of related questions in the works of A. N. Kolmogorov [3, 4], J. Moser [5], and V. I. Arnold [6—8].

Define the \(s\)-th step of the process of successive approximations. Suppose that at the \((s-1)\)-st step we have obtained the system

\[ \frac{d x_{is-1}}{dt}=\lambda_i x_{is-1}+f_{is-1}(x_{1s-1},\ldots,x_{ns-1}), \tag{2.1} \]

where

\[ f_{is-1}= \sum_{k_1+\cdots+k_n=m_{s-1}}^{\infty} M_{is-1}^{(k_1,\ldots,k_n)} x_{1s-1}^{k_1}\cdots x_{ns-1}^{k_n}. \tag{2.2} \]

Let \(\varphi_{is}(\xi_1,\ldots,\xi_n)\) be functions of the form

\[ \varphi_{is}= \sum_{k_1+\cdots+k_n=m_{s-1}}^{2m_{s-1}-2} L_{is}^{(k_1,\ldots,k_n)} \xi_1^{k_1}\cdots \xi_n^{k_n}, \tag{2.3} \]

satisfying the system of equations:

\[ \sum_{j=1}^{n}\lambda_j \xi_j \frac{\partial \varphi_{is}}{\partial \xi_j} -\lambda_i \varphi_{is} = \sum_{k_1+\cdots+k_n=m_{s-1}}^{2m_{s-1}-2} M_{is-1}^{(k_1,\ldots,k_n)} \xi_1^{k_1}\cdots \xi_n^{k_n}. \tag{2.4} \]

The change of variables

\[ x_{is-1}=\varphi_{is}(x_{1s},\ldots,x_{ns})\quad (i=1,\ldots,n) \tag{2.5} \]

will bring system (2.1) to the form

\[ \frac{d x_{is}}{dt}=\lambda_i x_{is}+f_{is}(x_{1s},\ldots,x_{ns}), \tag{2.6} \]

where the functions \(f_{is}\) are found from the equations:

\[ f_{is}+\sum_{j=1}^{n} f_{js}\frac{\partial \varphi_{is}}{\partial x_{js}} = f_{is-1}(x_{1s}+\varphi_{1s},\ldots,x_{ns}+\varphi_{ns}) - \]

\[ - \sum_{k_1+\cdots+k_n=m_{s-1}}^{2m_s-2} M_{is-1}^{(k_1,\ldots,k_n)} x_{1s}^{k_1}\cdots x_{ns}^{k_n}. \tag{2.7} \]

It is not difficult to see that the functions \(f_{is}\) are series in powers of the variables \(x_{1s},\ldots,x_{ns}\), beginning with terms of order not lower than \(m_s=2m_{s-1}-1\).

Our basic assumption concerning the right-hand sides of system (0.1) is that at each step there exist functions (2.3) which constitute a solution of system (2.4).

Let us analyze this assumption. The coefficients \(L_{is}^{(k_1,\ldots,k_n)}\) of the polynomials (2.3) are found, as is not difficult to see, from the equations:

\[ \left(\sum_{j=1}^{n} k_j\lambda_j-\lambda_i\right) L_{is}^{(k_1,\ldots,k_n)} = M_{is}^{(k_1,\ldots,k_n)}. \tag{2.8} \]

If

\[ \sum_{j=1}^{n} k_j\lambda_j-\lambda_i \ne 0, \]

then the coefficient \(L_{is}^{(k_1,\ldots,k_n)}\) is determined by the equa-

by (2.8). If, however,

\[ \sum_{j=1}^{n} k_j \lambda_j - \lambda_i = 0, \]

then we assume that \(M_{is}^{(k_1,\ldots,k_n)}=0\), and for such \(i, s, k_1,\ldots,k_n\), \(L_{is}^{(k_1,\ldots,k_n)}\) may be chosen arbitrarily. We shall suppose that in these cases \(L_{is}^{(k_1,\ldots,k_n)}=0\).

1. We shall now study the question of the convergence of the chosen process of successive approximations. To this end we make the assumption, usual in such cases, concerning the numbers \(\lambda_i\). We shall suppose that for all nonnegative integers \(k_1,\ldots,k_n\) for which the sum \(k=k_1+\cdots+k_n\) is sufficiently large, either

\[ \sum_{j=1}^{n} k_j \lambda_j - \lambda_i = 0, \tag{3.1} \]

or

\[ \left|\sum_{j=1}^{n} k_j \lambda_j - \lambda_i\right| > \frac{1}{k^\nu}, \tag{3.2} \]

where \(\nu \geq 2\).

Under this assumption one can carry out a finite number of steps in the process of successive approximations; therefore, without loss of generality, we may suppose that in system (0.1) the functions \(f_i\) are series whose lowest order of terms \(m\) is sufficiently high. In addition, we shall assume that the series representing \(f_i\) converge for \(|x_j|\leq 1\) and that for such \(x_j\), \(|f_i|<1\) \((j=1,\ldots,n)\). This assumption does not impair generality, since it can be achieved by a change of the form \(x_i=\alpha \xi_i\) with sufficiently small \(\alpha\).

Let us estimate the functions \(\varphi_{is}\) and \(f_{is}\) at each step. Since the left-hand side of equation (2.4) does not depend on the number of the step \(s\), we may confine ourselves to studying the first step, which brings system (0.1) to the form (1.5) by means of the substitution (1.4). We shall assume here that the series (1.1) converge for \(|x_i|\leq R\); for such \(x_j\), \(|f_i|<1\) \((i=1,\ldots,n)\), and

\[ \frac{1}{2}<R\leq 1. \]

The coefficients of the series (1.1) are estimated, as is well known, in the following way:

\[ \left|M_i^{(k_1,\ldots,k_n)}\right| < \frac{1}{R^k} \quad (i=1,\ldots,n,\; k=k_1+\cdots+k_n). \tag{3.3} \]

Since the number of coefficients of a form of degree \(k\) in \(n\) variables is less than \(k^n\), from the last inequality we obtain

\[ S_k(f_i)= \sum_{k_1+\cdots+k_n=m} \left|M_i^{(k_1,\ldots,k_n)}\right| < \frac{k^n}{R^n}. \tag{3.4} \]

The coefficients \(L_i^{(k_1,\ldots,k_n)}\) of the polynomials \(\varphi_i\) are determined from equations of the form (2.8). Moreover, if

\[ \sum_{j=1}^{n} \lambda_j k_j - \lambda_i = 0, \]

then, by virtue of our assumption, \(M_i^{(k_1,\ldots,k_n)}=0\), and we take \(L_i^{(k_1,\ldots,k_n)}=0\); but if

\[ \sum_{j=1}^{n} \lambda_j k_j - \lambda_i \ne 0, \]

then

\[ L_i^{(k_1,\ldots,k_n)} = -\frac{M_i^{(k_1,\ldots,k_n)}}{\displaystyle \sum_{j=1}^{n}\lambda_j k_j-\lambda_i}. \tag{3.5} \]

Hence, by virtue of assumption (3.2) and the estimates (3.3), we obtain

\[ \left|L_i^{(k_1,\ldots,k_n)}\right|<\frac{k^\nu}{R^k}, \tag{3.6} \]

therefore,

\[ S_k(\varphi_i)<\frac{k^{n+\nu}}{R^k}, \tag{3.7} \]

where the symbol \(S_k(\varphi_i)\) has the same meaning as in (3.4). From the last inequality we obtain

\[ S_k\left(\frac{\partial \varphi_i}{\partial y_j}\right)<\frac{k^{n+\nu+1}}{R^k}. \tag{3.8} \]

Introduce the notation:

\[ \varkappa=n+\nu+1, \]

\[ r=m^{-\frac{\varkappa}{m}}R,\qquad \rho=m^{-\frac{2\varkappa}{m}}R=m^{-\frac{\varkappa}{m}}r . \tag{3.9} \]

We shall show that, for \(k\geqslant m\geqslant 3\), the inequality

\[ \frac{k^\varkappa}{R^k}\leqslant \frac{1}{r^k}. \tag{3.10} \]

holds. To this end we prove that

\[ k^\varkappa\leqslant m^{\frac{k\varkappa}{m}} \tag{3.11} \]

or, what is the same,

\[ \ln k\leqslant \frac{k}{m}\ln m. \]

For \(k=m\) this inequality becomes an equality; differentiating it with respect to \(k\), we then obtain

\[ \frac{m}{k}\leqslant \ln m. \]

Since \(m\geqslant 3\), and \(k\geqslant m\), the last inequality is valid, and hence (3.11) is also valid. Dividing (3.11) by \(R^k\) and taking (3.9) into account, we obtain (3.10).

From the inequalities (3.7), (3.8), and (3.10) we obtain

\[ S_k(\varphi_i)<\frac{1}{r^k},\qquad S_k\left(\frac{\partial \varphi_i}{\partial y_j}\right)<\frac{1}{r^k}. \tag{3.12} \]

It follows from these inequalities that, for \(|y_i|\leqslant \rho\),

\[ |\varphi_i|\leqslant \sum_{k=m}^{\infty}\frac{\rho^k}{r^k} =\frac{\rho^m}{r^m}\,\frac{1}{1-\frac{\rho}{r}} =\frac{m^{-\varkappa}}{1-m^{-\frac{\varkappa}{m}}} \tag{3.13} \]

and

\[ \left|\frac{\partial \varphi_i}{\partial y_j}\right| < \frac{m^{-\varkappa}}{1-m^{-\frac{\varkappa}{m}}}. \tag{3.14} \]

We shall show that, for \(|y_i|\leqslant \rho\) and sufficiently large \(m\), the inequality

\[ |y_i+\varphi_i(y_1,\ldots,y_n)|\leqslant r. \tag{3.15} \]

holds.

In view of (3.13), for this it is enough to prove that, for sufficiently large \(m\),

\[ \rho+\frac{m^{-\chi}}{1-m^{-\frac{\chi}{m}}}<r \tag{3.16} \]

or

\[ \frac{m^{-\chi}}{1-m^{-\frac{\chi}{m}}}<r\left(1-m^{-\frac{\chi}{m}}\right); \]

this inequality is equivalent to the inequality

\[ \frac{m^{-\frac{\chi}{2}}}{1-m^{-\frac{\chi}{m}}}<r^{\frac12}. \tag{3.17} \]

It is not hard to see that

\[ \frac{m^{-\frac{\chi}{2}}}{1-m^{-\frac{\chi}{m}}} < \frac{m^{-\frac{\chi}{2}}}{1-e^{-\frac{\chi}{m}}}. \tag{3.18} \]

By l’Hôpital’s rule we have

\[ \lim_{m\to\infty}\frac{m^{-\frac{\chi}{2}}}{1-e^{-\frac{\chi}{m}}} = \lim_{m\to\infty}\frac{\chi m^{-\frac{\chi}{2}+1}}{2e^{-\frac{\chi}{m}}}. \]

Since \(\chi>2\), it follows from this and from (3.18) that

\[ \lim_{m\to\infty}\frac{m^{-\frac{\chi}{2}}}{1-m^{-\frac{\chi}{m}}}=0. \tag{3.19} \]

Consequently, for sufficiently large \(m\) we have

\[ \frac{m^{-\frac{\chi}{2}}}{1-m^{-\frac{\chi}{m}}}<\frac12, \tag{3.20} \]

and since \(\lim_{m\to\infty} m^{-\frac{\chi}{m}}=1\), for such \(m\) the inequality
\(m^{-\frac{\chi}{m}}>\frac12\) will hold and, hence, \(r^{1/2}>\frac12\). From (3.17) and (3.20) we obtain (3.16).

We now estimate \(f_i(y_1+\varphi_1,\ldots,y_n+\varphi_n)\) for \(|y_j|\le \rho\). From (3.4) and (3.15) we derive

\[ |f_i|\le \sum_{k=m}^{\infty} k^n\left(\frac{r}{R}\right)^k = \sum_{k=m}^{\infty} k^n m^{-\frac{nk}{m}} m^{-\frac{\nu+1}{m}k}. \]

Since \(k^n m^{-\frac{nk}{m}}\le 1\), the last inequality gives

\[ |f_i|\le \frac{m^{-(\nu+1)}}{1-m^{-\frac{\nu+1}{m}}}. \tag{3.21} \]

Since \(\nu \geqslant 2\), it is not difficult to see that

\[ \lim_{m\to\infty} \frac{m^{-(\nu+1)}}{1-m^{-\frac{\nu+1}{m}}}=0. \]

Hence, from (3.21), we obtain the inequalities

\[ |f_i|<\frac{1}{4}\quad (i=1,\ldots,n), \tag{3.22} \]

valid for \(|y_i|\leqslant \rho\) and sufficiently large natural \(m\).

Consider the linear algebraic system

\[ \alpha_i+\sum_{j=1}^{n} q_{ij}\alpha_j=\beta_i\quad (i=1,\ldots,n), \tag{3.23} \]

where \(|\beta_j|\leqslant \frac{1}{2}\). Then it is clear that there exists an \(\eta>0\) such that, for \(|q_{ij}|<\eta\), the solution of the system (3.23) satisfies the inequalities \(|\alpha_i|<1\).

From the inequalities (3.13), (3.14) and the relation (3.19) it follows that, for \(|y_j|\leqslant \rho\) and sufficiently large \(m\), the inequalities

\[ \left|\frac{\partial \varphi_i}{\partial y_j}\right|<\eta \tag{3.24} \]

and

\[ \left|\lambda_i\varphi_i-\sum_{j=1}^{n}\frac{\partial\varphi_i}{\partial y_j}\lambda_j y_j\right|<\frac{1}{4}. \tag{3.25} \]

hold.

Hence, from equations (1.3) and (1.6), we obtain the estimate

\[ |g_i(y_1,\ldots,y_n)|<1, \tag{3.26} \]

valid for \(|y_j|\leqslant \rho\) and sufficiently large \(m\).

Thus, the system (0.1), by means of the change of variables (1.4), is reduced to the form (1.5). The functions \(g_i(y_1,\ldots,y_n)\) are expanded in power series in \(y_1,\ldots,y_n\) beginning with terms of dimension not lower than \((2m-1)\), converging for \(|y_j|\leqslant \rho\), and for such \(y_j\) these functions satisfy the estimates (3.26). The functions \(\varphi_i\), for \(|y_j|\leqslant \rho\), as follows from (3.13) and (3.19), satisfy the inequalities:

\[ |\varphi_i(y_1,\ldots,y_n)|\leqslant m^{-\frac{\chi}{2}}. \tag{3.27} \]

4. Let us now prove that, under the assumptions made, the process of successive approximations described above converges.

First define the quantity \(m_s\)—the lowest order of the terms in the expansion of the functions \(f_{is}\) into series. As was shown, the recurrence relation \(m_s=2m_{s-1}-1\) holds. We shall prove that

\[ m_s=(m-1)2^s+1. \tag{4.1} \]

Suppose that this relation is valid, and find \(m_{s+1}\):

\[ m_{s+1}=2m_s-1=(m-1)2^{s+1}+1; \tag{4.2} \]

and since \(m_0=(m-1)+1=m\), by the principle of mathematical induction the last equality proves (4.1).

Suppose now that the series representing the functions
\(f_{is-1}(x_{1s-1}, \ldots, x_{ns-1})\) converge for \(|x_{js-1}| \leq \rho_{s-1}\), and that for such \(x_{js-1}\) these functions satisfy the estimates

\[ |f_{is-1}(x_{1s-1}, \ldots, x_{ns-1})| < 1; \tag{4.3} \]

then, as follows from the reasoning of the preceding subsection, the series representing the functions \(f_{is}(x_{1s}, \ldots, x_{ns})\) converge for \(|x_{js}| \leq \rho_s\), where

\[ \rho_s = m_{s-1}^{-\frac{2\chi}{m_{s-1}}}\rho_{s-1}, \tag{4.4} \]

and if \(\rho_{s-1} > \dfrac{1}{2}\), then for \(|x_{js}| \leq \rho_s\)

\[ |f_{is}(x_{1s}, \ldots, x_{ns})| < 1. \tag{4.5} \]

From our assumption concerning the functions \(f_i(x_1, \ldots, x_n)\) and from the recurrence relation (4.4) it follows that

\[ \rho_s = \prod_{k=0}^{s-1} m_k^{-\frac{2\chi}{m_k}} . \tag{4.6} \]

Hence it follows that

\[ \frac{1}{\rho_s} < \prod_{k=0}^{\infty} m_k^{\frac{2\chi}{m_k}} . \tag{4.7} \]

The infinite product on the right-hand side of this inequality converges. Indeed, the series

\[ \sum_{k=0}^{\infty} \frac{2\chi}{m_k}\ln m_k \]

converges, since

\[ \lim_{k\to\infty}\frac{\ln m_k\, m_k^{-1}}{m_k^{-1}\ln m_{k-1}}=\frac{1}{2}, \]

as follows from (4.1).

From the convergence of the product (4.7) it follows that, if \(m\) is chosen sufficiently large, then for all natural \(s\) the inequality \(\rho_s > \dfrac{1}{2}\) will hold. Therefore, for \(|\xi_j| \leq \dfrac{1}{2}\), the inequalities

\[ |f_{is}(\xi_1, \ldots, \xi_n)| < 1 \tag{4.8} \]

are satisfied.

Moreover, from inequality (3.27), proved at the end of the preceding subsection, the inequalities

\[ |\varphi_{is}(\xi_1, \ldots, \xi_n)| \leq m_s^{-\frac{\chi}{2}}, \tag{4.9} \]

follow for \(|\xi_j| \leq \dfrac{1}{2}\).

From equality (4.1) it follows that the series

\[ \sum_{k=0}^{\infty} m_k^{-\frac{\chi}{2}} \]

converges, and then, uniformly for \(|\xi_j| \leq \dfrac{1}{2}\), the series

\[ \sum_{k=0}^{\infty} \varphi_{ik}(\xi_1, \ldots, \xi_n) \]

also converge.

This latter circumstance proves that, under the assumptions made, the chosen process of successive approximations converges and that there exist analytic functions \(\psi_i(y_1,\ldots,y_n)\) such that the substitution (0.3) brings the system (0.1) to the form (0.2).

References

1. Siegel, C. L. Lectures on Celestial Mechanics. IL, Moscow, 1959.
2. Siegel, C. L. Collection of translations “Mathematics,” 5, no. 2, 1961.
3. Kolmogorov, A. N. DAN SSSR, 98, no. 4, 1954.
4. Kolmogorov, A. N. International Mathematical Congress in Amsterdam. Fizmatgiz, Moscow, 1961.
5. Moser, J. Collection of translations “Mathematics,” 6, no. 5, 1962.
6. Arnold, V. I. Izv. AN SSSR, Ser. Mat., 25, no. 1, 1961.
7. Arnold, V. I. UMN, 18, no. 5, 1963.
8. Arnold, V. I. UMN, 18, no. 6, 1963.

Received by the editors
December 1, 1964

Leningrad State
University