MATHEMATICS

A. D. BRYUNO

ASYMPTOTICS OF SOLUTIONS OF NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

(Presented by Academician I. G. Petrovsky, November 20, 1961)

The paper studies solutions of the system

\[ \frac{dx_1}{f_1(x_1,x_2,\ldots,x_n)} = \frac{dx_2}{f_2(x_1,x_2,\ldots,x_n)} = \cdots = \frac{dx_n}{f_n(x_1,x_2,\ldots,x_n)} \tag{1} \]

as functions \(X(\tau)=(x_1(\tau),\ldots,x_n(\tau))\) of a quite arbitrary parameter \(\tau\), approaching without bound a definite point \(X^0=(x_1^0,\ldots,x_n^0)\) with finite and infinite coordinates. Without loss of generality one may assume that all \(x_i^0\) are either zeros or infinities. A method is found that makes it possible to obtain asymptotics of such solutions of system (1), which, roughly speaking, enter \(X^0\) with a definite tangent direction.

1. Definition 1. A function \(\varphi(\tau)\), defined for large \(\tau\), has order

\[ p(\varphi(\tau))=\lim_{\tau\to\infty}\frac{\ln|\varphi(\tau)|}{\ln\tau}. \]

Definition 2. The functions \(\varphi(\tau)\) and \(\psi(\tau)\) are equivalent with respect to order \(p_0\), denoted \(\varphi_{p_0}\sim\psi\), if: 1) \(p(\varphi), p(\psi)\leqslant p_0\), 2) \(p(\varphi-\psi)<p_0\). The relation of equivalence with respect to order \(p_0\) possesses all the properties of equivalence and divides the set of all functions whose orders do not exceed \(p_0\) into equivalence classes.

Definition 3. The order of a vector-function \(\Phi(\tau)=(\varphi_1(\tau),\ldots,\varphi_n(\tau))\) is the vector \(P(\Phi)=(p(\varphi_1),\ldots,p(\varphi_n))=(p_1,\ldots,p_n)\).

2. Let

\[ f(X)=\sum_Q a_Q x_1^{q_1}x_2^{q_2}\cdots x_n^{q_n} =\sum_Q a_Q X^Q \]

be an absolutely convergent series in a neighborhood of the point \(X^0\), where \(Q=(q_1,q_2,\ldots,q_n)\) is a vector power of \(X\); the \(q_i\) are arbitrary real numbers. Upon substituting \(X=\Phi(\tau)\) into \(f(X)\) \((P(\Phi)=P)\), the different terms \(a_Q x_1^{q_1}\cdots x_n^{q_n}\) will have different orders with respect to \(\tau\).

If in

\[ f_i(X)=\sum_{Q'} a_{iQ'}x_1^{q'_1}\cdots x_n^{q'_n} =\sum_{Q'} a_{iQ'}X^{Q'}\quad (i=1,\ldots,n) \]

one retains only those terms \(a_{iQ'}x_1^{q'_1}\cdots x_n^{q'_n}\) for which \(a_{iQ'}\varphi_1^{q'_1}(\tau)\cdots\varphi_n^{q'_n}(\tau)/\varphi_i'(\tau)\) has the greatest order over all \(i\) and \(Q'\), then we obtain the truncation of system (1) with respect to order \(P\):

\[ \frac{dx_1}{\widetilde f_1(X)} = \frac{dx_2}{\widetilde f_2(X)} = \cdots = \frac{dx_n}{\widetilde f_n(X)}. \tag{2} \]

Here it is not excluded that \(\widetilde f_i\equiv 0\).

Let \(\mathfrak G_0\) be a set of functions that are representatives of certain equivalence classes with respect to order \(0\), possessing the properties:

1) If \(\varphi(\tau)\in\mathfrak G_0\), then \(\tau\varphi'(\tau)\in\mathfrak G_0\).

2) If \(\varphi,\psi\in\mathfrak G_0\), then \(\varphi+\psi\in\mathfrak G_0\).

3) If \(\varphi,\psi\in \mathcal G_0\), then \(\varphi\cdot\psi\in \mathcal G_0\) and \(\varphi\cdot\operatorname{const}\in \mathcal G_0\).

4) If \(\varphi\in \mathcal G_0\), then \(|\varphi|^\mu\in \mathcal G_0\), where \(\mu\) is any real number. We denote \(\mathcal G_\mu=\tau^\mu\mathcal G_0\).

All the properties of \(\mathcal G_0\) are possessed by the set of functions formed from \(\ln\tau\) and constants by a finite number of multiplications, additions, exponentiations, and logarithmations.

Theorem 1. Let \(\Phi(\tau)\) be a solution of system (1), \((P(\Phi)=P)\), such that
\[ \varphi_i \underset{p_i}{\sim} \psi_i,\qquad \varphi_i \underset{p_i-1}{\sim} \psi_i \quad (i=1,2,\ldots,n), \]
where \(\psi_i\in \mathcal G_{p_i}\). Then \(\Psi(\tau)\) is a solution of the truncation (2) of system (1) with respect to the order \(P\), if all \(\tilde f_i(X)\) are finite sums.

1. Theorem 1 reduces the study of solutions of the full system (1) to the study of solutions of the simpler truncated system (2). We indicate a geometric method for constructing truncated systems. Put
   \[ g_i(X)=\frac{f_i(X)}{x_i}\qquad (i=1,2,\ldots,n). \]
   System (1) will then be written as
   \[ \frac{d\ln|x_1|}{g_1(X)}=\frac{d\ln|x_2|}{g_2(X)}=\cdots=\frac{d\ln|x_n|}{g_n(X)}, \tag{1'} \]
   where
   \[ g_i(X)=\sum_Q a_{iQ}x_1^{q_1}\cdots x_n^{q_n}\equiv \sum_Q a_{iQ}QX \qquad (i=1,2,\ldots,n). \]
   Let
   \[ G(X)=(g_1(X),g_2(X),\ldots,g_n(X)) \]
   and
   \[ A_Q=(a_{1Q},\ldots,a_{nQ}). \]
   Then
   \[ G(X)=\sum_Q A_Q x_1^{q_1}\cdots x_n^{q_n}\equiv \sum_Q A_Q QX. \tag{3} \]

To each vector \(Q\) occurring in (3) we associate a point of the \(n\)-dimensional space \(\overline R\), whose \(i\)-th coordinate is \(q_i\) \((i=1,\ldots,n)\). The set of all such points will be called the diagram of system (1) and denoted by \(D(G)\). Under the substitution \(X=\Phi(\tau)\), \(P(\Phi)=P\), the term \(A_{Q_1}Q_1X\) has order
\[ \sum_{i=1}^n p_iq_i^1=(P,Q_1). \]
The terms of the sum (3) which, under this substitution, have order greater than \((P,Q_1)\) will correspond to points of the diagram lying in the positive half-space of the hyperplane \((P,Q)=(P,Q_1)=\operatorname{const}\) (here \(Q\) is the variable point); terms having smaller order correspond to points in the negative half-space of this hyperplane; terms having the same order correspond to points on the hyperplane. A hyperplane \((P,Q)=\operatorname{const}\) on which there are points of \(D(G)\), but in whose positive half-space there are none, will be called a plane of greatest order of the diagram \(D(G)\) with respect to the vector-order \(P\). The intersection of the nonpositive half-spaces for the planes of greatest order for different \(P\) will be called the polyhedron \(M(G)\) of system (1). If \(x_i^0=0\), then \(q_i\) must be bounded below; if \(x_j^0=\infty\), then \(q_j\) must be bounded above. Then for \(p_i\le 0\) and \(p_j\ge 0\) one can construct the polyhedron. The intersection of the polyhedron with a plane of greatest order will be called a quasi-face (vertices, edges, etc. are quasi-faces).

Each quasi-face \(\Gamma\) corresponds to a truncation \(G(X)\):
\[ \hat G(X)=\sum_{\hat Q} A_{\hat Q}\hat QX,\qquad \text{where } \hat Q\in D(G)\cap \Gamma, \]
and the truncated system
\[ \frac{d\ln|x_1|}{\tilde g_1(X)}=\frac{d\ln|x_2|}{\tilde g_2(X)}=\cdots=\frac{d\ln|x_n|}{\tilde g_n(X)}, \tag{4} \]
where
\[ \hat G(X)=(\tilde g_1(X),\tilde g_2(X),\ldots,\tilde g_n(X)). \]

The set \(S\) of order-vectors \(P\) for which system (4), corresponding to the quasiface \(\Gamma\), is a truncation of system (1′), will be called the sector of the quasiface \(\Gamma\) and denoted by \(S(\Gamma)\). Always \(S(\Gamma)\perp \Gamma\) and \(\Gamma=M(\hat G)\). Thus, the sector of a vertex, as a rule, is an \(n\)-dimensional pyramid, the sector of an edge an \((n-1)\)-dimensional pyramid, ..., the sector of a hyperface a ray orthogonal to it.

1. In what follows, a system of the form (1′) will be written as

\[ \frac{d\ln |X|}{dt}\equiv (\ln^{\cdot}|X|)=G(X)\equiv \sum_Q A_QQX, \tag{5} \]

\[ \ln |X|=(\ln |x_1|,\ \ln |x_2|,\ldots,\ln |x_n|). \]

Introduce the power transformation:

\[ y_i=x_1^{\alpha_{i1}}x_2^{\alpha_{i2}}\cdots x_n^{\alpha_{in}},\qquad i=1,2,\ldots,n, \tag{6} \]

with matrix \(\alpha=\|\alpha_{ik}\|\). In what follows we shall assume that \(|\alpha|\ne0\) and denote the transformation (6) by \(Y={}^{\alpha}X\). It is then proved that \(\ln |Y|=\ln |X|\cdot \alpha^*\), \(X={}^{\alpha^{-1}}Y\), and

\[ (\ln^{\cdot}|Y|)=G(X)\alpha^* =\sum_Q A_Q\alpha^*QX =\sum_Q A_Q\alpha^*Q\alpha^{-1}Y =G'(Y). \tag{7} \]

Thus, under the transformation (6), the exponent-vectors \(Q\) in system (5) are transformed linearly with matrix \(\alpha^{-1}\), while the coefficient-vectors \(A_Q\) are transformed linearly with matrix \(\alpha^*\). At the same time, the diagram and the polyhedron of system (5) are transformed affinely into the diagram and the polyhedron of system (7).

We shall call \(\dim \hat M(G)=d\) the dimension of system (5). The dimension of truncated systems is always less than \(n\).

Theorem 2. A system (5) of dimension \(d\), by a suitable power transformation, can be reduced to the form \((\ln^{\cdot}|Y|)=G'(y_1,y_2,\ldots,y_d)\). The transformation matrix can be constructed effectively by means of linear operations on the exponent-vectors \(Q\) of system (5).

Corollary. If \(d\le1\), then system (5) is solvable in quadratures.

Since \(A_Q\) and \(Q\) under the transformation (6) transform as conjugate vectors, we place them in conjugate spaces \(A_Q\in R,\ Q\in \hat R\). The linear subspace \(K\) in \(R\), generated by all coefficients \(A_Q\) from (5), will be called the subspace of coefficients of system (5). For the solutions \(\Phi(\tau)\) of system (5) discussed in Theorem 1, \(P({}^{\alpha}\Phi)=P(\Phi)\alpha^*\).

1. Let

\[ (\ln^{\cdot}|X|)=\hat G(X) \tag{8} \]

be the truncation of system (5) corresponding to the quasiface \(\Gamma\) of dimension \(d\). According to Theorem 1, to obtain the asymptotics of the solutions of system (5), it is necessary to investigate solutions \(\Psi(\tau)\) of system (8) such that \(P(\Psi(\tau))\in S(\Gamma)\). If this system can be solved directly, then the investigation is completed by verifying the indicated inclusion.

Criterion I. If \(S(\Gamma)\cap K(\Gamma)=0\), then the truncation (8), corresponding to the quasiface \(\Gamma\), has no solutions with order in \(S(\Gamma)\).

If system (8) cannot be solved directly, then we make the power transformation indicated in Theorem 2; then system (5) passes into

\[ (\ln^{\cdot}|Y|)=G'(Y), \tag{9} \]

and our truncation (8) into

\[ (\ln^{\cdot}|Y_1|)=\hat G'_1(Y_1), \tag{10¹} \]

\[ (\ln^{\cdot}|Y_2|)=\hat G'_2(Y_1), \tag{10²} \]

moreover, a transformation (6) is possible such that \(Y_2=X_2\). Here the following notation has been introduced:

\[ \begin{gathered} X_1=(x_1,\ldots,x_d),\qquad X_2=(x_{d+1},\ldots,x_n);\\ Y_1=(y_1,\ldots,y_d),\qquad Y_2=(y_{d+1},\ldots,y_n);\\ \hat G_1=(\tilde g_1,\ldots,\tilde g_d),\qquad \hat G_2=(\tilde g_{d+1},\ldots,\tilde g_n). \end{gathered} \]

Let \(X=\Psi(\tau)\) be a solution of system (8) and let \(\Psi(\tau)\to X^0\); then \(Y=\alpha\Psi(\tau)\) is a solution of system (10), but the limit for \(Y_1\) may already be an arbitrary set (stationary points, limit cycles, etc.). We shall restrict ourselves to solutions tending as \(\tau\to\infty\) to a definite limit \(Y_1^0\), with finite and infinite components; such solutions will be divided into two types: 1) at least one component of \(Y_1^0\) is zero or infinity; 2) all components of \(Y_1^0\) are finite and different from zero.

Theorem 3. All limits \(Y_1^0\) of solutions \(Y_1(\tau)\) of system \((10^1)\) of the second type satisfy the system \(\hat G_1'(Y_1^0)=0\).

Take one of such points \(Y_1^0\); two cases are possible: either \(\hat G_2'(Y_1^0)=0\), or \(\hat G_2'(Y_1^0)\ne0\).

Theorem 4. If \(\hat G_2'(Y_1^0)\ne0\), \(Y_1(\tau)\to Y_1^0\), then \(P(Y_2(\tau))=\hat G_2'(Y_1^0)\).

Since for solutions of the second type \(P_1(Y_1)=0\), in the case of Theorem 4 the order of the solutions is determined completely. In this case, after the change of variables

\[ Y_1=Y_1^0+\tilde Y_1,\qquad Y_2=\tilde Y_2 \tag{11} \]

in system (10) the problem is reduced to finding the asymptotics of solutions of the first type. If, however, \(\hat G_2'(Y_1^0)=0\), then the substitution (11) must be made in system (9) and the resulting system studied in a neighborhood of the point \(\tilde Y_1^0=0,\ \tilde Y_2^0=Y_2^0=X_2^0\) by the method described, i.e. constructing truncations, etc. As a rule, after a finite number of steps we arrive at the case \(\tilde G_2'(Y_1^0)\ne0\).

Finding solutions of the first type leads to the construction of truncations of system \((10^1)\) and their study by the methods presented. Here the order \(d\) of system \((10^1)\) is less than \(n\), and the dimension \(d_1\) of the truncation of this system is less than \(d\). Applying again power transformations and further truncations in no more than \(n\) steps, we arrive at the system

\[ \begin{aligned} (\ln^\cdot |W_1|)&=G_k^{(k)}=\mathrm{const},\\ (\ln^\cdot |W_2|)&=G_{k-1}^{(k)}(W_1),\\ (\ln^\cdot |V_2|)&=G_{k-2}^{(k)}(W_1,W_2),\\ &\ \ \vdots\\ (\ln^\cdot |Y_2|)&=G_0^{(k)}(W_1,W_2,V_2,\ldots,Z_2), \end{aligned} \tag{12} \]

which is integrated. The greater the number of steps in this process, the lower the accuracy, but the number of steps is determined by the complexity of the solution. Applying to the solutions of system (12) the transformations inverse to those by which this system was constructed, we obtain the asymptotics of the solutions of system (1).

Received
15 XI 1961