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Mathematics
A. A. SHESTAKOV
ON THE ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF MULTIDIMENSIONAL SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS HAVING A SINGULAR POINT OF HIGHER ORDER
(Presented by Academician I. G. Petrovskii, 18 XII 1959)
§ 1. This article is a continuation of the author’s earlier investigations, set forth in the notes \((^{1,2})\).
Consider a system of differential equations of the form
\[ \frac{dx}{dt}=X(x),\qquad x=(x_1,\ldots,x_n),\quad n\geqslant 3, \tag{1} \]
where the coordinates \(X_s(x)\) of the vector-function \(X(x)\) of the point \(x\) are power series in the coordinates of the point \(x\), and in these series the constant terms and the terms of first degree are absent.* In the article the index \(s\) everywhere takes the values \(1,2,\ldots,n\).
In \(x\)-space consider \(n\) classes \(L_s\) of points \((\alpha_{s1},\alpha_{s2},\ldots,\alpha_{s,s-1}, \alpha_{ss}-1,\alpha_{s,s+1},\ldots,\alpha_{sn})\), corresponding to all nonzero coefficients \(a_{s\alpha}\) in the expansions
\[ X_s(x)=\sum a_{s\alpha_1\alpha_2\ldots\alpha_n}x_1^{\alpha_{s1}}x_2^{\alpha_{s2}}\cdots x_n^{\alpha_{sn}}, \tag{2} \]
where \(a_{s\alpha}\) are constants and the summation extends over all positive integers \(\alpha_{sj}\). Points belonging to any class \(L_s\) lie inside or on the faces of the \(n\)-dimensional angle formed by the \(n\) planes—quadrants \(x_i=-1,\ x_j\geqslant -1,\ i\ne j,\ i,j=1,\ldots,n\), and having vertex at the point \((-1,-1,\ldots,-1)\). Suppose that there exists an \((n-1)\)-dimensional plane \(\pi\), intersecting all \(n\) edges of the \(n\)-dimensional angle in such a way that inside the resulting \((n+1)\)-dimensional polyhedron there are no points of any class \(L_s\), but on the plane \(\pi\) of this polyhedron there lies at least one point from each class \(L_s\).** The coordinates of the points of each class \(L_s\) lying on the plane \(\pi\) will be denoted by the letter \(\beta\), and the coordinates of all the remaining points by the letter \(\gamma\). We write the equation of the plane \(\pi\) in the form
\[ p_1x_1+\cdots+p_nx_n=p,\qquad p>0,\quad p_s>0. \]
Since the numbers \(p_s\) and \(p\) are rational, we shall assume that the coordinates \(p_s\) of the normal vector \((p_1,\ldots,p_n)\) of the plane \(\pi\) and the free term \(p\) have no common divisor other than one. Under this condition the numbers \(p_s,\ p\) are uniquely determined. By virtue of the properties of the plane
* The results set forth in the article are extended without difficulty to algebraic systems of differential equations, but for simplicity we restrict ourselves to equations with analytic right-hand sides.
** Since \(n\) points uniquely determine an \((n-1)\)-dimensional plane, in the general case, by a corresponding displacement of a plane passing through at least one point of each class \(L_s\), we obtain the plane \(\pi\). Therefore the case under consideration is general.
\(\pi\) we shall have the relations
\[ \sum_k(\beta_{sk}-\delta_{sk})p_k=p,\qquad \sum_k(\gamma_{sk}-\delta_{sk})p_k>p. \tag{3} \]
§ 2. Consider the “truncated” system of differential equations of the form
\[ \frac{dx_s}{dt}=\varphi_s(x_1,\ldots,x_n), \tag{4} \]
where the polynomials \(\varphi_s(x)\) are defined by the formulas
\[ \varphi_s(x)=\sum a_{s\beta_1\beta_2\ldots\beta_n} x_1^{\beta_{s1}}x_2^{\beta_{s2}}\cdots x_n^{\beta_{sn}}, \tag{5} \]
the summation being extended over the points lying in the plane \(\pi\). By direct verification we are convinced of the validity of the identities
\[ \varphi_s(\lambda^{p_1}x_1,\ldots,\lambda^{p_n}x_n) =\lambda^{p+p_s}\varphi_s(x_1,\ldots,x_n) \tag{6} \]
for any real number \(\lambda\in(-\infty,+\infty)\).
We shall seek a solution of system (4) in the form
\[ x_s^*=\omega_s(c+t)^{rp_s}, \tag{7¹} \]
\[ \overline{x}_s=\overline{\omega}_s(c-t)^{rp_s}, \tag{7²} \]
where \(\omega_s,\ \overline{\omega}_s,\ r,\ c>0\) are certain constants.
Substituting (7¹) and (7²) into system (4), respectively, we obtain
\[ rp_s\omega_s(c+t)^{rp_s-1} =(c+t)^{r(p+p_s)}\varphi_s(\omega_1,\ldots,\omega_n), \]
\[ -rp_s\overline{\omega}_s(c-t)^{rp_s-1} =(c-t)^{r(p+p_s)}\varphi_s(\overline{\omega}_1,\ldots,\overline{\omega}_n). \]
The last relations will be satisfied if and only if one sets
\(rp_s-1=r(p+p_s)\), i.e. \(r=-1/p\), and if
\(\omega=(\omega_1,\ldots,\omega_n)\) and
\(\overline{\omega}=(\overline{\omega}_1,\ldots,\overline{\omega}_n)\)
are solutions, respectively, of the systems of algebraic equations
\[ p_s\omega_s+p\varphi_s(\omega_1,\ldots,\omega_n)=0, \tag{8} \]
\[ p_s\overline{\omega}_s-p\varphi_s(\overline{\omega}_1,\ldots,\overline{\omega}_n)=0. \tag{9} \]
Equations (8) and (9) will be called determining, and the real solutions of system (4) of the form (7¹) and (7²)—simplest solutions.
For the simplest solutions (7¹) and (7²) we have:
\(\|x^*\|\to0,\ t\to+\infty,\ \|\overline{x}\|\to0,\ t\to-\infty\), i.e. the solution (7¹) is a \(0^+\)-curve, and the solution (7²) is a \(0^-\)-curve. In \(x\)-space the simplest solutions will be parabolas with parameter \(t\).
- In order to investigate the solutions of system (1) that tend, as \(t\to\pm\infty\), to the simplest solutions (7¹) and (7²) of the first approximation system (4), introduce into system (1) a new independent variable
\[ \tau=(c+t)^{-1/p},\qquad \overline{\tau}=(c-t)^{-1/p}, \tag{10¹} \]
respectively, and make the substitution of the unknown functions
\[ x_s=(y_s+\omega_s)\tau^{p_s},\qquad x_s=(\overline{y}_s+\overline{\omega}_s)\overline{\tau}^{p_s}, \tag{10²} \]
where \(\omega\) and \(\overline{\omega}\) are real solutions of the determining equations (8) and (9). Then, for the determination of the new unknown functions \(y_s\) and \(\overline{y}_s\), we obtain the following systems of equations of Poincaré–Lyapunov type*:
\[ \tau\frac{dy_s}{d\tau} =-p\sum_j\varphi_{sj}(\omega)y_j-p_sy_s +Y_s(\tau,y_1,\ldots,y_n); \tag{11} \]
* By a system of Poincaré–Lyapunov type we mean a system whose right-hand side contains linear terms, and such that the characteristic numbers of the matrix of the system have nonzero real part.
\[ \bar{\tau}\frac{d\bar{y}_s}{d\bar{\tau}} = p\sum \varphi_{sj}(\bar{\omega})\,\bar{y}_j - p_s\bar{y}_s + \bar{Y}_s(\bar{\tau}, \bar{y}_1,\ldots,\bar{y}_n), \tag{12} \]
whose matrices respectively have the form
\[ \left\|-p\varphi_{sj}(\omega)-\delta_{sj}p_s\right\|, \tag{13} \]
\[ \left\|p\varphi_{sj}(\bar{\omega})-\delta_{sj}p_s\right\|, \tag{14} \]
and the functions \(Y_s\) and \(\bar{Y}_s\) satisfy the conditions
\[ |Y_s(\tau,y)|<\alpha_1 h+\beta_1,\qquad 0\leq \tau\leq \tau_0,\qquad \|y\|\leq h, \tag{15^1} \]
\[ |\bar{Y}_s(\bar{\tau},\bar{y})|<\alpha_2 h+\beta_2,\qquad 0\leq \bar{\tau}\leq \tau_0,\qquad \|\bar{y}\|\leq h, \tag{15^2} \]
where \(\alpha_1,\alpha_2,\beta_1,\beta_2\) are positive constants which may be made arbitrarily small depending on the choice of \(\tau_0\) and \(h\).
Theorem 1. If the number \(p\) is even, all the numbers \(p_s\) are odd, and \(k\) characteristic numbers of the matrix (13) (respectively, of the matrix (14)) have positive real part, then along the parabola \(x_s=\omega_s\tau^{p_s}\), \(\tau>0\) (respectively, \(x_s=\bar{\omega}_s\bar{\tau}^{p_s}\), \(\bar{\tau}>0\)) a family of \(0^+\)-curves (respectively, \(0^-\)-curves) of dimension \(k\) enters the point \(0\), and along the parabola \(x_s=-\omega_s\tau^{p_s}\), \(\tau>0\) (respectively, \(x_s=-\bar{\omega}_s\bar{\tau}^{p_s}\), \(\bar{\tau}>0\)) a family of \(0^+\)-curves (respectively, \(0^-\)-curves) of the same dimension \(k\) enters the point \(0\).
Theorem 2. If all the numbers \(p\) and \(p_s\) are odd and \(k\) characteristic numbers of the matrix (13) have positive real part, then along the parabola \(x_s=\omega_s\tau^{p_s}\), \(\tau>0\), a family of \(0^+\)-curves of dimension \(k\) enters the point \(0\), and along the parabola \(x_s=-\omega_s\tau^{p_s}\), \(\tau>0\), a family of \(0^-\)-curves of dimension \(k\) enters the point \(0\).
Proof. Putting \(\lambda=-1\) in the relations (6), we obtain
\[ \varphi_s\bigl(\omega_1(-1)^{p_1},\ldots,\omega_n(-1)^{p_n}\bigr) = (-1)^{p+p_s}\varphi_s(\omega_1,\ldots,\omega_n). \tag{16} \]
Let the number \(p\) be even and all the numbers \(p_s\) odd. Then it follows from (16) that the functions \(\varphi_s(x)\) are odd, and the vectors \(\omega\) and \(-\omega\) will be solutions of the determining equations (8). On the other hand, it is easy to show that the functions \(\varphi_s(x)\) satisfy the relations
\[ p_1x_1\varphi_{s1}(x)+\cdots+p_nx_n\varphi_{sn}(x) = (p+p_s)\varphi_s(x), \tag{17} \]
which are a generalization of Euler’s formula for homogeneous functions. In view of (17), the partial derivatives \(\varphi_{sk}(x)\) are even functions and, consequently, \(\varphi_{sk}(-\omega)=\varphi_{sk}(\omega)\).
Let all the numbers \(p,p_s\) be odd. If the vector \(\omega\) is a solution of equations (8), then the vector \(-\omega\) will be a solution of equations (9). Performing successively in system (1) the changes of variables
\[ \tau=(c+t)^{-1/p};\qquad x_s=(y_s+\omega_s)\tau^{p_s};\qquad t\to+\infty,\ \tau\to 0, \tag{18^1} \]
\[ \bar{\tau}=(c-t)^{-1/p};\qquad x_s=(\bar{y}_s-\omega_s)\bar{\tau}^{p_s};\qquad t\to-\infty,\ \bar{\tau}\to 0, \tag{18^2} \]
we obtain, for determining the functions \(y_s\), a system of the form (11), and, for determining the functions \(\bar{y}_s\), a system of Poincaré–Lyapunov type
\[ \bar{\tau}\frac{d\bar{y}_s}{d\bar{\tau}} = p\sum \varphi_{sj}(-\omega)\bar{y}_j - p_s\bar{y}_s + Y_s^{*}(\bar{\tau},\bar{y}_1,\ldots,\bar{y}_n), \tag{19} \]
whose matrix has the form
\[ \left\|p\varphi_{sj}(-\omega)-\delta_{sj}p_s\right\| \tag{20} \]
and the functions \(Y_s^{*}(\bar{\tau},\bar{y})\) satisfy conditions of type (15).
By virtue of relations (16) and (17), the functions \(\varphi_s(x)\) are even, and all partial derivatives \(\varphi_{sj}(x)\) are odd. Consequently, \(\varphi_{sj}(-\omega)=-\varphi_{sj}(\omega)\), and the matrices (13) and (20) coincide.
It is obvious that if any one of the systems of differential equations (11), (12), (19) has a family of \(0\)-curves \(y_s=y_s(\tau)\) of dimension \(k\), then system (1) has a family of \(0\)-curves \(x_s=x_s(t)\) of the same dimension \(k\), possessing the property \(x_s-\omega_s\tau^{p_s}=o(\tau^{p_s})\). This proves Theorems 1 and 2.
§ 4. Let us now consider the general case, when among the numbers \(p_s\) there are both even and odd ones.
First case. \(p\) is an even number, and among the \(p_s\) there are both even and odd numbers. In this case, to a solution \((\omega_1,\ldots,\omega_n)\) of equations (8) (respectively, to a solution \((\bar\omega_1,\ldots,\bar\omega_n)\) of equations (9)) there corresponds, by virtue of (16), the solution \(\bigl((-1)^{p_1}\omega_1,\ldots,(-1)^{p_n}\omega_n\bigr)\) of equations (8) (respectively, the solution \(\bigl((-1)^{p_1}\bar\omega_1,\ldots,(-1)^{p_n}\bar\omega_n\bigr)\) of equations (9)).
To the solutions now under consideration there correspond the matrices
\[ \left\|-p\varphi_{sj}\bigl((-1)^{p_1}\omega_1,\ldots,(-1)^{p_n}\omega_n\bigr)-\delta_{sj}p_s\right\|, \tag{13¹} \]
\[ \left\|\rho\varphi_{sj}\bigl((-1)^{p_1}\bar\omega_1,\ldots,(-1)^{p_n}\bar\omega_n\bigr)-\delta_{sj}p_s\right\|. \tag{14¹} \]
The analogue of Theorem 1 will be the following proposition:
Theorem 3. If \(\rho\) is an even number, and the matrices (13) and (13¹) (respectively, the matrices (14) and (14¹)) have respectively \(k_1\) and \(k_2\) characteristic numbers with positive real parts, then along the parabola \(x_s=\omega_s\tau^{p_s}\), \(\tau>0\) (respectively, \(x_s=\bar\omega_s\bar\tau^{p_s}\), \(\bar\tau>0\)) there enters the point \(0\) a family of \(0^+\)-curves (respectively, \(0^-\)-curves) of dimension \(k_1\), while along the parabola \(x_s=(-1)^{p_s}\omega_s\tau^{p_s}\), \(\tau>0\) (respectively, \(x_s=(-1)^{p_s}\bar\omega_s\bar\tau^{p_s}\), \(\bar\tau>0\)) there enters the point \(0\) a family of \(0^+\)-curves (respectively, \(0^-\)-curves) of dimension \(k_2\).
Second case. \(p\) is an odd number, and among the \(p_s\) there are both even and odd numbers. In this case, to a solution \((\omega_1,\ldots,\omega_n)\) of equations (8) (respectively, to a solution \((\bar\omega_1,\ldots,\bar\omega_n)\) of equations (9)) there corresponds, by virtue of (16), the solution \(\bigl((-1)^{p_1}\omega_1,\ldots,(-1)^{p_n}\omega_n\bigr)\) of equations (9) (respectively, the solution \(\bigl((-1)^{p_1}\bar\omega_1,\ldots,(-1)^{p_n}\bar\omega_n\bigr)\) of equations (8)). Along with the matrix (13¹), consider the matrix
\[ \left\|\rho\varphi_{sj}\bigl((-1)^{p_1}\omega_1,\ldots,(-1)^{p_n}\omega_n\bigr)-\delta_{sj}p_s\right\|. \tag{20¹} \]
If in § 3 the matrices (13) and (20) coincided, then now the matrices (13¹) and (20¹), generally speaking, are different. Therefore the analogue of Theorem 2 will be the following.
Theorem 4. If \(p\) is an odd number and the matrices (13¹) and (20¹) have respectively \(k_1\) and \(k_2\) characteristic numbers with positive real parts, then along the parabola \(x_s=\omega_s\tau^{p_s}\), \(\tau>0\), there enters the point \(0\) a family of \(0^+\)-curves of dimension \(k_1\), while along the parabola \(x_s=(-1)^{p_s}\omega_s\tau^{p_s}\), \(\tau>0\), there enters the point \(0\) a family of \(0^-\)-curves of dimension \(k_2\).
All-Union Correspondence Institute
of Railway Transport Engineers
Received
15 XII 1959
REFERENCES
- A. A. Shestakov, DAN, 79, No. 2 (1951).
- A. A. Shestakov, DAN, 79, No. 1 (1951).