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UDC 517.925
MATHEMATICS
D. M. GROBMAN
A NEIGHBORHOOD OF A SINGULAR POINT OF A SYSTEM OF DIFFERENTIAL EQUATIONS WITH SMALL NONLINEARITIES
(Presented by Academician I. G. Petrovskii, 31 III 1966)
1°. Consider the systems
\[ x' = Ax + F(t,x), \tag{1} \]
\[ y' = Ay, \tag{2} \]
where \(A\) is a constant \(n \times n\)-matrix; \(x, y, F\) are \(n\)-dimensional vectors. Suppose that
\[ F(t,0)=0 \tag{3} \]
and in the domain \(|x| \le r_0 < 1\) the condition
\[ |F(t,x_1)-F(t,x_2)| \le L(r)|x_1-x_2|, \tag{4} \]
is satisfied, where \(r=\max(|x_1|,|x_2|)\); \(L(r)\) decreases monotonically as \(r \to 0\).
In note \((^1)\), under the assumption that the vector \(F(t,x)\) is given only for \(t \ge 0\), several propositions were formulated concerning the asymptotic behavior of solutions of system (1) as \(t \to +\infty\).
If one assumes that the vector \(F(t,x)\) is defined for all \(t\) and has properties (3) and (4), then the results of \((^1)\) can be strengthened by considering the behavior of solutions of system (1) both as \(t \to +\infty\) and as \(t \to -\infty\).
There are propositions analogous to Theorems 1–3 of note \((^1)\). In order not to repeat ourselves, we shall give the formulation of only one of them, for example the analogue of Theorem 3 from \((^1)\).
2°. Let us recall some definitions needed for the formulation of the theorem.
The exponent of the vector \(x(t)\) is the number
\[ \chi(x)=\varlimsup_{t\to+\infty}\frac{1}{t}\ln |x(t)|. \]
The minus-exponent \(\bar{\chi}(x)\) of the vector \(x(t)\) is defined by the equality
\[ \bar{\chi}(x)=\varlimsup_{t\to-\infty}\frac{1}{-t}\ln |x(t)|. \]
The exponent and minus-exponent characterize the exponential growth of \(x(t)\) as \(t \to +\infty\) and \(t \to -\infty\).
For a more precise estimate of the growth of \(x(t)\), introduce other characteristics. The second exponent of the vector \(x(t)\) with exponent \(\omega\) is the number
\[ \chi_2(x)=\varlimsup_{t\to+\infty}\frac{\ln\left(e^{-\omega t}|x(t)|\right)}{\ln t}. \]
The second minus-exponent \(\bar{\chi}_2(x)\) of the vector \(x(t)\) with minus-exponent \(\omega\) will be defined by the formula
\[ \bar{\chi}_2(x)=\varlimsup_{t\to-\infty}\frac{\ln\left(e^{-\omega |t|}|x(t)|\right)}{\ln |t|}. \]
The ratio \(|x(t)-y(t)|/|y(t)|\) will be called the deviation of the vector \(x\) from the vector \(y\). If the deviation of \(x\) from \(y\) tends to zero as \(t\to+\infty\) \((t\to-\infty)\), then we shall say that \(x\) and \(y\) are analogous as \(t\to+\infty\) \((t\to-\infty)\).
3°. Introduce the following notation: \(\omega_1<\omega_2<\cdots<\omega_s\) are the distinct real parts of the eigenvalues of the matrix \(A\); \(m_k+1\) is the maximum of the orders of those boxes in the Jordan form of the matrix \(A\) which correspond to the number \(\omega_k\). We shall assume that \(\omega_1<0,\ \omega_s>0\), since the case \(\omega_1\omega_s>0\) was essentially considered in note (1).
Let the function \(L(r)\) be given by the equalities
\[ L(0)=0,\qquad L(r)=\varepsilon(r)r^\lambda|\ln r|^\mu,\qquad r\in(0,r_0], \tag{5} \]
where \(\lambda>0\), \(\mu\) is an arbitrary number, \(\varepsilon(r)\geq 0\) increases monotonically together with \(r\), and
\[ \int_0^{r_0}\frac{\varepsilon(r)}{r|\ln r|}<+\infty . \tag{6} \]
Consider the numbers \(\omega_0<\omega_1(1+\lambda)\), \(\omega_{s+1}>\omega_s(1+\lambda)\), and for each \(k=1,2,\ldots,s\) find an index \(\tilde k\) from the inequalities
\[ \omega_{\tilde k-1}<(1+\lambda)\omega_k\leq \omega_{\tilde k},\quad \text{if }\omega_k\leq 0, \tag{7} \]
\[ \omega_{\tilde k-1}\leq(1+\lambda)\omega_k<\omega_{\tilde k},\quad \text{if }\omega_k>0. \tag{8} \]
By \(m_{\tilde k}^{0}\) denote the number which is equal to 0 if the inequality (7) or (8) corresponding to the index \(k\) is strict, and equal to \(m\) in the contrary case.
Define the function \(\rho(t)\) by the equalities
\[ \rho(t)= \begin{cases} 1, & \text{for } |t|\leq 1,\\ |t|, & \text{for } |t|>1. \end{cases} \]
4°. Theorem. Suppose that conditions (3)—(6) are satisfied. Then:
1) Every solution \(x(t)\) of system (1) with exponent (minus-exponent) \(\omega<0\) is analogous to some solution \(y(t)\) of system (2), and the deviation of \(x\) from \(y\) is
\[ o\left(e^{\lambda\omega|t|}|t|^{m_{\tilde k}^{0}+1+\lambda l+\mu}\right) \quad \text{as } t\to+\infty\quad (t\to-\infty), \tag{9} \]
where \(l\) is the second exponent (second minus-exponent) of \(x\) and \(y\).
2) For every solution \(x(t)\) of system (1) which at \(t=0\) passes through a sufficiently small neighborhood \(S^*\) of the origin and has a negative exponent (minus-exponent) \(\omega\), the following inequality holds for \(t\geq 0\) \((t\leq 0)\):
\[ |x(t)|\leq M|x(0)|e^{\omega|t|}\rho^l(t); \]
\(M>0\) does not depend on \(x\); \(l\) is the second exponent (second minus-exponent) of \(x\).
3) There exists a homeomorphism \(\Phi^*\), mapping the neighborhood \(S^*\) onto some domain \(G\), possessing the following properties:
a) \(\Phi(0)=0\);
b) if \(y(0)\in G\) and the solution \(y(t)\) of system (2) has exponent (minus-exponent) \(\omega<0\), then through the point \(\Phi^{*-1}(y(0))\) at \(t=0\) there passes a solution \(x(t)\) of system (1), analogous to \(y(t)\) as \(t\to+\infty\) \((t\to-\infty)\) and with deviation (9); moreover, for \(t\geq 0\) \((t\leq 0)\),
\[ |x(t)-y(t)|\leq |x(0)|^{1+\lambda}\psi(t,|x(0)|)e^{(1+\lambda)\omega|t|}\rho(t)^{m_{\tilde k}^{0}+(1+\lambda)l+1+\mu}, \tag{10} \]
and for \(\mu\leq 0\)
\[ \psi(t,r)\to 0,\quad \text{when } |t|+r^{-1}\to\infty, \]
for $\mu>0$
\[ \psi(t,r)=|\ln r|^\mu \varepsilon(t,r), \]
where $\varepsilon(t,r)\to 0$ when $|t|+r^{-1}\to\infty$;
c) the homeomorphisms $\Phi^*$ and $\Phi^{*-1}$ satisfy the Lipschitz condition and have the form
\[ \Phi^*(x)=x+\varphi^*(x),\qquad \Phi^{*-1}(x)=x+\psi^*(x), \]
where
\[ |\varphi^*(x)|=o\bigl(|x|^{1+\lambda}\bigr)\quad \text{as } x\to 0,\quad \text{if } \mu\le 0, \]
\[ |\varphi^*(x)|=o\bigl(|\ln |x||^\mu |x|^{1+\lambda}\bigr)\quad \text{as } x\to 0,\quad \text{if } \mu>0; \]
\[ |\psi^*(x)|/|\varphi^*(x)|\to 1,\qquad \text{when } x\to 0. \]
4) If none of the numbers $\omega_i\ne 0$ $(i=1,2,\ldots,s)$, then through the points corresponding, under the mapping $\Phi^*$, to $t=0$ there pass either saddle solutions $x(t)$ and $y(t)$ of systems (1) and (2), or 0-curves; in the latter case all assertions of item 3 b) are valid for $x$ and $y$.
Institute of Electronic
Control Machines
Received
23 III 1966
CITED LITERATURE
- D. M. Grobman, DAN, 166, No. 1, 15 (1966).