Abstract Generated abstract
This paper studies the local behavior near the origin of a nonautonomous system of differential equations whose nonlinear term is small in a Lipschitz sense, comparing it with the corresponding linear system with constant matrix. Extending earlier one-sided asymptotic results to functions defined for all time, it introduces exponents, minus-exponents, and second-order growth characteristics to describe solutions as time tends to both positive and negative infinity. Under logarithmic smallness and integrability conditions on the nonlinearity, the main theorem establishes asymptotic analogy between nonlinear and linear solutions with negative exponents, gives quantitative deviation estimates, and constructs a local homeomorphism relating initial data for the two systems with Lipschitz control and higher-order error bounds.
Full Text
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).