Abstract
The question of the equivalence of systems of two differential equations \cite{4}
\begin{gather}\frac{dx}{dt}=X[R(t)+Q(t)],\label{1}\\frac{dx}{dt}=YR(t)\label{2}\end{gather}
in the sense of $x=YA(t)$, where $A(t)\to A=\operatorname{const}$ as $t\to\infty$, is considered. Here $R+Q=P_0+P_1t^{-1}+P_2t^{-2}+\cdots$, where $P_k$ are constant second-order matrices, and $R(t)$ is a segment of this series. Bibliography: 7 items.
Full Text
Preamble
In this section, we investigate the asymptotic behavior of the solutions to the system of differential equations as $t \to \infty$. We consider the relationship between the matrices $R(t)$, $Q(t)$, and $X$, specifically focusing on the case where $X' = X[R(t) + Q(t)]$ and $Y' = YR(t)$. We assume that $Y^{-1} X = A(t)$, where $A(t) \to A = \text{const}$ as $t \to \infty$.
§ 1. Case of Distinct Real Eigenvalues
Let the matrix $P_0$ have distinct real eigenvalues $\alpha_1$ and $\alpha_2$, with $\alpha_1 > \alpha_2$. Suppose the matrix $R(t)$ can be expanded in a series of the form:
$$R(t) = P_0 + P_1 t^{-1} + P_2 t^{-2} + \dots = \sum_{k=0}^{\infty} P_k t^{-k} \qquad \text{(1.1)}$$
where $P_k$ are constant matrices. Following the methods established in \cite{5}, the fundamental matrix $Y$ can be represented as:
$$Y = e^{\text{diag}[\alpha_1, \alpha_2]t} Z(t) \qquad \text{(1.2)}$$
where $Z(t) = I + O(t^{-1})$ as $t \to \infty$. If we define $A(t) = Y^{-1} X$, and assume $A(t) \to A = \text{const}$, then the matrix $A$ must be diagonal, $A = \text{diag}[b_{11}, b_{22}]$. If $b_{11} b_{22} \neq 0$, then the matrix $X$ exhibits the same asymptotic growth as $Y$.
§ 2. Case of Purely Imaginary Eigenvalues
Consider the case where $P_0$ has purely imaginary eigenvalues $P_0 = \text{diag}[i\alpha, -i\alpha]$. If the first-order perturbation term $P_1$ vanishes ($P_1 = 0$), the asymptotic behavior is determined by higher-order terms. Under these conditions, the transformation $Y = e^{\text{diag}[i\alpha, -i\alpha]t} Z(t)$ yields a matrix $A(t)$ that approaches a constant diagonal matrix $B = \text{diag}[b_{11}, b_{22}]$. If $b_{11} b_{22} \neq 0$, the system remains stable, and the solutions are bounded.
§ 3. Case of Complex Conjugate Eigenvalues
When $P_0$ has complex conjugate eigenvalues of the form $\alpha \pm i\alpha$, we utilize the representation:
$$Y = e^{\alpha t} e^{\text{diag}[i\alpha, -i\alpha]t} Z(t) \qquad \text{(3.1)}$$
The analysis follows a similar logic to the previous sections. The matrix $A(t)$ converges to a constant matrix $A$ as $t \to \infty$. The specific form of $A$ depends on the coefficients $P_k$ of the expansion of $R(t)$. If $P_1 = 0$, the convergence rate is determined by $O(t^{-2})$.
§ 4. Case of Multiple Eigenvalues
In the case where $P_0$ has a multiple eigenvalue $\alpha$ with a non-trivial Jordan block:
$$P_0 = \begin{pmatrix} \alpha & 1 \ 0 & \alpha \end{pmatrix} \qquad \text{(4.1)}$$
the asymptotic expansion of the solution $X$ involves logarithmic terms. Specifically, the matrix $Y$ can be expressed as:
$$Y = e^{\alpha t} \begin{pmatrix} 1 & t \ 0 & 1 \end{pmatrix} Z(t) \qquad \text{(4.4)}$$
where $Z(t) = I + O(t^{-1})$. If $A(t) = Y^{-1} X \to A$, then $A$ must commute with the Jordan form of $P_0$.
For the general case where $R(t) = P_0 + \sum_{k=1}^{\infty} P_k t^{-k}$, we define the matrices $L_k$ based on the coefficients $P_k$. If the determinant $D(S) \neq 0$ and the structural conditions on the elements $s_{21}, s_{22}$ are met, the solution $X$ can be represented as:
$$X = t^C Z(t) S^{-1} \text{diag}[t, 1] e^{\alpha t} \qquad \text{(4.18)}$$
where $C$ is a constant matrix related to the eigenvalues of the perturbation.
If the leading perturbation term $P_1$ satisfies certain nullity conditions, the logarithmic growth is suppressed, and $A(t)$ converges to a constant matrix $B$ more rapidly. Specifically, if $b \neq 0$ and the resonance conditions are avoided, the solution maintains the form:
$$A(t) = \begin{pmatrix} t^{-1} & -\ln t \ 0 & 1 \end{pmatrix} Z^{-1}(t) B X Z(t) S^{-1} \begin{pmatrix} t & 1 \ 0 & 1 \end{pmatrix} \qquad \text{(4.37)}$$
This indicates that the interaction between the Jordan structure of the unperturbed system and the power-law decay of the perturbation $R(t) - P_0$ leads to a variety of asymptotic regimes, including pure power-law growth, logarithmic corrections, or convergence to a steady state.
References
- Bellman, R. Stability Theory of Differential Equations. McGraw-Hill, 1953.
- Gel'fand, I. M. Lectures on Linear Algebra. Moscow, 1951.
- Coddington, E. A., and Levinson, N. Theory of Ordinary Differential Equations. McGraw-Hill, 1955.
- Erugin, N. P. Linear Systems of Ordinary Differential Equations. Minsk, 1963.
- Harris, W. J. Asymptotic Expansions for Ordinary Differential Equations. 1966.
- Rapoport, I. M. On Some Asymptotic Methods in the Theory of Differential Equations. Kiev, 1954.
- Shchelkanovtsev, N. M. Journal of Mathematics, Vol. 10, No. 6, 1966.