Preamble

This work continues the investigations into the properties of linear differential systems initiated in 1967 (see [1]). We consider the relationship between the stability of solutions and the structural properties of the system matrix, building upon the foundations laid in [5–8] and subsequent developments in [9, 10]. Let the function $\phi(t)$ be defined such that its mean value is given by the limit $\Phi_{cp} = \lim_{k \to \infty} \phi(t_k)$, as discussed in [3, p. 535]. Following the methodology in [3, pp. 533–534], we analyze the behavior of the fundamental matrix $U(t)$ for the system $x' = A(t)x$.

By applying the transformation $x = U(t)u$, the original system is reduced to the form $u' = P(t)u$ (1), where $P(t) = U^{-1}(t)A(t)U(t) - U^{-1}(t)U'(t)$. According to the results in [2, pp. 261–266], if the norm $|U(t)|$ remains bounded by a constant, the stability properties of the transformed system are preserved. Specifically, if $U(t)$ is a unitary matrix such that $U^{-1}(t) = U^*(t)$, then $|U(t)| = |U^{-1}(t)| = 1$. Under these conditions, the matrix $P(t)$ remains bounded, provided that the original matrix $A(t)$ and the derivative $U'(t)$ are bounded.

As demonstrated in [4, p. 43], for any sequence $t_k \to \infty$, we can extract a subsequence such that $U(t_k + t) \to V(t)$ and $A(t_k + t) \to \bar{A}(t)$ in the topology of uniform convergence on compact intervals. Consequently, the limit system for (1) can be expressed as:
$$\begin{aligned} P(t_k + t) = U^{-1}(t_k + t)A(t_k + t)U(t_k + t) - U^{-1}(t_k + t)U'(t_k + t) \end{aligned}$$
Taking the limit as $k \to \infty$, we obtain the limiting matrix $Q(t) = V^{-1}(t)\bar{A}(t)V(t) - V^{-1}(t)V'(t)$, which defines the dynamics of the limit system $v' = Q(t)v$. This relationship ensures that the asymptotic behavior of the original system $x' = A(t)x$ is reflected in the properties of the transformed system $u' = P(t)u$.

1. Properties of the Mean Value $\int P(l)dl$

We examine the integral properties of the matrix $P(t)$. For any $\epsilon > 0$, there exists a $T > 0$ such that for any interval $[ \tau, t ]$ with $t - \tau > T$, the average value of the function $p(t)$ satisfies:
$$\frac{1}{t - \tau} \int_{\tau}^{t} p(s) ds = \bar{p} \pm \epsilon$$
This property is crucial for establishing the existence of Lyapunov exponents. Following the construction in [4, p. 43], we can define a sequence of intervals $[ \tau_k, t_k ]$ where the local average of $p(t)$ deviates from the global mean by a controlled amount. By carefully selecting these intervals, we can construct a function $p(t)$ that oscillates between prescribed bounds, allowing us to analyze the sensitivity of the system to small perturbations.

Specifically, let $f_\delta(T)$ be a modulus of continuity such that $|p(\tau + t) - p(t)| < \delta$ for sufficiently small $\tau$. By applying the results from [2, p. 276], we can show that the characteristic exponents of the system $x' = A(t)x$ are determined by the limit of the integral of the diagonal elements of the transformed matrix $P(t)$. If the system is regular, these exponents coincide with the mean values of the coefficients.

2. Stability and Perturbations

Consider the perturbed system $y' = A(t)y + B(t)y$. If the norm of the perturbation matrix $|B(t)|$ is sufficiently small for $t > 0$, the stability of the original system $x' = A(t)x$ is preserved. This is consistent with the classical theorems of Lyapunov and Perron. For a system $x' = A(t)x$, the adjoint system is given by $x' = -A^*(t)x$. As noted in [2, pp. 272–273], the relationship between the characteristic exponents of the original and adjoint systems provides a criterion for regularity.

If $A(t)$ is a limiting matrix obtained from a sequence $A(t_k + t)$, then the properties of the system $x' = A(t)x$ are inherited from the asymptotic behavior of the original system. In particular, the transformation $x = U(t)u$ leads to a diagonal or triangular form $P(t)$, where the diagonal entries $p_{ii}(t)$ represent the local growth rates of the solutions. The existence of the limit:
$$\lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} p_{ii}(t) dt$$
is a necessary condition for the system to possess a well-defined spectral structure.

In conclusion, the analysis of the transformed matrix $P(t)$ and its integral properties allows for a comprehensive description of the stability regions and the behavior of solutions under small perturbations. These results extend the findings of Vinograd, Erugin, and other researchers in the field of linear differential equations.

References

1. Vinograd, R. E., & Izobov, V. V. Methods of Lyapunov Exponents. Moscow, 1963.
2. Erugin, N. P. Linear Systems of Ordinary Differential Equations. Minsk, 1963.
3. Bylov, B. F., Vinograd, R. E., Grobman, D. M., & Nemytskii, V. V. Theory of Lyapunov Exponents. Moscow, Nauka, 1966.
4. Bourbaki, N. Topologie Générale. Chapitre 10. Espaces fonctionnels. Paris, 1949.
5. [Journal of Differential Equations], vol. 2, pp. 14–24, 1962.
6. [Additional Reference], 1957.
7. Lillo, J. C. Proc. Amer. Math. Soc., vol. 12, no. 1, pp. 400–407, 1961.
8. [Doklady Akademii Nauk], vol. 139, no. 2, pp. 313–315, 1961.
9. [Mathematical Notes], vol. 66, no. 2, pp. 215–229, 1965.