Preamble

In this section, we consider the differential equation of the form:
$$\begin{aligned} x''' + P_1(t)x'' + P_2(t)x' + P_3(t)x = 0 \end{aligned}$$
where $x = x(t)$ and the coefficients $P_i(t)$ are periodic functions. Following the methodology established in \cite{1}, we analyze the fundamental system of solutions $x_1(t), x_2(t), x_3(t)$ satisfying the initial conditions $x_i^{(s-1)}(0) = \delta_{is}$ for $i, s = 1, 2, 3$.

We define the auxiliary functions $u(t)$, $u'(t)$, and $u''(t)$ as follows:
$$\begin{aligned} u''(t) &= \exp \left( -\int_0^t P_1(\tau) d\tau \right) \ u'(t) &= \int_0^t u''(\tau) d\tau \ u(t) &= \int_0^t u'(\tau) d\tau = \int_0^t dx \int_0^x \exp \left( -\int_0^{x_2} P_1(\tau_2) d\tau_2 \right) dx_1 \end{aligned}$$
Under the assumption that $\int_0^t P_1(\tau) d\tau > 0$, these functions characterize the growth and stability of the solutions.

2. Properties of the Integral Operators

Let $P(t)$ be a function defined on $[0, t]$. We assume $\phi(t) \geq 0$ and $P(t)\phi(t)dt > 0$ on the interval $[0, \infty)$. For $0 < \tau < t$, we define the kernel functions:
$$\begin{aligned} \Phi(\tau) = \tau u'(\tau) - u(\tau) > 0 \end{aligned}$$
Given that $u(0) = 0$ and $u'(0) = 0$, it follows from the monotonicity of $u''(\tau)$ that $\Phi(\tau)$ is strictly increasing. Furthermore, we establish the inequality:
$$\begin{aligned} u'(\tau)[u(t) - u(\tau)] > u'(\tau) - u(\tau) - t u'(\tau) + u(t) \end{aligned}$$
This relationship is critical for bounding the integral terms in the subsequent stability analysis.

3. Constraints on the Coefficients

We assume the following conditions on the coefficients $P_i(t)$ for $t \in [0, \infty)$:
- (a) $P_1(t)$ is such that $u''(t)$ is well-defined and integrable.
- (b) $P_2(t) > 0$ for $t < \omega/2$ and $P_2(t)$ remains bounded.
- (c) $P_3(t) < 0$ on specific sub-intervals, satisfying $P_2(t) + P_3(t)t > 0$.
- (d) $P_2(t)u'(t) + P_3(t)u(t) > 0$.

These conditions ensure that the operator $F_{ji}(t)$, defined by the iterative relation
$$\begin{aligned} F_{ji}(t) = P_2(t)x_{j,i-1}'(t) + P_3(t)x_{j,i-1}(t) \end{aligned}$$
remains positive, which is a sufficient condition for the non-oscillation of the solutions.

4. Stability and Periodicity

For a periodic system with period $\omega$, the solutions at $t + k\omega$ can be expressed via the values in the initial period $[0, \omega]$. Specifically, for $k = 0, 1, 2, \dots$:
$$\begin{aligned} u(k\omega + t) = u(k\omega) + u'(k\omega)t + u(t) \end{aligned}$$
The characteristic equation for the system is given by $\lambda^3 + A_1\lambda^2 + A_2\lambda + A_3 = 0$. As shown in \cite{1}, for stability, the coefficients must satisfy specific bounds related to the trace of the monodromy matrix. We define:
$$\begin{aligned} \Gamma = -[x_1(\omega) + x_2'(\omega) + x_3''(\omega)] \end{aligned}$$
If $-3 < \Gamma < 1$, the system exhibits stable behavior. The values of $x_{is}(\omega)$ are calculated using the integral representations:
$$\begin{aligned} x_{11}(\omega) = 1 + \int_0^\omega u''(\tau) F_{11}(\tau) [u'(\omega) - u'(\tau)] d\tau \end{aligned}$$
By applying the mean value theorem to the integral terms and utilizing the inequalities derived in Section 2, we can provide upper bounds for the solutions $x_{is}(t)$.

5. Numerical Example

Consider the third-order equation:
$$\begin{aligned} x''' + \mu \sin^2 t x'' + (\lambda^2 + \delta \cos 2t) x' + \eta \sin^3 2t x = 0 \end{aligned}$$
where $\mu, \gamma > 0$. For small values of the parameters $\mu, \lambda, \delta, \eta$, we can verify the stability conditions. Using the derived formulas:
$$\begin{aligned} u''(t) = \exp(-\mu \sin^2 t), \quad u'(t) = \int_0^t \exp(-\mu \sin^2 \tau) d\tau \end{aligned}$$
The stability region in the parameter space is determined by the condition $\Gamma(\mu, \lambda, \delta, \eta) < 1$. Numerical results indicate that for $\lambda^2 > |\delta|$, the system remains stable provided the damping coefficient $\mu$ is sufficiently large to satisfy the integral inequalities.

References

1. Starzhinskii, V. M., Bulletin of Moscow State University, Mathematics and Mechanics Series, No. 6, pp. 818–825, 1959. (Received May 28, 1966).