Introduction

In 1967, A. M. Samoilenko [1, 2] proposed and investigated a numerical-analytical method for studying periodic solutions of the system:
$$\frac{dx}{dt} = f(t, x)$$
where $f(t, x)$ is a periodic function in $t$. In these works, the existence of periodic solutions was established, and an algorithm for their construction was provided. The present paper extends this method to systems of the form:
$$\frac{d^2x}{dt^2} = f\left(t, x, \frac{dx}{dt}\right) \tag{1.1}$$
where the function $f(t, x, y)$ is periodic in $t$ with period $T$.

In Section 1, we prove the existence of periodic solutions for system (1.1). Section 2 provides a method for constructing these solutions, while Section 3 discusses the practical application of the results obtained in Section 1. We demonstrate that the periodic solution of (1.1) can be found as the limit of a sequence of periodic functions.

Section 1. Existence of Periodic Solutions

Consider the second-order differential equation (1.1). We assume that the function $f(t, x, y)$ is defined in the domain:
$$-\infty < t < \infty, \quad a \leq x \leq b, \quad c \leq y \leq d \tag{1.2}$$
and is periodic in $t$ with period $T$. Furthermore, we assume $f$ satisfies the Lipschitz condition:
$$|f(t, x, y) - f(t, x', y')| \leq K_1|x - x'| + K_2|y - y'| \tag{1.3}$$
where $K_1$ and $K_2$ are positive constants.

Following the methodology in [1], we define the operator $L$ as:
$$Lf(t) = \int_0^t [f(\tau) - \bar{f}] d\tau \tag{1.4}$$
where $\bar{f}$ is the average value of the function:
$$\bar{f} = \frac{1}{T} \int_0^T f(t) dt \tag{1.5}$$
We further define the iterated operator $L^2f(t) = L(Lf)$. It was shown in [1] (Lemma 1) that the following estimate holds:
$$|Lf(t)| \leq \alpha_1(t) |f|_0 \tag{1.7}$$
where $\alpha_1(t) = 2t(1 - t/T)$ for $0 \leq t \leq T$, and $| \cdot |_0$ denotes the maximum norm.

Theorem 1. Suppose the function $f(t, x, y)$ satisfies the Lipschitz condition (1.3) and is bounded by $M$ in the domain (1.2). If the constants $a, b, c, d, M, K_1, K_2$ satisfy the conditions:
$$(A) \quad b - a \geq \dots, \quad d - c \geq \dots$$
$$(B) \quad \alpha_1(t)(K_1 + K_2) < 1$$
then there exists a sequence of functions $x_n(t, x_0)$ defined by the recurrence relation:
$$x_{n+1}(t, x_0) = x_0 + L^2 f(t, x_n(t, x_0), \dot{x}n(t, x_0)) \tag{I}$$
which converges uniformly to a periodic function $x\infty(t, x_0)$. This limit function satisfies the integro-differential equation:
$$x(t, x_0) = x_0 + L^2 f(t, x(t, x_0), \dot{x}(t, x_0)) \tag{III}$$
The proof proceeds by induction. For $m=0$, we choose $x_0(t, x_0) = x_0$. From (I), we obtain:
$$|x_1(t, x_0) - x_0| \leq \alpha_1(t) M$$
Given the constraints on the domain, we ensure that $x_m(t, x_0)$ remains within the interval $[a, b]$ and its derivative $\dot{x}_m(t, x_0)$ remains within $[c, d]$ for all $m$.

To prove convergence, we examine the difference between successive approximations:
$$|x_{m+1}(t) - x_m(t)| \leq |L^2 [f(t, x_m, \dot{x}m) - f(t, x}, \dot{x{m-1})]| \tag{1.9}$$
Applying the Lipschitz condition and the properties of the operator $L$, we derive:
$$|x(t) - x_m(t)|0 \leq Q_0 |x_m(t) - x(t)|0 \tag{1.14}$$
where $Q_0 < 1$ is a constant derived from condition (B). This ensures the uniform convergence of the sequence $x_m(t)$ to $x\infty(t)$ and $\dot{x}m(t)$ to $\dot{x}\infty(t)$.

Section 2. Relationship to the Original Differential Equation

The limit function $x_\infty(t, x_0)$ obtained in Section 1 is a solution to the original differential equation (1.1) if and only if the following condition is satisfied:
$$\Delta(x_0) = \frac{1}{T} \int_0^T f(t, x_\infty(t, x_0), \dot{x}\infty(t, x_0)) dt = 0 \tag{2.5}$$
This condition represents the "averaging" of the perturbation over one period. If there exists a value $x_0 = x^$ such that $\Delta(x^) = 0$, then $x\infty(t, x^)$ is a periodic solution of (1.1) with initial condition $x(0) = x^$.

Theorem 3. If the function $\Delta(x_0)$ changes sign or vanishes at some point $x_0 \in [a, b]$, then the system (1.1) possesses a periodic solution. In practice, we approximate $\Delta(x_0)$ using the $m$-th iteration:
$$\Delta_m(x_0) = \frac{1}{T} \int_0^T f(t, x_m(t, x_0), \dot{x}_m(t, x_0)) dt \tag{2.7}$$
The error of this approximation is bounded by:
$$|\Delta(x_0) - \Delta_m(x_0)| \leq \delta_m \tag{2.8}$$
where $\delta_m \to 0$ as $m \to \infty$.

Section 3. Numerical Application and Examples

The numerical-analytical method is particularly effective for systems with symmetries.

Theorem 5. If the function $f(t, x, y)$ satisfies the symmetry condition:
$$f(-t, x, -y) = -f(t, x, y)$$
then the system (1.1) has a periodic solution passing through $x_0 = 0$, provided the domain conditions are met. In this case, the approximating functions $x_m(t, 0)$ are odd, and the integral condition $\Delta_m(0) = 0$ is satisfied automatically for all $m$.

As an application, consider the motion of a mechanical system described by:
$$\frac{d^2\theta}{dt^2} + \nu^2 \theta = F(t) \tag{3.3}$$
where $F(t)$ is a periodic forcing function. Using the proposed method, we can determine the stability regions and the amplitude of periodic oscillations by analyzing the zeros of the functional $\Delta(x_0)$.

References

1. Samoilenko, A. M., Numerical-analytical method for investigating periodic solutions, Ukrainian Mathematical Journal, Vol. XVII, No. 4, 1965.
2. Samoilenko, A. M., On the periodic solutions of non-linear systems, Ukrainian Mathematical Journal, Vol. XVIII, No. 2, 1966.
3. Bogolyubov, N. N., Mitropolsky, Y. A., Asymptotic Methods in the Theory of Non-linear Oscillations, Moscow, 1963.