Preamble

This study investigates the dynamics of a coupled system of differential equations, building upon the foundational work established in \cite{1} and \cite{2}. We consider a system of the form:

$$\begin{aligned} \ddot{x} + a\dot{x} + f(x) &= N(Q - x) + kF(y - x) \ \ddot{y} + a\dot{y} + f(y) &= N(Q - y) - kF(y - x) \end{aligned} \tag{1}$$

Under the assumptions (A), let $f(x)$ be a periodic function such that $f(x_1) = f(x_2) = 0$ for $x_1 - x_2 = 2\pi$. We assume $f'(0) > 0$ and $f(x) > 0$ for $x \in (0, x_1)$. The parameters $N, Q,$ and $k$ are constants. The coupling function $F(z)$ is assumed to be bounded, $|F(z)| \leq 1$, and we specifically consider cases where $f(x) = \sin x$. This system describes the synchronization of coupled oscillators under external torque $N(Q-x)$, as discussed in \cite{3}.

By introducing the variables $u = \dot{x}$ and $v = \dot{y}$, we can rewrite the system (1) as a set of first-order equations. Following the methodology in \cite{4}, we define the auxiliary function $\Phi(x)$ to simplify the analysis of the phase trajectories:

$$\Phi(x) = f(x) - NQ + k \tag{4}$$

We assume the existence of a periodic solution such that $\Phi(x + 2\pi) = \Phi(x)$ and $\int_0^{2\pi} \Phi(x) dx < 0$ (6), (7). For the system to exhibit stable equilibrium points, the parameters $Q, N,$ and $k$ must satisfy the condition:

$$0 < Q < \frac{\max f(x) \pm k}{N} \tag{10}$$

As shown in [FIGURE:1], the phase portrait of the system is characterized by the behavior of the function $\Phi(x)$. Let $\eta_0$ be a root such that $\Phi(\eta_0) = 0$. We define the regions of stability based on the sign of $\Phi(x)$ relative to the equilibrium points $\eta_1$ and $\eta_2$, where $\eta_1 - \eta_2 = 2\pi$.

The existence of limit cycles and the stability of the rotational motions are governed by the energy integrals $S_1$ and $S_2$ (see [FIGURE:2]). According to the criteria in \cite{2}, the condition for the absence of out-of-phase rotations is given by the inequality:

$$S_1(\eta_0) > S_2(\eta_0) \tag{14}$$

For the trajectories $u = S_1(x)$ in the interval $\eta_0 < x < \eta_1$, we can derive the following upper bound:

$$S_1(x) < (a + N)(\eta_1 - x) + \sqrt{2 \int_x^{\eta_1} \Phi(x) dx} \tag{16}$$

Conversely, for the trajectory $u = S_2(x)$ in the interval $\eta_2 < x < \eta_0$, the lower bound is expressed as:

$$S_2(x) > (a + N)(\eta_2 - x) + \sqrt{2 \int_{\eta_2}^x \Phi(x) dx} \tag{20}$$

Based on these estimates, we can formulate several theorems regarding the global stability of the system.

Theorem 1. If the parameters satisfy the inequality:
$$a > \frac{N(-x_2)}{\ln(a+N)} + \sqrt{2J} + \dots \tag{21}$$
then the system (1) does not possess limit cycles of the second kind, and all trajectories converge to the equilibrium state (7).

Theorem 2. For the specific case $f(x) = \sin x$, the condition for synchronization (the absence of asynchronous rotations) is satisfied if:
$$a > \frac{\sqrt{2J} - (a+N)\pi}{\pi} + \dots \tag{24}$$
where $J$ represents the integral of the restoring force over a period.

Further analysis of the phase space shows that when $k < NQ$, the coupling term $kF(y-x)$ acts as a restorative force that facilitates synchronization. If the damping coefficient $a$ is sufficiently large, specifically satisfying:
$$\frac{\sqrt{2J} - 4(a+N)\pi - (a+N)x_1}{\sqrt{2J} - 4(a+N)\pi} > 0 \tag{30}$$
then the system (1) is guaranteed to reach a synchronous state regardless of the initial conditions. These results extend the classical stability criteria for pendulum-like systems to coupled oscillators with linear dissipation and constant external torque.

References

1. V. S. [Author], Journal of Moscow University, No. 4, pp. 122–125, 1965.
2. V. M. [Author], Journal of Applied Mathematics, 5 (24), pp. 61–68, 1961.
3. A. [Author] and Khaikin, S. E., Theory of Oscillations, 1959.
4. V. S. [Author] and E. A. [Author], Journal of Technical Physics, No. 2 (21), pp. 137–146, 1961.