Preamble

This paper, published in 1967 (Vol. III, No. 9), investigates the qualitative behavior of a system of differential equations:
$$\begin{aligned}
\dot{\theta} &= x \
\dot{x} &= -ax - \gamma \sin \theta + y^2 \sin \theta \cos \theta + L \
\dot{y} &= -ay - 2xy \cot \theta
\end{aligned}$$
where the variables are defined in the domain $0 < \theta < \pi$, $-\infty < x < +\infty$, and $y > 0$. The analysis considers three distinct cases for the parameters: a) $a = L = 0$; b) $a \neq 0, L = 0$; and c) $a \neq 0, L \neq 0$. The study builds upon the foundational methods established in \cite{1}.

Section 1. The Case $a = L = 0$

In the absence of damping ($a=0$) and external torque ($L=0$), the system (1) possesses a first integral. Specifically, in the domain $D$, there exists a function $S$ such that:
$$S = x^2 + y^2 \sin^2 \theta + \sin^2 \frac{\theta}{2} = h$$
The trajectories of the system (1) are constrained to the surfaces defined by (3). Analysis of the phase portrait in the $\theta y$-plane shows that for $x > 0$, the trajectories move toward increasing $\theta$, while for $x < 0$, they move toward decreasing $\theta$.

The stability of the equilibrium points and the behavior of the solutions $x(t)$ and $y(t)$ as $t \to +\infty$ are governed by the conservation of $S$. Using the methods described in \cite{2}, we can demonstrate that the integral surfaces (3) are closed in the phase space. Furthermore, the symmetry of the system implies that if $\theta(t)$ is a solution, then $\theta(-t)$ and $y(-t)$ also relate to the system's dynamics through $x(-t)$.

Section 2. The Case $a \neq 0, L = 0$

When damping is introduced ($a > 0$) but the external torque remains zero ($L = 0$), the function $V$ (analogous to the energy integral) serves as a Lyapunov function. We define:
$$V = x^2 + y^2 \sin^2 \theta + \sin^2 \frac{\theta}{2}$$
Taking the derivative of $V$ along the trajectories of (1), we obtain:
$$\dot{V} = -2a(x^2 + y^2 \sin^2 \theta) \leq 0$$
This inequality holds for $0 < \theta < \pi$ and $y > 0$. Since $\dot{V}$ is negative semi-definite, the system is dissipative. For any initial condition in the domain (4), the limit of the trajectory as $t \to \infty$ approaches the equilibrium point $(0, 0, 0)$.

By integrating the relation for $\dot{V}$, we can establish bounds on the rate of decay:
$$V(t) \leq V(t_0) \exp[-2a(t - t_0)]$$
This ensures that all trajectories starting within the region defined by (4) eventually converge to the origin. To further analyze the asymptotic behavior, we introduce a coordinate transformation $u = x \sin \theta$. The transformed system allows us to apply the comparison theorems from \cite{2} to show that $\sup |u(t)| \to 0$ and $\inf y(t) > 0$ under specific conditions.

Section 3. The Case $a \neq 0, L \neq 0$

In the general case where both damping and external torque are present ($a > 0, L \neq 0$), we assume $L < \gamma$. The system possesses two equilibrium points in the domain:
- Point $A$: $(\theta_0, 0, 0)$, where $\theta_0 = \arcsin(L/\gamma)$
- Point $B$: $(\pi - \theta_0, 0, 0)$

Point $A$ is a stable equilibrium. By linearizing the system around $A$, we find that the eigenvalues of the Jacobian matrix have negative real parts, confirming local asymptotic stability. Conversely, point $B$ is an unstable equilibrium of the saddle type.

The phase space is divided into two regions of attraction, $D_1$ and $D_2$. Trajectories in $D_1$ converge to the stable equilibrium $A$ as $t \to +\infty$. In $D_2$, the behavior is more complex; however, it can be shown that for certain initial conditions, $\theta(t) \to \pi$ and $x(t) \to 0$, while $y(t)$ may grow or diminish depending on the specific parameters of the damping.

The integral curves in the vicinity of the unstable point $B$ are characterized by the function:
$$x^2 + y^2 \sin^2 \theta + 2 \int f(u) du = 0$$
where $f(u) = \gamma \sin u - L$. This analysis confirms that the presence of the torque $L$ shifts the equilibrium positions and alters the global topology of the phase portrait compared to the $L=0$ case.

References

1. [Author Name], [Journal/Book], 1949.
2. [Author Name], [Journal/Book], 1960.
3. [Author Name], [Journal/Book], Vol. 86, No. 3, pp. 453–456, 1952.