Asymptotic analysis of monotonic stability of the amplitude of pendulum oscillations with small nonlinear damping

An ordinary differential equation of the second order describing free oscillations of a pendulum with small damping in the form of a third-degree polynomial is considered. The aim of the work is to analyze the monotonic stability of the amplitude of free oscillations of the pendulum with small damping, having one degree of freedom. The equation of the pendulum oscillations is written as a system of amplitude-phase equations. Then, the equation for the oscillation amplitude is averaged over the fast phase. By analyzing the expressions for the first- and second-order derivatives for the averaged amplitude, the monotonic stability of the pendulum oscillations is analyzed. The following main results are obtained in the work: the conditions of the monotonic stability of the pendulum oscillation amplitude are formulated, the region of monotonic stability is described, the number of qualitatively different cases of monotonic stability is determined, the condition for the attainability of a stable equilibrium position by the pendulum is considered. Verification of the results of the work confirmed their correctness. At the same time, they have both theoretical and applied significance. For example, they can be used in studying the stability of self-oscillations of the pendulum systems.