Abstract:

The Damped Oscillatory Logistic Growth (DOLG) Model is introduced as a novel hybrid framework that integrates oscillatory dynamics, damping, and logistic growth into a single differential equation. This model extends classical systems such as the harmonic oscillator, logistic growth equation, and damped systems by combining their key features into a unified framework. Numerical solutions reveal rich dynamical behaviors, including damped oscillations, stabilization to carrying capacity, and phase-dependent growth patterns. The system’s stability is analytically and numerically confirmed, with trajectories converging to the non-trivial equilibrium (x,v) = (K,0) for all parameter regimes. The effects of damping, growth rate, and oscillation frequency are explored through time series and phase portraits, demonstrating the model’s versatility in capturing complex phenomena. Potential applications span ecology, economics and engineering, offering new insights into oscillating populations, cyclical growth, and mechanical systems with growth constraints. This study lays the groundwork for future research on hybrid dynamical systems and their interdisciplinary applications.