(untitled) (European Journal of Pure and Applied Mathematics, 2025)
This research shows an analytical and numerical analysis of a nonlinear three-dimensional dynamical system controlled by a unique control parameter, named α. The system exhibits self-excited oscillations through local bifurcations for negative α, including saddle-node and supercritical Hopf bifurcations, leading to periodic orbits and a sequence of period-doubling transitions into chaos. On the other hand, when α is negative and there are no equilibrium points, the system shows long term oscillations that last for a long time through hidden attractors—bounded chaotic dynamics with basins of attraction that are not connected to any equilibrium. Numerical continuation and Lyapunov spectrum analysis confirm the simultaneous existence of periodic, quasiperiodic, and chaotic regimes. The results demonstrate the intricate interplay between local bifurcations and global nonlinear frameworks, emphasizing the distinctive routes to chaos and the emergence of hidden dynamics in systems lacking stability.