Chaotic and Resonant Behaviors in a Coupled Spring–Pendulum System: A Numerical Study

We investigate the dynamics of a coupled spring–pendulum in a specific laboratory configuration: a mass constrained to move vertically by a linear spring and connected to a pendulum swinging beneath it. The system is cast in a Hamiltonian framework, and the first-order canonical equations are integrated numerically with tight tolerances. Diagnostics combine time-series readouts, phase-space portraits, and Poincaré sections. Chaos is confirmed via the maximum Lyapunov exponent using a two-trajectory method with periodic renormalization. Varying only the initial values reveals a coherent transition pathway. At small angles and low energy, the response is nearly harmonic with thin, regular phase orbits. Near the internal resonance, envelope modulation and structured intermodal energy exchange emerge. Added initial momentum and slight detuning, the system crosses into a chaotic regime marked by a positive Lyapunov exponent and thickened, filamentary phase traces. Beyond these findings, the study contributes a reproducible and transferable workflow. Phase–time juxtapositions are used to tie geometric features to their temporal causes, improving interpretability. The Hamiltonian form provides a transparent check on conserved energy, offering a solver-agnostic pass criterion for numerical validation. Practically, the results show how initial values and detuning act as operational knobs, allowing one to induce or avoid transitions without hardware changes. Pedagogically, the system serves as an accessible bridge from linear oscillations to nonlinear dynamics in classroom or laboratory settings.