Energy-based analysis of quadratically coupled double pendulum with internal resonances

This article presents an energy-based method for examining the dynamics of an autoparametric double pendulum with quadratic coupling. The method is based on decomposing the total energy (Hamiltonian) into partial terms and studying these terms in the time domain. The article then demonstrates using this method to analyse internal resonances in various regimes, including regular periodic, quasiperiodic, and chaotic motion. This study also explores higher-order resonances linked to the geometric tuning parameter and initial energy, which enrich the complexity of the system. The work focuses on identifying conditions that lead to the development of internal resonances and analysing energy exchange mechanisms at various initial energy levels. This analysis uses Fast Lyapunov Indicators and Poincaré maps for these higher energy levels to identify chaotic regimes. Within these regimes, the assessment of coupling energy contributes to a deeper understanding of the studied system. Moreover, the research demonstrates how energy-based methods can be used to investigate auto-parametric systems.