Phase-space organization of the elastic pendulum: chaotic fraction, energy exchanges, and the order-chaos-order transition

Abstract

We study the phase-space organization of the planar elastic pendulum as a function of its two dimensionless control parameters: the reduced energy $R$ and the squared frequency ratio $μ$. By randomly sampling the isoenergetic volume to classify trajectories as oscillatory, rotational, or chaotic across the $(μ, R)$ parameter plane, we obtain a global portrait of the coexistence and competition between dynamical regimes. The chaotic fraction is not uniformly distributed across the parameter plane but concentrates in a well-defined central cloud whose ridge follows a linear relation in the $(μ, R)$ plane and whose maximum does not exceed $70\%$ of the available phase space. The order-chaos-order transition is not a global property of the parameter plane but occurs specifically in the central region surrounding this cloud: along paths that traverse it, oscillatory orbits progressively give way to chaotic trajectories, which in turn yield to rotational orbits as the energy grows, revealing a clear sequential mechanism underlying the transition. The onset of rotational motion is gradual rather than sharp, reflecting a strong dependence on initial conditions. By decomposing the total energy into spring-like, pendulum-like, and coupling contributions, we establish a direct correspondence between the coupling power and the abundance of chaotic trajectories, showing that enhanced inter-mode energy exchange is a reliable indicator of dynamical complexity. These results provide a comprehensive and quantitative map of the dynamical regimes of the elastic pendulum, clarifying the structure of the chaotic cloud and connecting it to the underlying mode-coupling mechanisms.