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.

Figures (9)

• Figure 1: The planar elastic pendulum composed of a spring of natural length $l_0$ and stiffness $k$. Throughout this paper Cartesian canonical coordinates are used.
• Figure 2: Example of trajectories in configuration space $(q_1, q_2)$. Panel (a), (c) and (e) illustrates oscillations, (b) and (e) show a rotation trajectories where the pendulum completes a full circle. Panel (d) exhibits an example of a chaotic trajectory. Parameters are $\mu = 4$ and $R = 1$ while initial conditions are $q_1 = p_2 = 0$ and $q_2 =$$-0.7905$ for (a), $0.75$ for (b), $-2.505$ for (c), $0.5$ for (d), $-1.0$ for (e) $-1.5$ for (f).
• Figure 3: Three-dimensional visualization of the phase space volume for parameters $\mu = 4$ and $R = 1$ in the $(q_1, q_2, p_2)$ subspace. The central 3D scatter plot illustrates the spatial distribution of oscillating (blue), circular/rotating (green), and chaotic (red) trajectories. The surrounding two-dimensional panels display horizontal cross-sections of this volume at various values of $q_1$, highlighting the internal structure and boundaries of the different dynamics.
• Figure 4: Poincaré sections at the plane $q_1 = 0$ with $p_1 > 0$, illustrating the phase space structure for different parameter combinations. The panels correspond to $(\mu, R)$ values of: (a) $(4, 1)$, (b) $(2.5, 6)$, (c) $(6, 6)$, and (d) $(7, -0.5)$. The diagrams reveal the coexistence of regular regions (invariant tori corresponding to oscillating or rotating motion in blue or green respectively) and chaotic seas (in red), which vary significantly depending on the chosen parameters.
• Figure 5: Fraction of oscillatory, rotational, and chaotic trajectories across the $(\mu, R)$ parameter plane. Each panel shows the fraction of trajectories of a given type, estimated from $10^4$ initial conditions sampled randomly within the isoenergetic volume for each parameter pair; the color scale ranges from $0$ to $1$. The dotted line marks the minimum energy threshold for rotational motion, given by Eq. \ref{['eq:rotation_energy']}. The dashed line indicates the location of maximum chaotic fraction in the parameter plane, whose linear fit is discussed in the text, Eq. \ref{['eq:maximums']}.