Extreme rotational events in a forced-damped nonlinear pendulum (2304.00039v1)

Abstract: Since Galileo's time, the pendulum has evolved into one of the most exciting physical objects in mathematical modeling due to its vast range of applications for studying various oscillatory dynamics, including bifurcations and chaos, under various interests. This well-deserved focus aids in comprehending various oscillatory physical phenomena that can be reduced to the equations of the pendulum. The present article focuses on the rotational dynamics of the two-dimensional forced damped pendulum under the influence of the ac and dc torque. Interestingly, we are able to detect a range of the pendulum's length for which the angular velocity exhibits a few intermittent extreme rotational events that deviate significantly from a certain well-defined threshold. The statistics of the return intervals between these extreme rotational events are supported by our data to be spread exponentially. The numerical results show a sudden increase in the size of the chaotic attractor due to interior crisis which is the source of instability that is responsible for triggering large amplitude events in our system. We also notice the occurrence of phase slips with the appearance of extreme rotational events when phase difference between the instantaneous phase of the system and the externally applied ac torque is observed.

• The paper identifies extreme rotational events triggered by an interior crisis in a forced-damped nonlinear pendulum system.
• It uses numerical analysis to show that return intervals between extreme events follow an exponential distribution, indicating non-random behavior.
• The paper highlights that phase slips during transitions from librational to rotational motion contribute significantly to system instability.

Extreme Rotational Events in a Forced-Damped Nonlinear Pendulum

The paper "Extreme Rotational Events in a Forced-Damped Nonlinear Pendulum" investigates the manifestation of intermittent extreme rotational events in a two-dimensional forced damped pendulum system, under the influence of alternating current (ac) and direct current (dc) torque. The authors explore how chaotic dynamics influence the pendulum’s angular velocity, leading to intermittent large amplitude events, termed extreme rotational events (EREs).

Summary of Results

The paper examines chaotic dynamics and interior crisis in a nonlinear pendulum model to quantify the emergence of EREs and their statistical characteristics. The authors detect a specific range of pendulum lengths where angular velocities deviate intermittently and significantly from a defined threshold, exhibiting EREs. Numerical analysis suggests that the return intervals between these events follow an exponential distribution, indicating the presence of underlying patterns in what may initially appear as stochastic behavior.

Key findings include:

• Chaotic Dynamics and Interior Crisis: The pendulum system undergoes sudden changes in its chaotic attractor size due to an interior crisis. This phenomenon is identified as the source of the instability responsible for triggering large amplitude events.
• Phase Slips: The transition of the system between librational and rotational motion during the appearance of EREs is associated with phase slips, denoting significant changes in the phase difference between the system's instantaneous phase and the external ac torque.
• Statistical Regularities: The statistical analysis reveals that the inter-event intervals of EREs fit the exponential distribution, further confirming the presence of nonlinear dynamical processes governing their occurrence.

Implications and Future Directions

From a theoretical perspective, this work contributes insights into the behavior of nonlinear dynamical systems subject to deterministic chaos and interior crises. Understanding these events could significantly impact fields where dynamic stability under perturbative forces is critical, such as engineering and climate systems.

Practical implications of this research can potentially extend to designing more robust systems that anticipate and mitigate the risks associated with these irregular extremes, akin to mitigating rogue waves or market crashes through enhanced understanding of system dynamics.

Future work could investigate coupled nonlinear pendula to characterize chaos-induced extreme events in more complex systems. Moreover, verifying the experimental observability of such extreme events could deepen our understanding of their mechanisms and lead to real-world applications.

In summary, the paper offers valuable observations on the emergence and statistical attributes of extreme rotational events in nonlinear pendular dynamics, presenting a robust framework for analyzing similar phenomena across various scientific disciplines.