Fractional damping enhances chaos in the nonlinear Helmholtz oscillator
Abstract
The main purpose of this paper is to study both the underdamped and the overdamped dynamics of the nonlinear Helmholtz oscillator with a fractional-order damping. For that purpose, we use the Grünwald–Letnikov fractional derivative algorithm in order to get the numerical simulations. Here, we investigate the effect of taking the fractional derivative in the dissipative term in function of the parameter . Our main findings show that the trajectories can remain inside the well or can escape from it depending on which plays the role of a control parameter. Besides, the parameter is also relevant for the creation or destruction of chaotic motions. On the other hand, the study of the escape times of the particles from the well, as a result of variations of the initial conditions and the undergoing force F, is reported by the use of visualization techniques such as basins of attraction and bifurcation diagrams, showing a good agreement with previous results. Finally, the study of the escape times versus the fractional parameter shows an exponential decay which goes to zero when is larger than one. All the results have been carried out for weak damping where chaotic motions can take place in the non-fractional case and also for a stronger damping (overdamped case), where the influence of the fractional term plays a crucial role to enhance chaotic motions. We expect that these results can be of interest in the field of fractional calculus and its applications.
Description
This paper aims to explore both underdamped and overdamped dynamics in the nonlinear Helmholtz oscillator with fractional-order damping. Utilizing the Grünwald–Letnikov fractional derivative algorithm, numerical simulations are conducted to investigate the impact of the fractional derivative in the dissipative term concerning the parameter α. Results demonstrate that trajectories may either remain within the well or escape depending on α, acting as a control parameter, and also influence the creation or suppression of chaotic motions. Visualization techniques such as basins of attraction and bifurcation diagrams are employed to analyze the escape times of particles from the well due to variations in initial conditions and external force F, consistent with prior findings. Additionally, the study reveals an exponential decay in escape times with respect to the fractional parameter α, converging to zero for α greater than one. Notably, the results are obtained for weak damping scenarios where chaotic motions occur in the non-fractional case, as well as for stronger damping situations (overdamped case), where the fractional term significantly influences chaotic behaviors. These findings hold implications for the field of fractional calculus and its practical applications.
Collections
- Artículos de Revista [4507]
Los ítems de digital-BURJC están protegidos por copyright, con todos los derechos reservados, a menos que se indique lo contrario