Fractional damping enhances chaos in the nonlinear Helmholtz oscillator
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ORIGINAL PAPER
Fractional damping enhances chaos in the nonlinear Helmholtz oscillator Adolfo Ortiz · Jianhua Yang · Mattia Coccolo · Jesús M. Seoane · Miguel A. F. Sanjuán
Received: 30 July 2020 / Accepted: 2 November 2020 © Springer Nature B.V. 2020
Abstract The main purpose of this paper is to study both the underdamped and the overdamped dynamics of the nonlinear Helmholtz oscillator with a fractionalorder 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 α. Our main findings show that the trajectories can remain inside the well or can escape from it depending on α which plays the role of a control parameter. Besides, the parameter α 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 A. Ortiz Centro de Investigación en Micro y Nanotecnología, Facultad de Ingeniería, Universidad Veracruzana, Calz. Ruiz Cortinez 455, CP 94294 Boca del Río, Veracruz, México J. Yang School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China M. Coccolo (B) · J. M. Seoane · M. A. F. Sanjuán Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, Móstoles, 28933 Madrid, Spain e-mail: [email protected]
results. Finally, the study of the escape times versus the fractional parameter α shows an exponential decay which goes to zero when α 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. Keywords Nonlinear oscillations · Delay systems · Resonance · Fractional derivatives
1 Introduction Fractional calculus is an area of the mathematical analysis. It studies the possibilities of taking real or even complex numbers as orders of the integral and derivatives on a known or unknown function. Such operators are rather useful in science and engineering. Also, fractional differential operators involve defined integrals over a time domain, and this poses significant memory effects as shown in Ref. [1]. For these reasons, the fractional calculus has gained much attention and relevance in the past few years due to its applications to several research fields such as control systems, nonlinear oscillators, potential fields, diffusion problems, viscoelasticity and rheo
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