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ORIGINAL RESEARCH article

Front. Phys.
Sec. Interdisciplinary Physics
Volume 12 - 2024 | doi: 10.3389/fphy.2024.1529644
This article is part of the Research Topic Nonlinear Vibration and Instability in Nano/Micro Devices: Principles and Control Strategies View all 7 articles

Chaotic and Fractal Maps in Higher-Order Derivative Dynamical Systems

Provisionally accepted
Rami Ahmad El-Nabulsi Rami Ahmad El-Nabulsi 1,2*Waranont Anukool Waranont Anukool 1
  • 1 Chiang Mai University, Chiang Mai, Thailand
  • 2 Czech Academy of Science, Cezke Budejovice, Czechia

The final, formatted version of the article will be published soon.

    Hamiltonian maps are considered a class of dynamical systems that hold meticulous properties used to model a large number of complex dynamical systems. When time flows in two-dimensional degrees of freedom dynamical systems, the trajectories in phase space can be analyzed within bidimensional surfaces known as Poincaré sections. The Chirikov-Taylor standard map for two canonical dynamical variables (momentum and coordinate) is the most renewed map characterized by a family of area-preserving maps with a single parameter that control the degree of chaos. In this study, a generalization of the standard map for two different problems is presented and discussed. The first problem deals with higher-order derivative Hamiltonian system (up to the 4 th -order) since the 4 th -order character provides the possibility of chaotic behavior at all scales including nanoscales where high-order derivatives take place in nanosystems. The second problem concerns the time-dependent δ-kicked rotor in fractal dimensions characterized by a time-dependent potential due to its important implications in quantum chaos. This study shows that higher-order derivatives maps and fractal dimensional δ-kicked rotor maps apparently exhibit a large number of chaotic orbits and fractal patterns including the spiral fractals patterns comparable to Julia set. Moreover, these problems are characterized by additional parameters which can be used to control chaos. Some of these parameters lead to chaos and others lead to fractal patterns.

    Keywords: Standard map, δ-kicked rotor, higher-order derivative Hamiltonians, Fractal dimensions, chaos PACS Classification Numbers: 05.45.-a, 05.45.Df

    Received: 17 Nov 2024; Accepted: 10 Dec 2024.

    Copyright: © 2024 El-Nabulsi and Anukool. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

    * Correspondence: Rami Ahmad El-Nabulsi, Chiang Mai University, Chiang Mai, Thailand

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