A New No Equilibrium Fractional Order Chaotic System, Dynamical Investigation, Synchronization, and Its Digital Implementation

Abstract

In this paper, a new fractional order chaotic system without equilibrium is proposed, analytically and numerically investigated, and numerically and experimentally tested. The analytical and numerical investigations were used to describe the system’s dynamical behaviors including the system equilibria, the chaotic attractors, the bifurcation diagrams, and the Lyapunov exponents. Based on the obtained dynamical behaviors, the system can excite hidden chaotic attractors since it has no equilibrium. Then, a synchronization mechanism based on the adaptive control theory was developed between two identical new systems (master and slave). The adaptive control laws are derived based on synchronization error dynamics of the state variables for the master and slave. Consequently, the update laws of the slave parameters are obtained, where the slave parameters are assumed to be uncertain and are estimated corresponding to the master parameters by the synchronization process. Furthermore, Arduino Due boards were used to implement the proposed system in order to demonstrate its practicality in real-world applications. The simulation experimental results were obtained by MATLAB and the Arduino Due boards, respectively, with a good consistency between the simulation results and the experimental results, indicating that the new fractional order chaotic system is capable of being employed in real-world applications.

Uruk 4d Discrete Chaotic Map for Secure Communication Applications

“…Dong [6] described a new autonomous chaotic system with two stable node-foci capable of producing double-wing hidden chaotic attractors by modifying a simple three-dimensional continuous quadratic dynamical system. Rahman et al [7] proposed a new fractional order chaotic system devoid of equilibrium. Due to its lack of equilibrium, it has the ability to elicit subtle, untidy attractants.…”

Analysis of the dynamics of new models of nonlinear systems with state variable damping and elastic coefficients

“…Owing to their broad importance in many fields and their numerous engineering applications, nonlinear oscillators have recently attracted the attention of a large number of researchers due to their extremely rich dynamics ( Joshi, 2021 ; Dashkovskiy and Pavlichkov, 2020 ; Ahmed et al, 2017 ; Cheng and Zhan, 2020 ; Kudryashov, 2018 ; Tang et al, 2020 ; Dashkovskiy and Pavlichkov, 2020 ; Fonkou et al, 2022 ; Han et al, 2019 ; Ramirez et al, 2020 ; FitzHugh, 1961 ; Rahman et al, 2021a , Rahman et al, 2021b , Rahman et al, 2021c , Rahman et al, 2021d ). Their fields of application are between others: seismology, communication and neurophysiology ( Rahman et al, 2021a , Rahman et al, 2021b , Rahman et al, 2021c , Rahman et al, 2021d ; Lucero and Schoentgen, 2013 ; Rowat and Selverston, 1993 ; Balachandran and Kandiban, 2009 ). These oscillators exhibit rich dynamics among which limit cycle oscillations of sinusoidal and relaxation nature, since one of their important characteristic is their capacity to present limit cycle behaviors which is an important criterion in the characterization of the artificial pacemaker ( Steeb, 1977 ; Hochstadt and Stephan, 1967 ; D'Heedene, 1996 ; Steeb and Kunick, 1987 ; Steeb et al, 1983 ).…”

“…When subjected to an external periodic excitation, numerical studies and singular point analysis have revealed chaotic behaviors, allowing the analysis of phenomena such as control and cardiac activity with numerous technological applications. ( Steeb and Kunick, 1982 ; Steeb and Kunick, 1983 ; Forger, 1999 ; Enrique et al, 2020 ; Rahman et al, 2019 ; Kai and Tomita, 1979 ; Rahman et al, 2021a , Rahman et al, 2021b , Rahman et al, 2021c , Rahman et al, 2021d ; Van der Pol and Van der Mark, 1926 ; Van der Pol and Van der Mark, 1928 ; Alhasnawi et al, 2021 ).…”

Multistability route in a PWL multi-scroll system through fractional-order derivatives

“…With the recent increase of studies and experiments with fractional order systems, the possibilities of finding new behaviors and better descriptions of natural phenomena are a recurring theme in the literature [22][23][24][25][26][27][28][29][30]. However, the use of this numerical tool has been neglected because it is used as a dynamical validation mechanism and the effects and physical implications associated with the use of fractional order derivatives are ignored.…”