Eur. Phys. J. Spec. Top. (2022) 231: 1931-1968
https://doi.org/10.1140/epjs/s11734-022-00468-y

Regular Article

Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique

¹ Escuela Nacional de Estudios Superiores Unidad Juriquilla, Universidad Nacional, Autónoma de México, Boulevard Juriquilla 3001, Juriquilla La Mesa, 76230, Juriquilla, QRO, México
² Université de Technologie de Compiègne, CNRS, UMR 7253 Heudiasyc, Compiègne Cedex, France
³ CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, 62490, Cuernavaca, MOR, México
⁴ Department of Mathematics, Faculty of Science, King Khalid University, 61413, Abha, Kingdom of Saudi Arabia
⁵ Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt

^c jose.ga@cenidet.tecnm.mx

Received: 27 March 2021
Accepted: 13 January 2022
Published online: 16 February 2022

Abstract

This research deals with a comparative numerical analysis of chaos in two systems with non-integer derivatives. The one-scroll system and circle equilibrium system with different hidden attractors are simulated considering the fractal derivative, Khalil and Atangana conformable derivatives, and the truncated M-derivative considering a constant and variable-order. Phase portraits are shown, as well as bifurcation diagrams, and Lyapunov exponents are obtained. Later, 0–1 test, dynamic death analysis, and sensitivity to initial conditions are considered to choose which derivative produces richer chaotic behaviors. According to those mentioned above, we could observe that the M-derivative not only generalizes Khalil’s type conformable derivative but also its two non-integer orders produce interesting dynamic behaviors compared to the remaining derivatives. In the numerical results, we observe that the variable order makes the system more sensitive to the change in the initial conditions. The new chaotic behaviors with constant and variable order are used to develop a synchronization scheme of two identical one-scroll systems via the backstepping technique with the truncated M-derivative involved.