Eur. Phys. J. Plus (2020) 135: 32
https://doi.org/10.1140/epjp/s13360-019-00006-1

Regular Article

Dynamics of distributed-order hyperchaotic complex van der Pol oscillators and their synchronization and control

¹ Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71516, Egypt
² Department of Basic Science, College of Computer and Information Sciences, Majmaah University, Al-Majmaah, 11952, Saudi Arabia
³ Mathematics Department, Faculty of Science, King Khalid University, Abha, 61418, Kingdom of Saudi Arabia
⁴ Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, 71511, Egypt
⁵ Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China
⁶ Department of Mathematics, Faculty of Science, Sohag University, Sohâg, 82524, Egypt

^* e-mail: gmahmoud@aun.edu.eg
^** e-mail: gmahmoud_56@yahoo.com

Received: 22 July 2019
Accepted: 4 September 2019
Published online: 9 January 2020

Abstract

The distributed-order hyperchaotic unforced and forced complex van der Pol oscillators with complex parameter are introduced and investigated in this paper. The basic dynamical properties including equilibrium point and its stability and chaotic behavior of the unforced oscillator are studied. The intervals of the parameters values at which this oscillator has periodic, chaotic, and hyperchaotic behaviors are calculated using Lyapunov exponents. These intervals of chaotic and hyperchaotic behaviors can be used in many applications such as secure communication and electronic circuits. Using the linear feedback control, the control of solutions of our oscillator(unforced) converge to a fixed point are studied. We state a scheme to achieve the complete synchronization between two distributed-order hyperchaotic unforced complex van der Pol oscillators. The analytical formula of the controller is derived and used to achieve synchronization. Secure communications via hyperchaotic masking for a text which contains alphabets, numbers, space, and symbols are investigated using the proposed scheme of this work. The dynamics of the distributed-order hyperchaotic forced complex van der Pol oscillator with complex parameter is investigated. Synchronization and secure communications can be similarly studied for the forced oscillator.