ech T Press Science Computers, Materials & Continua DOI:10.32604/cmc.2021.013395 Article Synchronization Phenomena Investigation of a New Nonlinear Dynamical System 4D by Gardano’s and Lyapunov’s Methods Abdulsattar Abdullah Hamad 1 , Ahmed S. Al-Obeidi 2 , Enas H. Al-Taiy 2 , Osamah Ibrahim Khalaf 3 and Dac-Nhuong Le 4,5, * 1 School of Mathematics, Madurai Kamaraj University, Madurai, 625021, India 2 Specialty of Mathematics, Gifted School of Nineveh, Mosul, Iraq 3 Al-Nahrain University, Baghdad, 70030, Iraq 4 Institute of Research and Development, Duy Tan University, Danang, 550000, Vietnam 5 Faculty of Information Technology, Duy Tan University, Danang, 550000, Vietnam * Corresponding Author: Dac-Nhuong Le. Email: ledacnhuong@duytan.edu.vn Received: 05 August 2020; Accepted: 07 September 2020 Abstract: Synchronization is one of the most important characteristics of dynamic systems. For this paper, the authors obtained results for the nonlinear systems controller for the custom Synchronization of two 4D systems. The fndings have allowed authors to develop two analytical approaches using the second Lyapunov (Lyp) method and the Gardano method. Since the Gardano method does not involve the development of special positive Lyp functions, it is very effcient and convenient to achieve excessive system SYCR phenomena. Error is overcome by using Gardano and overcoming some problems in Lyp. Thus we get a great investigation into the convergence of error dynamics, the authors in this paper are interested in giving numerical simulations of the proposed model to clarify the results and check them, an important aspect that will be studied is Synchronization Complete hybrid SYCR and anti-Synchronization, by making use of the Lyapunov expansion analysis, a proposed control method is developed to determine the actual. The basic idea in the proposed way is to receive the evolution of between two methods. Finally, the present model has been applied and showing in a new attractor, and the obtained results are compared with other approximate results, and the nearly good coincidence was obtained. Keywords: Chaos; Lu model; anti-synchronization; hybrid synchronization; Gardano’s method; nonlinear dynamical system 1 Introduction Chaotic systems with real state variables are boing found and studied with increased attention in several aspects of nonlinear dynamical systems, the frst physical and mathematical model of a chaotic system is the system of Lorenz, which only includes real variables discovered in 1963 and opens the way to other chaos systems such as the system of Chen, Lu’s system (2002), Liu’s This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.