## 1 Introduction  
Chaos is an emerging interdisciplinary field developed with the rapid development of modern science and technology, especially with the emergence and widespread use of computer technology. The dynamic behavior of a chaotic system drastically depends on its initial conditions [1]. A new system is derived in this condition, even if there is a small change in the initial values. The chaotic system appears to be a random behavior, but in reality, this behavior of chaos follows a natural order. Chaotic systems are difficult to control due to their random dynamic behavior, heavy dependence on initial conditions, and pseudo-randomness [2]. Many researchers have extended their fascinating exploration of mathematics and found a variety of new simple systems that can exhibit chaotic states [3; 4; 5; 6; 7]. For example, logistic [8], [PERSON]'s [9], Chen [10], and Lu [11] systems. In these maps, initial conditions and unstable fixed points of the chaotic attractors are located nearby. Moreover, [PERSON] found in his book some new systems, simpler models of chaotic systems that are more concise in terms of the values of some system parameters, special symmetries, and dynamics [7]. Chaotic phenomena exist in the scientific fields of geology, biology, and social science. The phenomenon has ergodicity, and is nonperiodic, such as noise-like characteristics [12], making the issue of chaotic synchronous control issue become a study in the field of nonlinear science. Since the synchronization of chaos was first discovered by [PERSON] and [PERSON] in 1990 [13], there has been considerable interest in this research for its potential usage in secure communications, radars, chemical processes, and engineering applications. These depend on the designing of a strategic approach that enables the dynamic behavior of the master system to control the dynamic behavior of the slave system [14; 15; 16]. Various control methods have been proposed for synchronization of chaos, such as sliding synchronous mode control [17], the sliding-mode control approach [18], adaptive control methods [19], adaptive observer-based synchronous strategies [20], linear feedback controllers with unknown parameters [21], predictive methods of a hyperchaotic system [22], backstepping synchronization based on the equivalent transfer function method [23], and a time-delay control [24]. In [15; 25], the authors analyzed the exponential synchronization for time-delay perturbation chaotic systems with multiple positive Lyapunov exponents and the applications to secure communications. In practical engineering applications, such as communications, automation, biology, and chemistry, uncertainties due to time delays cannot be avoided, leading to multiple variations in the dynamic behavior of the actual system and increasing the uncertainty and complexity of the system model. In such time-delayed systems, the dynamic behavior of the system is more difficult to control and apply in practice. Hence the problem of stability analysis and control of systems with time delays has become a considerable and topical area for researchers in this field. Therefore, researchers have conducted various experiments on uncertainty [26; 27] and systems with time delay [28; 29; 30; 31; 32]. In [27], the nonlinear control hyperchaotic Rossler system with uncertain parameters is proposed. In [26], an adaptive synchronous strategy is introduced to control uncertain dynamical system time delays based on parameter identification. Ref. [11] presents a new state feedback control method to generate a hyperchaotic lv attractor. A delayed model with a third-degree exponential polynomial is proposed and used to control testosterone secretion [32]. Different synchronous mechanisms in nonidentical time-delay maps are studied [31]. By a fuzzy fractional-order neural network, an adaptive synchronization for a class of fractional-order time-delay uncertain chaotic systems is yielded [30]. In [29], analytical estimation is applied to synchronization in coupled time-delay systems.