## 1 Introduction
An iterative learning control strategy has been proposed to synchronize two chaotic systems with a free couple and free time delay [28]. On the other hand, this has given rise to a variety of chaotic signal synchronous control mechanisms and programs, with only one positive Lyapunov exponent of low-dimensional chaotic system complexity, resulting in a less secure confidential system [33]. However, the chaotic time-delay system possesses an infinite-dimensional state space and abundant dynamic behaviors in nonlinear fields. For example, Lyapunov exponents for complex systems with delayed feedback are studied [34]. [PERSON] et al. proposed a novel control method for heterogeneous uncertain chaotic systems with time delay, in which a robust framework for synchronous error estimation is presented [35]. Moreover, it has been used for secure communications through multiple heterogeneous chaotic systems in engineering applications. Using synchronized chaotic systems for real-world applications largely relies on circuit implementations. Thus, the chaotic system of circuit simulation has widespread concern from researchers in this field [36; 37]. Ref. [37] proposed a new four-dimensional chaotic system of a circuit simulation program to map out the trajectory of the high-dimensional chaotic attractor. In [38], the author presented numerical simulations that assume a crucial role in their investigation due to chaotic systems being so challenging to understand analytically. Numerous researchers are confident in the relevance of system modeling and circuit simulation, developing new models with components that can present unpredictable, chaotic behavior [39; 40; 41]. Models of systems and circuits are applicable in different fields and initiated the exploration of other chaotic oscillation circuits [42; 43; 44]. [PERSON] designed a simple chaotic jerk circuit in 2011 [40], which includes only a few electronic components, such as one inductor, two capacitors, one adaptive resistor, and a nonlinear resistor. His works illustrated similar chaotic behavior to the trajectories around the equilibriums of the system. [PERSON] was the first to propose continuous-time chaotic systems [45]. After that, various discrete-time chaotic hyperchaotic systems have been proposed in the literature [46; 47; 48; 49; 50], while typical ordinary differential equations are used to represent most types. Chaotic behavior involves exploiting the time-delay paths in autonomous Boolean networks [51; 52]. Furthermore, chaos was initially implemented with electronic logic circuits (logic gates and field-programmable gate arrays FPGAs) [53; 54]. In [55], dynamical behaviors are analyzed using two new asymmetric chaos circuit oscillators with no dependence on incommensurate time delays. One of the items that can be studied for uncertain time-delay systems is the broadband, nonperiodic and unpredictable time-delay system. On the other hand, time-delay chaos systems can exhibit more complex hyper-chaotic behaviors. Many positive Lyapunov exponents could be generated even if a low-dimensional time-delay system is used. Those properties offered by chaotic time-delay signals are suitable for signals used in electronic engineering and information processing in particular secure communication systems. Furthermore, compared to the typical low-dimensional chaotic system well studied, the chaotic time-delay system, which has more complex hyper-chaotic behaviors, is rarely reported. Significantly, the chaotic time-delay circuit hinders the practical application of the randomness time-delay systems in secure communication. Accordingly, this research aims to develop a novel chaotic time-delay system and asymptotical synchronous conditions based on the Lyapunov-Krasovskii functional theory.