## 7 Conclusions  
In this paper, approximate LS and AS in two unidirectionally coupled hyperchaotic Chen systems without time-delay coupling are analytically investigated. Firstly, the synchronization condition for exact LS between two unidirectionally coupled hyperchaotic Chen systems with time delay in signal transmission is obtained. Differently from other studies, LS is viewed as a special type of GS, which can be analyzed by using the auxiliary system approach. Based on the Laplace transform and the successive approximation method, the synchronization condition for exact LS is analytically obtained, and its effectiveness is verified using numerical simulations. Our analysis shows that the condition for exact LS is valid for any value of the time delay in signal transmission. Under such conditions, approximate LS and AS in two unidirectionally coupled hyperchaotic Chen systems without time-delay coupling are discussed by replacing the true time-delay terms with their Taylor expansions up to the third order. For a small time delay, the approximate LS and AS are precise enough. Our research indicates the possibility of recreating the past signals or predicting the future signals of a hyperchaotic Chen system using its current signals.  
In previous research studies, the true time-delay terms are first replaced by using their first-order approximations. Then, synchronization conditions for approximate LS and AS are discussed by considering two chaotic systems with parameter mismatch. The lag and anticipation time of approximate LS and AS have to be involved in the synchronization condition, which are difficult to change within a wide range. In our study, the synchronization condition for exact LS between the drive and response systems with time-delay coupling is firstly obtained, which has nothing to do with the time delay. That is, the condition for exact LS is valid for any value of the time delay. Then, the value of the lag and anticipation time in approximate LS and AS can be individually varied. It is theoretically possible that any value of the lag and anticipation time in approximate LS and AS can be reached as long as there are enough truncated terms in Equation (23). The calculation of terms, of course, is tedious with the increase in truncated terms. In addition, it will become difficult to physically realize such a controller in the real world [31, 32]. This is the disadvantage of the proposed scheme achieving approximate LS and AS between two unidirectionally coupled hyperchaotic Chen systems without time-delay coupling.  
Our research provides a simple and effective way to achieve approximate LS and AS between two hyperchaotic systems without time-delay coupling, which can be applied to eliminate the negative effects of the time delay in signal transmission.  
**Author Contributions:** Conceptualization, L-J.O. and B.-Q.X.; methodology, B.D.; software, B.-Q.X.; validation, B.-Q.X.; formal analysis, L-J.O.; investigation, B.D.; resources, B.D.; data curation, B.-Q.X.; writing--original draft preparation, L-J.O.; writing--review and editing, B.D.; visualization, B.-Q.X.; supervision, B.D.; project administration, [PERSON]; funding acquisition, B.D. All authors have read and agreed to the published version of the manuscript.  
**Funding:** The Zhejiang Provincial Natural Science Foundation of China (Nos:LY20E080017), the National Natural Science Foundation of China (Nos:51708349) and the Service Innovation Project of the Wenzhou Association for Science and Technology, China. (Grant No.KJFW53).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable  
**Data Availability Statement:** The study did not report any data.  
**Acknowledgments:** The authors greatly appreciate the anonymous referees for their valuable suggestions and questions.  
**Conflicts of Interest:** The authors declare no conflict of interest.