# Synchronization and Application of a Novel Hyperchaotic System Based on Adaptive Observers
## 1 Introduction  
The understanding of chaotic behavior was greatly advanced in the 1960s when [PERSON] developed a computer program to simulate airflow, which is when chaos theory first emerged [1]. Extensive study was conducted in the 1970s when [PERSON] discovered that even minimal changes in the initial circumstances might have a significant influence on weather forecasts. According to this study, system states varied significantly based on the basic characteristic of chaotic systems, which is the smallest deviation between two beginning circumstances [2]. The study of hyperchaotic systems was further developed in 1979 when [PERSON] discovered a four-dimensional hyperchaotic system [3]. Multidimensional chaotic and hyperchaotic systems became more well-known in the following years, providing useful applications in science and engineering [4]. Recently, there are many studies to be found in the literature on the dynamic analysis of chaotic systems [5, 6, 7]. In contrast to chaotic systems, hyperchaotic systems, which have multiple positive Lyapunov exponents, are more unpredictable and resistant to control. This property has drawn interest from the encryption and communication industries [8]. Multiple attractors improve security by making it more difficult for an attacker to determine the precise state of the system, which makes hyperchaotic systems more resistant to disruptions [9]. Secure applications of hyperchaotic systems have been a topic of great interest recently [10, 11]. Circuit simulations of hyperchaotic systems facilitate real-world exploration and analysis of their dynamics, validating theoretical models and providing insights into system behavior under different conditions [12; 13]. The implementation of hyperchaotic systems in circuits offers practical applications in secure communications, random number generation and nonlinear dynamics research [14; 15]. Synchronization has gained attention in the context of hyperchaotic systems, especially with the growing focus on secure communication applications [16]. In order to attain coherence, synchronization entails changing the dynamic behaviors of two systems, referred to as the driver and reaction [17]. In an effort to compel the response system to synchronize with the driving system, the notion of synchronization in chaotic systems was first presented in 1990 [18]. To this end, a number of approaches have been put forth, such as adaptive design techniques and sampled data feedback synchronization [19; 20]. Observer-based hyperchaotic system synchronization involves designing an observer (state estimator) and control strategy to achieve synchronization between two or more hyperchaotic systems [21]. Observers play a crucial role in estimating unmeasurable states, overcoming challenges posed by limited sensors or measurements, especially in hyperchaotic systems. Adaptive observers address uncertainties and variations in system parameters, enhancing the robustness and effectiveness of synchronization strategies [22]. The synchronization of adaptive observer-based hyperchaotic systems holds particular importance in secure communication systems [23]. The complex dynamics of hyperchaotic systems offer potential for encryption. Decryption and synchronization of these systems are vital for establishing secure communication channels [24]. The inclusion of adaptive observers adds robustness to the synchronization process, addressing uncertainties and variations in real-world applications [25]. The synchronization and implementation of adaptive observer-based hyperchaotic systems present a promising approach for secure communication. This approach, encompassing dynamic encryption keys, adaptation to changing conditions and resistance to attacks, proves valuable for applications where secure and robust communication is imperative. Typical examples are military communications, financial transactions, or confidential data transfer [26]. [PERSON] et al. [27] proposed a new 4-dimensional hyperchaotic system.