# Coherent Chaotic Communication Using Generalized Runge-Kutta Method
###### Abstract  
Computer simulation of continuous chaotic systems is usually performed using numerical methods. The discretization may introduce new properties into finite-difference models compared to their continuous prototypes and can therefore lead to new types of dynamical behavior exhibited by discrete chaotic systems. It is known that one can control the dynamics of a discrete system using a special class of integration methods. One of the applications of such a phenomenon is chaos-based communication systems, which have recently attracted attention due to their high covertness and broadband transmission capability. Proper modulation of chaotic carrier signals is one of the key problems in chaos-based communication system design. It is challenging to modulate and demodulate a chaotic signal in the same way as a conventional signal due to its noise-like shape and broadband characteristics. Therefore, the development of new modulation-demodulation techniques is of great interest in the field. One possible approach here is to use adaptive numerical integration, which allows control of the properties of the finite-difference chaotic model. In this study, we describe a novel modulation technique for chaos-based communication systems based on generalized explicit second-order Runge-Kutta methods. We use a specially designed test bench to evaluate the efficiency of the proposed modulation method and compare it with state-of-the-art solutions. Experimental results show that the proposed modulation technique outperforms the conventional parametric modulation method in both coverage and noise immunity. The obtained results can be efficiently applied to the design of advanced chaos-based communication systems as well as being used to improve existing architectures. chaos; chaotic signal modulation; coherent communication system; Runge-Kutta method method 2  
The main idea behind chaotic communication systems is to use chaotic signals, also referred to as chaotic carriers, to encode and decode the transmitted information [11]. Chaotic signals possess several desirable properties for covert communication purposes, such as broad bandwidth, robustness to noise and interference, and low correlation with other signals. Moreover, chaotic signals can provide a natural way of encrypting the information to protect from possible tampering or eavesdropping, as they are difficult to distinguish from noise, predict, or reconstruct without knowing the exact parameters and initial conditions of the underlying chaotic system [12]. Therefore, chaotic communication systems can offer a high level of security and privacy for the transmitted data. However, chaotic communication systems (CCSs) also face several important limitations that need to be addressed and overcome for successful design. One of the main challenges in constructing so-called coherent CCSs is the issue of synchronization between the transmitter and the receiver, which is essential for coherent detection and decoding of the information [13]. Synchronization is the process of achieving identical or clearly correlated dynamics between two or more chaotic systems and can be achieved by various methods such as direct coupling by the Pecora-Carroll technique [14], feedback control [15], or parameter adaptation [16]. It should be noted that, despite the use of chaotic carriers allowing the creation of robust, secure, and covert communication systems, several pitfalls prevent the widespread adoption of chaotic systems for this type of communication. First, the generation, transmission, and denoising of chaotic signals are complicated tasks that require special mathematics and should be approached with care [17]. Second, the sensitive nature of chaotic dynamics increases the impact of discretization on the properties of the finite difference model. The interesting fact is that the discrete model of a continuous chaotic system may not only lack some of the known properties of the original system but may also exhibit new behavior that was not present in the original system [18].