# Coherent Chaotic Communication Using Generalized Runge-Kutta Method
###### Abstract
[PERSON] [35] showed that a numerical method based on trigonometric polynomials provides better accuracy than forward Euler and, as also shown herein, requires lower FPGA resources compared to fourth-order RK. The proposed FPGA hardware realization for the new Wang chaotic system includes integer, floating-point, and fixed-point components. This implementation achieves a significant reduction in hardware resources: approximately 10 times less than in previous works. Speaking of coherent chaotic systems, the synchronization of chaotic signals becomes a keystone procedure. Since [PERSON] and [PERSON] [36] demonstrated that it was possible to synchronize chaotic circuits, there has been a significant amount of research dedicated to studying the phenomenon of synchronization. [PERSON] et al. examined the potential for synchronization between analog and discrete models of the Vilnius chaotic oscillator [37] and evaluated the efficiency of the Pecora-Carroll synchronization method for achieving synchronization between analog and discrete chaos oscillators [38]. [PERSON] et al. demonstrated the practical possibility of synchronization between digital and analog chaotic circuits, achieving synchronization accuracy higher than in the analog-to-analog case [39]. [PERSON] et al. created a chaotic synchronization scheme based on electro-optical hybrid entropy sources [40]. In our previous papers, we presented a new modulation principle that utilizes the variable properties of a discrete integration operator for signal modulation. [PERSON] et al. [22] introduced symmetry coefficient modulation (SCM), which uses the unique properties of symmetric numerical integration methods with adaptive symmetry. In [41; 42], [PERSON] et al. described an approach to adaptive synchronization by controlling the symmetry coefficient. [PERSON] et al. [43] proposed effective numerical estimation of CCS secrecy based on a modified return map analysis. Previous works have shown high potential for CCSs that use signal modulation methods based on the controllable properties of the discrete operator. However, using symmetric integration in embedded solutions is not always suitable due to the possible implicitness of the resulting finite-difference models, which, in turn, causes extra computational costs. Therefore, in this paper, we present a simple and reliable CCS based on a generalized explicit second-order Runge-Kutta solver. The main contributions of this study can be summarized as follows:  
1. We investigate the nonlinear properties of a finite-difference model of the Gokylidirim et al. system obtained using a generalized explicit second-order Runge-Kutta solver. The discovered phenomenon of artificial multistability is explored using the Lyapunov spectrum, bifurcation analysis, and basin of attraction analysis. We present a novel modulation technique called alpha modulation (\(\alpha\)M), which is suitable for constructing coherent chaos-based communication systems. The proposed technique assumes the controllable value of \(\alpha\) in the generalized explicit second-order Runge-Kutta solver and corresponding discrete chaotic map obtained from a continuous system. The prototype system is based on the Gokyildirim et al. chaotic system. The experiments were performed using the Arduino-based CCS test bench developed by the authors. 3. The experimental results indicate that data transfer using \(\alpha\)M is feasible. The system's performance was evaluated by comparing several important properties and included noise resistivity and secrecy superior to those of parameter modulation (PM) implemented on the same hardware. 4. The study employed the quantified return map analysis (QRMA) technique to evaluate the secrecy of the communication system. We discovered that while the noise resistivity among all the investigated modulation techniques is similar, the symmetry modulation technique exhibits greater secrecy than parametric modulation and variable midpoint modulation. The rest of the paper is organized as follows.