# Chaotic Communication System with Symmetry-Based Modulation
## 1 Introduction  
Digital modulation is a widely used technique in modern communication systems, for example, in a direct down-conversion software-defined radio (DDC SDR) which is in high demand on today's market [1]. Digital modulation provides opportunities that are not possible for neither analog nor binary modulation. For example, it allows using arbitrary carriers, such as ultra-wideband signals and, which is of particular interest, chaotic signals, as well as applying advanced digital signal processing algorithms. The idea of using chaotic oscillators to implement communication systems goes back to the discovery of the chaotic synchronization phenomenon. In the early work by [PERSON] et al. published in 1993 [2], a secure communication system is proposed using based on two circuits implementing the Lorenz chaotic oscillator. The authors present two possible data transmission schemes: masking of the analog signal by chaotic signal, and bifurcation parameter modulation suitable for binary data transfer. This work stated the fundamental possibility of such type of communication. A similar solution based on the Chua circuit was described by [PERSON] et al. [3]. One of the first papers that used chaotic communication for the transmission of complex signals such as speech was published by the Saratov nonlinear dynamics team [4]. Since these pioneering works, a large number of methods for modulating chaotic signals had been developed, based on different possible types of synchronization, including full, generalized, and phase synchronization. The prospective application of chaotic communication include wired [5], wireless [6], optical [7], and underwater acoustic [8] communication. The influence of noise and multipath propagation on the chaotic communication quality and the transfer rate had been extensively studied in the last three decades. [PERSON] et al. [9] have shown analytically that the spectra of Lyapunov exponents preserve in a chaotic signal after being transmitted through a wireless channel. Authors state that this physical property provides excellent performance of the fully chaos-based wireless communication system in the presence of bounded noise and the relative motion between transmitter and receiver. [PERSON] et al. [7] listed some established advantages of direct chaotic communication, such as security, secrecy, potentially high data transfer rate, and the existence of a larger number of addresses in the channel, which allows increasing the transmission energy without a significant increase in interference. The high transfer rate may be achieved due to the possibility to encode several different symbols in a short waveform. The authors mentioned an experimentally reached rate of 1 Gbps with a bit error rate (BER) of \(10^{-7}\) when transferring data over optical fiber. Besides, many researchers have focused their research exclusively on the possibility of secure and covert data transmission using chaos [5,10]. Among the proposed chaotic communication techniques, two main classes may be distinguished--coherent and non-coherent systems. Coherent systems are based on the synchronization of chaotic oscillators, while non-coherent systems perform data recovery by detecting the features of the received chaotic signal without reproducing the signal on the receiver side. For both of these classes, several different modulation methods have been invented. Advanced non-coherent modulation is robust against linear and non-linear waveform distortions [8,11]. Coherent modulation systems are more sensitive to distortions and disturbances, but provide simpler ways of data transferring, as well as a higher potential level of security and secrecy. Therefore the enhancing of coherent data modulation schemes is an essential task in chaos applications [11]. In this paper, we propose a new coherent chaotic transmission method by modulating the symmetry parameter of the finite-difference model of a continuous chaotic system, obtained by symmetric semi-implicit integration, which is used for chaotic waveform generation.