## 1 Introduction  
Chaos-based digital communication systems have been researched for several decades. The development of the first electronic circuits capable of generating chaotic oscillations in the 90s raised a huge interest in the scientific community and paved the way for many creative ideas about how chaotic signals can be employed in communications. The problem of digital communication using chaos has been addressed in many doctoral theses [1, 2], survey papers [3] and regular scientific articles. In the following paragraphs, authors will pay attention to the latest developments in spread spectrum (SS) communication systems that employ chaos. There are several approaches how to employ chaos for SS communications. In most of the approaches, chaos is employed as a means for the masking or scrambling of the transmitted signal. Broadband, noise-like nature of the chaotic signals makes them very well suited to these tasks. To decipher the transmitted information, it is necessary to employ synchronization techniques or apply differential encoding methods. One of the most straightforward and widely-used approaches is to employ discrete chaotic sequences for spreading information bits over a longer time interval and widerfrequency range by multiplying information symbols by chaotic sequences. This technique, called direct sequence spread spectrum (DSSS), allows to decrease SNR of the received signal at the expense of frequency bandwidth. This, in turn, allows for maintaining communication over longer distances. Moreover, it is possible to use spreading sequences for multiple access. However, practical implementation of coherent chaotic DSSS requires complex and computation-demanding synchronization loops [4] for code acquisition [5; 6] and tracking. It is worth mentioning, that in vast chaotic DSSS papers, the synchronization problem is skipped. To overcome the synchronization issue, a non-coherent differential chaos shift keying (DCSK) communication scheme was proposed. In this very popular modulation scheme, the receiver does not need to synchronize code with the transmitter as the reference codes are transmitted along with the information-carrying codes, inherently wasting up to half of the available bandwidth. Recently, several mechanisms how for reducing synchronization overhead were proposed. In quadrature DCSK schemes [7; 8; 9], the reference signals get transmitted in the quadrature channel, whereas useful information is sent over in-phase (I) channel. Further advances in DCSK development employ Walsh codes [10], permutation matrices [11; 12], unitary transforms [13], Hilbert transform [14], multi-carrier techniques [15], combining direct and time-reversed sequences [16]. There are researches devoted to the employment of DCSK [17] in conjunction with Reconfigurable Intelligent Surface (RIS) which has a great potential for terrestrial and space communications [18]. Many recent papers propose to combine DCSK and index modulations [19; 20]  
If chaotic spreading is applied in the frequency domain, the resulting spread spectrum modulation is referred to as chaotic FHSS [21]. It is worth mentioning that DSSS can be easily converted into FHSS by applying frequency modulation (FM) [22]. In our research, we also use this technique. Spread spectrum techniques can employ not only discrete-time sequences but also continuous-time waveforms. For example, chaotic spreading sequence (CSS) modulation technique [23], which is employed in the LoRa PHY, can be considered as frequency domain spreading using sawtooth waveforms. Moreover, in a recently published highly-cited paper [24], it was shown that nonlinear chirps provide even better performance compared to CSS. There are few papers [25] addressing the employment of continuous-time chaos for spreading, as there is a huge challenge with the implementation of spreading waveform synchronization. If the spreading waveform is purely chaotic, correlation-based techniques can not be employed.