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One of the applications of dynamical systems with chaotic behavior is data encryption. Chaos-based cryptography uses chaotic dynamical systems as the basis for creating algorithms. The present article discusses a new dynamical system called M-map with its analysis: fixed points, bifurcation diagram, Lyapunov exponent, and invariant density. The obtained bifurcation diagram and the plot of the Lyapunov exponent (with a minimum value of ln2 and a maximum value of ln4) suggest that the so-called robust chaos characterizes this map. Moreover, the obtained results are compared with other dynamical systems used in cryptography. Additionally, the article proposes a new image encryption algorithm. It uses, among others, cyclically shifted S-box or saving encrypted pixels on the first or last free space in the cipher-image. The conducted analysis shows that the cipher-images are characterized by an entropy value close to 8, a correlation of adjacent pixels value close to 0, or values of Number of Pixel of Change Rate (NPCR) and Unified Average Changing Intensity (UACI) measures close to 100% and 33%, respectively.

The Delphi method is one of the basic tools for forecasting values in various types of issues. It uses the knowledge of experts, which is properly aggregated (e.g., in the form of descriptive statistics measures) and returns to the previous group of experts again, thus starting the next round of forecasting. The multi-stage prediction under the Delphi method allows for better stabilization of the results, which is extremely important in the process of forecasting. Experts in the forecasting process often have access to time series forecasting software but do not necessarily use it. Therefore, it seems advisable to add to the aggregate the value obtained using forecasting software. The advantage of this approach is in saving the time and costs of obtaining a forecast. That should be understood as a smaller burden on data analysts and the value of their work. According to the above mentioned key factors, the main contribution of the article is the use of a virtual expert in the form of a computer-enhanced mathematical tool, i.e., a programming library for a forecasting time series. The chosen software tool is the Prophet library—a Facebook tool that can be used in Python or R programming languages.

The adaptive approach of strongly non-linear fast-changing signals identification is discussed. The approach is devised by adaptive sampling based on chaotic mapping "in yourself" of a signal. Presented sampling way may be utilized online in the automatic control of chemical reactor (throughout identification of concentrations and temperature oscillations in real-time), in medicine (throughout identification of ECG and EEG signals in real-time), etc. In this paper, we presented it to identify the Weierstrass function and ECG signal.

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Based on Chebyshev polynomials, one can create an asymmetric cryptosystem that allows for secure communication. Such a cryptosystem is based on the fact that these polynomials form a semi-group due to the composition operation. This article presents two new cryptosystems based on modifications of Chebyshev's polynomials. The presented analysis shows that their security is the same as in the case of algorithms associated with the problem of discrete logarithms. The article also shows methods that allow for the faster calculation of Chebyshev polynomials.

In (Lawnik M., Generation of numbers with the distribution close to uniform with the use of chaotic maps, In: Obaidat M.S., Kacprzyk J., Ören T. (Ed.), International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH) (28-30 August 2014, Vienna, Austria), SCITEPRESS, 2014) Lawnik discussed a method of generating pseudo-random numbers from uniform distribution with the use of adequate chaotic transformation. The method enables the " attening" of continuous distributions to uniform one. In this paper a inverse process to the above-mentioned method is presented, and, in consequence, a new manner of generating pseudo-random numbers from a given continuous distribution. The method utilizes the frequency of the occurrence of successive branches of chaotic transformation in the process of " attening". To generate the values from the given distribution one discrete and one continuous value of a random variable are required. The presented method does not directly involve the knowledge of the density function or the cumulative distribution function, which is, undoubtedly, a great advantage in comparison with other well-known methods. The described method was analysed on the example of the standard normal distribution.