Chaotic systems have been extensively applied in image encryption as a source of randomness. However, dynamical degradation has been pointed out as an important limitation of this procedure. To overcome this limitation, this paper presents a novel image encryption scheme based on the pseudo-orbits of 1D chaotic maps. We use the difference of two pseudo-orbits to generate a random sequence. The generated sequence has been successful in all NIST tests, which implies it has adequate randomness to be employed in encryption process. Confusion and diffusion requirements are also effectively implemented. The usual low key space of 1D maps has been improved by a novelty procedure based on multiple perturbations in the transient time. A factor using the plain image is one of the perturbation conditions, which ensures a new and distinct secret key for each image to be encrypted. The proposed encryption scheme has been efficaciously verified using the Lena, Baboon, and Barbara test images.

This work proposes a modified logistic map based on the system previously proposed by Han in 2019. The constructed map exhibits interesting chaos related phenomena like antimonotonicity, crisis, and coexisting attractors. In addition, the Lyapunov exponent of the map can achieve higher values, so the behavior of the proposed map is overall more complex compared to the original. The map is then successfully applied to the problem of random bit generation using techniques like the comparison between maps, $XOR$ , and bit reversal. The proposed algorithm passes all the NIST tests, shows good correlation characteristics, and has a high key space.

In this paper, we consider nonlinear integration techniques, based on direct Padé approximation of the differential equation solution, and their application to conservative chaotic initial value problems. The properties of discrete maps obtained by nonlinear integration are studied, including phase space volume dynamics, bifurcation diagrams, spectral entropy, and the Lyapunov spectrum. We also plot 2D dynamical maps to enlighten the features introduced by nonlinear integration techniques. The comparative study of classical integration methods and Padé approximation methods is given. It is shown that nonlinear integration techniques significantly change the behavior of discrete models of nonlinear systems, increasing the values of Lyapunov exponents and spectral entropy. This property reduces the applicability of numerical methods based on Padé approximation to the chaotic system simulation but it is still useful for construction of pseudo-random number generators that are resistive to chaos degradation or discrete maps with highly nonlinear properties.

The synchronization between chaotic systems implemented in similar ways—e.g., computer models or circuits—is a well-investigated topic. Nevertheless, in many practical applications, such as communication, identification, machine sensing, etc., synchronization between chaotic systems of different implementation types—e.g., between an analog circuit and computer model—might produce fruitful results. In this research, we study the synchronization between a circuit modeling the Rössler chaotic system and a computer model using the same system. The theoretical possibility of this kind of synchronization is proved, and experimental evidence of this phenomenon is given with special attention paid to the numerical methods for computer model simulation. We show that synchronization between a circuit with uncertain parameters and a computer model is possible, and the parameters obtained from the synchronized computer model are in high correspondence with the circuit element specification. The obtained results establish the possibility of using adaptive generalized synchronization for the parameter identification of real systems. It was also found that Heun’s method yielded the most accurate results in synchronization among second-order numerical integration methods. The best among the first-order methods appears to be the Euler–Cromer method, which can be of interest in embedded applications.