Chaotic systems depict complex dynamics, thanks to their nonlinear behaviors. With recent studies on fractional-order nonlinear systems, it is deduced that fractional-order analysis of a chaotic system enriches its dynamic behavior. Therefore, the investigation of the chaotic behavior of a 4D memristive Chen system is aimed in this study by taking the order of the system as fractional. The nonlinear behavior of the system is observed numerically by comparing the fractional-order bifurcation diagrams and Lyapunov Exponents Spectra with 2D phase portraits. Based on these analyses, two different fractional orders (i.e., q = 0.948 and q = 0.97) are determined where the 4D memristive system shows chaotic behavior. Furthermore, a single state fractional-order sliding mode controller (FOSMC) is designed to maintain the states of the fractional-order memristive chaotic system on the equilibrium points. Then, control method results are obtained by both numerical simulations and different illustrative experiments of microcontroller-based realization. As expected, voltage outputs of the microcontroller-based realization are in good agreement with the time series of numerical simulations.

Fractional-order (FO) chaotic systems exhibit richer and more complex dynamic behaviors compared to integer-order ones. This inherent richness and complexity enhance the security of FO chaotic systems against various attacks in image cryptosystems. In the present study, a comprehensive examination of the dynamical characteristics of the fractional-order Arneodo (FOAR) system with cubic nonlinearity is conducted. This investigation involves the analysis of phase planes, bifurcation diagrams, Lyapunov exponential spectra, and spectral entropy. Numerical studies show that the Arneodo chaotic system exhibits chaotic behavior when the lowest fractional-order (FO) value is set to 0.55. In this context, the aim is to securely encrypt biomedical images based on the Arneodo chaotic system with the lowest FO value using the Nvidia Jetson Nano development board. However, though the lowest FO system offers enhanced security in biomedical image encryption due to its richer dynamic behaviors, it necessitates careful consideration of the trade-off between high memory requirements and increasing complexity in encryption algorithms. Within the scope of the study, a novel random number generator (RNG) is designed using the FOAR chaotic system. The randomness of the random numbers is proven by using internationally accepted NIST 800-22 and ENT test suites. A biomedical image encryption application is developed using pseudo-random numbers. The images obtained as a result of the application are evaluated with tests such as histogram, correlation, differential attack, and entropy analyses. As a result of the study, it has been shown that encryption and decryption of biomedical images can be successfully performed on a mobile Nvidia Jetson Nano development card in a secure and fast manner.

Interest in studies on fractional calculus and its applications has greatly increased in recent years. Fractional-order analysis has the potential to enhance the dynamic structure of chaotic systems. This study presents the implementation of a lower-order fractional electronic circuit using standard components for the Sprott K system. To our knowledge, there are no chaotic circuit realizations in the literature where the value of a fractional-order parameter is approximately 0.8, making this study pioneering in this aspect. Additionally, various numerical analyses of the system are conducted, including chaotic time series and phase planes, Lyapunov exponents, spectral entropy (SE), and bifurcation diagrams, in order to examine its dynamic characteristics and complexity. As anticipated, the voltage outputs obtained from the oscilloscope demonstrated good agreement with both the numerical analysis and PSpice simulations.