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In this paper, we introduce a novel integer-order memristor-modified Shinriki circuit (MMSC). We investigate the dynamic properties of the MMSC system and the existence of chaos is proved with positive largest Lyapunov exponent. Bifurcation plots are derived to analyze the parameter dependence of the MMSC system. The fractional-order model of the MMSC system (FOMMSC) is derived and the bifurcation analysis of the FOMMSC system with the fractional orders is carried out. Fractional-order adaptive sliding-mode controllers (FOASMCs) and genetically optimized PID controllers are designed to synchronize identical FOMMSC systems with unknown parameters. Numerical simulations are conducted to validate the theoretical results. FPGA implementation of the FOASMC controllers is presented to show that the proposed control algorithm is hardware realizable. MMSC has trigonometric functions which make the system more complex and the optimization and synchronization of such systems in the integer order itself are harder, so the paper does the same in fractional order. The proposed system is a memristive circuit which can show special features such as multistability, hyperchaos, and multiscroll attractor. Such a system with these features is very rare in the literature.

We announce a new 4D hyperchaotic system with four parameters. The dynamic properties of the proposed hyperchaotic system are studied in detail; the Lyapunov exponents, Kaplan-Yorke dimension, bifurcation, and bicoherence contours of the novel hyperchaotic system are derived. Furthermore, control algorithms are designed for the complete synchronization of the identical hyperchaotic systems with unknown parameters using sliding mode controllers and genetically optimized PID controllers. The stabilities of the controllers and parameter update laws are proved using Lyapunov stability theory. Use of the optimized PID controllers ensures less time of convergence and fast synchronization speed. Finally the proposed novel hyperchaotic system is realized in FPGA.