In this work, the study of an innovative chaotic system made from two memristors with symmetric bifurcation and multistability is presented. Within a four dimensional chaotic framework, the system is architected with two flux controlled memristors. Computational simulations reveal intricate dynamical phenomena such as symmetric bifurcations, multistability, and very large sensitivity to initial conditions. Using Lyapunov exponents, bifurcation diagrams and phase portraits, we investigate the system's ability to produce chaotic attractors of pronounced multistability. Pinched hysteresis loop and system offset investigation at different frequencies are investigated for possible applications in neuromorphic computing, random number generation and secure communication protocols. We further advance our understanding of the complex dynamical properties of memristive chaotic systems. A representative analog circuit corroborates their numerical findings.

A phase-shifting chaotic oscillator is constructed by memristive coupling. The introduced memristor revises the frequency response as the core of the frequency selection network in the oscillator. A controlled memristor is derived to maintain a stable amplification. Thus, the oscillator has two independent offset boosting voltages, and the voltages of two capacitors can be effectively controlled by cancellation. More conveniently, the simultaneous and proportional change of the op-amp supply voltage and the memristor in the oscillator also rescales the capacitors’ voltages in the same proportion. Finally, a memristor-equivalent circuit with the feedback from AD633 for division operation greatly reduces the cost of components. The hardware experiment confirms the theoretical analysis and numerical simulations.

By introducing fractional exponentiation into a three-dimensional chaotic system, a jerk system with only six terms is designed. It has the property of total amplitude control, where a single non-bifurcation parameter can directly rescale all system variables without affecting the dynamics. It also features two-dimensional offset boosting, where a single parameter can realize direct offset boosting while another provides interlocked cross-dimensional offset boosting. Furthermore, this jerk system has a parameter-dominated symmetric attractor, which means that symmetric attractors appear successively as the parameter changes from positive to negative. Circuit experiments confirm the feasibility of analog fractional exponentiation using the 444 circuit and the complete control, including amplitude control and offset boosting, of the resulting system. The proposed circuit may facilitate applications of chaotic signal generators where signal versatility is important and exemplifies the generative potential of analog fractional exponentiation.