A discrete memristor is introduced into the Rulkov neuron to mimic biological neuronal synapse and modify firing dynamics. In the memristive Rulkov neuron, chaotic firing with local amplitude control is obtained, where the range of chaotic bursting can be modified by two independent controllers. These two independent bifurcation parameters provide direct amplitude/frequency control. Furthermore, offset boosting-entangled complex dynamics are captured, where the initial condition of the membrane potential can visit any of the self-reproducing attractors and even modify the complex firing, indicating the coexistence of homogeneous and heterogeneous multistabilities. Consequently, a CH32-based circuit is developed to verify various firing activities. The pseudo-random number generator results are explored based on the National Institute of Standards and Technology showing its higher performance in secure optical communication, which is further proved in the seven-core 2-km communication setup.

A chaotic system with a hyperbolic function flux-controlled memristor is designed, which exhibits conditional symmetry and attractor growing. The newly introduced cosine function keeps the polarity balance when some of the variables get polarity inversed and correspondingly conditional symmetric coexisting chaotic attractors are coined. Due to the periodicity of the cosine function, the memristive system with infinitely many coexisting attractors shows attractor growing in some special circumstances. Analog circuit experiment proves the theoretical and numerical analysis.

A comprehensive review of symmetry and conditional symmetry is made from the core conception of symmetry and conditional symmetry. For a dynamical system, the structure of symmetry means its robustness against the polarity change of some of the system variables. Symmetric systems typically show symmetrical dynamics, and even when the symmetry is broken, symmetric pairs of coexisting attractors are born, annotating the symmetry in another way. The polarity balance can be recovered through combinations of the polarity reversal of system variables, and furthermore, it can also be restored by the offset boosting of some of the system variables if the variables lead to the polarity reversal of their functions. In this case, conditional symmetry is constructed, giving a chance for a dynamical system outputting coexisting attractors. Symmetric strange attractors typically represent the flexible polarity reversal of some of the system variables, which brings more alternatives of chaotic signals and more convenience for chaos application. Symmetric and conditionally symmetric coexisting attractors can also be found in memristive systems and circuits. Therefore, symmetric chaotic systems and systems with conditional symmetry provide sufficient system options for chaos-based applications.

Typically, a chaotic system with quadratic terms is realized based on integral summation circuits, where however operational amplifiers and multipliers are usually overused. In this brief, a new approach for simplifying such circuit implementation is proposed utilizing the inherent characteristics of a multiplier. As a result, a chaotic system with quadratic terms is realized using only a few multipliers combined with some passive linear elements. In some circumstances, an operational amplifier is employed with simple linear feedback in case the multiplier is overloaded. This simplification of chaotic circuits makes chaos generation more easily implementable, cost-effective, and hence practical for chaos-based communications.