From a pedagogical point of view, the memristor is defined in this tutorial as any 2-terminal device obeying a state-dependent Ohm’s law. This tutorial also shows that from an experimental point of view, the memristor can be defined as any 2-terminal device that exhibits the fingerprints of “pinched” hysteresis loops in the v–i plane. It also shows that memristors endowed with a continuum of equilibrium states can be used as non-volatile analog memories. This tutorial shows that memristors span a much broader vista of complex phenomena and potential applications in many fields, including neurobiology. In particular, this tutorial presents toy memristors that can mimic the classic habituation and LTP learning phenomena. It also shows that sodium and potassium ion-channel memristors are the key to generating the action potential in the Hodgkin-Huxley equations, and that they are the key to resolving several unresolved anomalies associated with the Hodgkin-Huxley equations. This tutorial ends with an amazing new result derived from the new principle of local activity, which uncovers a minuscule life-enabling Goldilocks zone, dubbed the edge of chaos, where complex phenomena, including creativity and intelligence, may emerge. From an information processing perspective, this tutorial shows that synapses are locally-passive memristors, and that neurons are made of locally-active memristors.

Memristor, Hodgkin-Huxley, and Edge of Chaos.

In this paper, a fractional-order chaotic circuit with different coupled memristors is established. The dimensionality of the system is reduced by the flux-charge analysis method and the stability of the equilibrium points is analyzed by the fractional-order stability theory. Then, the complex dynamic behaviors, including periodic and chaotic attractors, period doubling bifurcation orbit, coexistence bifurcation, and asymmetric coexisting attractors, are studied by phase diagrams, bifurcation portraits, Lyapunov exponent spectra, and attractive basins. Moreover, the analog circuit of the fractional-order coupled system is constructed and the results validate the correctness of the theoretical analysis. Finally, a novel encryption scheme based on the fractional-order coupled memristive system combined with Josephus traversal and DNA operations is proposed. The simulation results show that this algorithm has a good effect.

Multiple coexisting analysis of a fractional-order coupled memristive system and its application in image encryption

Resistive switching devices (memristors) constitute an emerging device technology promising for a variety of applications that are currently being studied. In this context, the use of memristors as coupling elements of the dynamics of chaotic circuits for adaptive synchronization purposes, was recently proposed and the passive crossbar array was evaluated as target interconnect medium. Nonetheless, memristors may suffer from defects and degradation. Therefore, this work evaluates the impact of memristor switching faults in an adaptive chaotic synchronization scheme, exploring at the same time the fault-tolerance of the crossbar architecture. Moreover, inspired from our observations in the stuck-at-OFF fault analysis of the memristive crossbar, some alternative scalable memristive interconnect patterns are suggested, whose performance is found independent of the number of interconnected chaotic circuits, requiring a much smaller number of total memristors than the crossbar array. All simulations are based on an accurate physics-based model of a bipolar memristor with filamentary switching mechanism. Based on our results, using the alternative topologies instead of the crossbar array leads to significant savings in the synchronization time that increase with the number of interconnected chaotic units, at the cost of more limited scaling capability and fault-tolerance.

Alternative memristor-based interconnect topologies for fast adaptive synchronization of chaotic circuits

To achieve control and synchronization of chaotic electronic circuits, a nonlinear optimal (H-infinity) control method is developed and is tested on Chua’s circuit. Although this electronic circuit is deterministic, for specific values of its parameters its phase diagrams may change in a random-like manner, thus exhibiting a chaotic behavior. In the article’s control approach, an approximate linearization procedure is applied first to the dynamic model of the circuit. The linearization takes place around a temporary operating point which is recomputed at each iteration of the control method. It actually uses Taylor series expansion and the computation of the system’s Jacobian matrices. At a next stage, an H-infinity feedback controller is developed for the approximately linearized model of the circuit. This controller is obtained after solving an algebraic Riccati equation at each time step of the control method. To prove the stability properties of the control scheme and the elimination of the synchronization error, Lyapunov analysis is used. The proposed control scheme is demonstrated to satisfy the H-infinity tracking performance condition, and this indicates elevated robustness against model uncertainty and external perturbations. Moreover, the global asymptotic properties of the control method are proven. Finally, under the proposed nonlinear optimal control approach, it is shown that different Chua’s circuits can get synchronized and that chaotic behavior can be replicated by them.

Nonlinear optimal control and synchronization for chaotic electronic circuits

In the paper, the memristive two-port is defined via its constitutive relation as a two-dimensional surface in a four-dimensional space of the port charges and fluxes. It follows from six possible representations of this two-port that it can be modeled via classical two-terminal memristors /memductors, complemented by four types of memristive controlled sources, each of them being controlled by two state variables. Two examples of the memristive two-port are analyzed, viz. the circuit with mutual memristors and the memristive transformer.

Memristive two-ports

The main purpose of this paper is to explore the spontaneous synchronization of two Chua's circuits based on coupled memristors with cubic nonlinearity. An analog circuit is designed to simulate the magnetic flux controlled memristor with coupling effects and then applied to implement the coupled Chua's circuit. Simulation is carried out to numerically exhibit the strange attractors and spontaneous synchronization caused by increasing the coupling strength.

Spontaneous synchronization of two Chua's circuits based on coupled memristors

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Spontaneous synchronization of two Chua's circuits based on coupled memristors