This tutorial clarifies the axiomatic definition of (v(α); i(β)) circuit elements via a lookup table dubbed an A-pad, of admissible (v; i) signals measured via Gedanken probing circuits. The (v(α); i(β)) elements are ordered via a complexity metric. Under this metric, the memristor emerges naturally as the fourth element, characterized by a state-dependent Ohm's law. A logical generalization to memristive devices reveals a common fingerprint consisting of a dense continuum of pinched hysteresis loops whose area decreases with the frequency ω and tends to a straight line as ω ~ ∞, for all bipolar periodic signals and for all initial conditions. This common fingerprint suggests that the term memristor be used hence-forth as a moniker for memristive devices.

The Fourth Element.

In this article, size-dependent modeling and analysis of thermoelastic coupling effect on the oscillations of Timoshenko nanobeams are carried out. Small-scale effect on the nanostructure and heat conduction is taken into account with the aid of nonlocal strain gradient theory (NSGT) together with dual-phase-lag (DPL) heat conduction model. In order to illustrate the influence of nonclassical scale parameters on the coefficients of governing equations, the normalized forms of size-dependent equations of motion and heat conduction are established by definition of some dimensionless parameters. These coupled differential equations are then solved in the Laplace domain to attain the analytical thermoelastic responses of a simply supported Timoshenko nanobeam subjected to a dynamic load. Through several numerical examples, a detailed parametric study is performed to illuminate the decisive role of nonlocal, strain gradient and phase lag parameters in thermoelastic behavior of Timoshenko nanobeams. Furthermore, comparing the results corresponding to various relative magnitudes of nonlocal and strain gradient length scale parameters confirms the potential of NSGT for covering both hardening and softening characteristic of nanoscale structures.

Size-dependent thermoelastic vibrations of Timoshenko nanobeams by taking into account dual-phase-lagging effect

Due to the potential difference between two neurons and that between the inner and outer membranes of an individual neuron, the neural network is always exposed to complex electromagnetic environments. In this paper, we utilize a hyperbolic-type memristor and a quadratic nonlinear memristor to emulate the effects of electromagnetic induction and electromagnetic radiation on a simple Hopfield neural network (HNN), respectively. The investigations show that the system possesses an origin equilibrium point, which is always unstable. Numerical results uncover that the HNN can present complex dynamic behaviors, evolving from regular motions to chaotic motions and finally to regular motions, as the memristors' coupling strength changes. In particular, coexisting bifurcations will appear with respect to synaptic weights, which means bi-stable patterns. In addition, some physical results obtained from breadboard experiments confirm Matlab analyses and Multisim simulations.

Complex dynamics in a Hopfield neural network under electromagnetic induction and electromagnetic radiation.

This paper devotes to the adaptive globally synchronization within predefined-time of two time-delayed fractional-order chaotic systems. Firstly, through fractional calculus, two novel different fractional-order systems with time-delay are proposed, whose convergence is guaranteed and phase trajectory is given. Secondly, by exploiting the non-negative Lyapunov function and inequality theorem, a novel global predefined-time stability theorem is proposed, which can ensure the settling time tunable. And the upper bound of the settling time estimation is more accurate compared with the classical results. With the help of novel predefined-time stability theorem, two active controllers are designed, namely the fixed-time synchronization controller and predefined-time synchronization controller, to achieve the fixed-time synchronization and the predefined-time synchronization of two different time-delayed fractional-order chaotic systems respectively. Finally, several numerical simulations are presented in order to show the effectiveness of the proposed methods.

/pdf/adaptive-predefined-time-synchronization-of-two-different-293dooyf84.pdf

Adaptive Predefined-Time Synchronization of Two Different Fractional-Order Chaotic Systems With Time-Delay

Since the Lorenz chaotic system was discovered in 1963, the construction of chaotic systems with complex dynamics has been a research hotspot in the field of chaos. Recently, memristive Hopfield neural networks (MHNNs) offer great potential in the design of complex, chaotic systems because of their special network structures, hyperbolic tangent activation function, and memory property. Many chaotic systems based on MHNNs have been proposed and exhibit various complex dynamical behaviors, including hyperchaos, coexisting attractors, multistability, extreme multistability, multi-scroll attractors, multi-structure attractors, and initial-offset coexisting behaviors. A comprehensive review of the MHNN-based chaotic systems has become an urgent requirement. In this review, we first briefly introduce the basic knowledge of the Hopfiled neural network, memristor, and chaotic dynamics. Then, different modeling methods of the MHNN-based chaotic systems are analyzed and discussed. Concurrently, the pioneering works and some recent important papers related to MHNN-based chaotic systems are reviewed in detail. Finally, we survey the progress of MHNN-based chaotic systems for application in various scenarios. Some open problems and visions for the future in this field are presented. We attempt to provide a reference and a resource for both chaos researchers and those outside the field who hope to apply chaotic systems in a particular application.

/pdf/a-review-of-chaotic-systems-based-on-memristive-hopfield-2ga3q8bb.pdf

A Review of Chaotic Systems Based on Memristive Hopfield Neural Networks

The firing patterns of each bursting neuron are different because of the heterogeneity, which may be derived from the different parameters or external drives of the same kind of neurons, or even neurons with different functions. In this paper, the different electromagnetic effects produced by two fractional-order memristive (FOM) Hindmarsh–Rose (HR) neuron models are selected for characterizing different firing patterns of heterogeneous neurons. Meanwhile, a fractional-order memristor-coupled heterogeneous memristive HR neural network is constructed via coupling these two heterogeneous FOM HR neuron models, which has not been reported in the adjacent neuron models with memristor coupling. With the study of initial-depending bifurcation behaviors of the system, it is found that the system exhibits abundant hidden firing patterns, such as periods with different topologies, quasiperiodic firings, chaos with different topologies, and even hyperchaotic firings. Particularly, the hidden hyperchaotic firings are perfectly detected by two-dimensional Lyapunov stability graphs in the two-parameter space. Meanwhile, the hidden coexisting firing patterns of the system are excited from two scattered attraction domains, which can be confirmed from the local attraction basins. Furthermore, the color image encryption based on the system and the DNA approach owns great keyspace and a good encryption effect. Finally, the digital implementations based on Advanced RISC Machine are in good coincidence with numerical simulations.

Hidden coexisting firings in fractional-order hyperchaotic memristor-coupled HR neural network with two heterogeneous neurons and its applications.

This paper addresses the finite-time synchronization problem for fractional-order memristor-based neural networks (FMNNs) with discontinuous activations, in which multiple delays are considered. Fi...

Finite-time synchronization for fractional-order memristor-based neural networks with discontinuous activations and multiple delays

Coexisting multi-stability of Hopfield neural network based on coupled fractional-order locally active memristor and its application in image encryption

In order to describe the complex dynamical behaviors of neurons with electromagnetic effect, this paper presents a fractional-order memristive (FOM) Hindmarsh–Rose (HR) neuron model by introducing a magnetic flux-controlled memristor to the three-dimensional HR neuron model. The proposed FOM HR neuron model without equilibrium point shows complex hidden dynamical behaviors, such as periodic orbits, chaotic behaviors, period-doubling bifurcations, and coexisting asymmetric phenomena, which are revealed by numerical simulations of local attraction basins, Lyapunov exponents, bifurcation diagrams, phase portraits, and so on. Furthermore, since most biomedical diseases are caused by abnormal discharge of neurons, a sliding mode control strategy is applied to suppress the chaotic behaviors of the FOM HR neuron model, which is helpful to prevent and treat diseases. Finally, we introduce the circuit designs of the FOM HR neuron model in detail, and the simulation results captured by oscilloscope in circuit implementation match well with the theoretical analysis.

Hidden dynamical behaviors, sliding mode control and circuit implementation of fractional-order memristive Hindmarsh−Rose neuron model

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.

Yongbing Hu

Papers

Hidden coexisting firings in fractional-order hyperchaotic memristor-coupled HR neural network with two heterogeneous neurons and its applications.

Finite-time synchronization for fractional-order memristor-based neural networks with discontinuous activations and multiple delays

Coexisting multi-stability of Hopfield neural network based on coupled fractional-order locally active memristor and its application in image encryption

Hidden dynamical behaviors, sliding mode control and circuit implementation of fractional-order memristive Hindmarsh−Rose neuron model

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