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This paper presents a multistable discrete memristor that is based on the discretization of a continuous-time model. It has been observed that the discrete memristor model is capable of preserving the characteristics of the continuous memristor model. Furthermore, a three-dimensional memristor discrete-time FitzHugh–Nagumo model is constructed by integrating the discrete memristor into a two-dimensional FitzHugh–Nagumo (FN) neuron model. Subsequently, the dynamic behavior of the proposed neuron model is analyzed through Lyapunov exponents, phase portraits, and bifurcation diagrams. The results show multiple kinds of coexisting hidden attractor behaviors generated by this neuron model. The proposed approach is expected to have significant implications for the design of advanced neural networks and other computational systems, with potential applications in various fields, including robotics, control, and optimization. [ABSTRACT FROM AUTHOR]

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections. [ABSTRACT FROM AUTHOR]

Some endeavors have been recently dedicated to explore the dynamic properties of the fractional-order discrete-time chaotic systems. To date, attention has been mainly focused on fractional-order discrete-time systems with "self-excited attractors." This paper makes a contribution to the topic of fractional-order discrete-time systems with "hidden attractors" by presenting a new 2-dimensional discrete-time system without equilibrium points. The conceived system possesses an interesting property not explored in the literature so far, i.e., it is characterized, for various fractional-order values, by the coexistence of various kinds of chaotic attractors. Bifurcation diagrams, computation of the largest Lyapunov exponents, phase plots, and the 0-1 test method are reported, with the aim to analyze the dynamics of the system, as well as to highlight the coexistence of chaotic attractors. Finally, an entropy algorithm is used to measure the complexity of the proposed system. [ABSTRACT FROM AUTHOR]

This article investigates a non-equilibrium chaotic system in view of commensurate and incommensurate fractional orders and with only one signum function. By varying some values of the fractional-order derivative together with some parameter values of the proposed system, different dynamical behaviors of the system are explored and discussed via several numerical simulations. This system displays complex hidden dynamics such as inversion property, chaotic bursting oscillation, multistabilty, and coexisting attractors. Besides, by means of adapting certain controlled constants, it is shown that this system possesses a three-variable offset boosting system. In conformity with the performed simulations, it also turns out that the resultant hidden attractors can be distributively ordered in a grid of three dimensions, a lattice of two dimensions, a line of one dimension, and even arbitrariness in the phase space. Through considering the Caputo fractional-order operator in all performed simulations, phase portraits in two- and three-dimensional projections, Lyapunov exponents, and the bifurcation diagrams are numerically reported in this work as beneficial exit results. [ABSTRACT FROM AUTHOR]

Dynamical systems described by fractional-order difference equations have only been recently introduced inthe literature. Referring to chaotic phenomena, the type of the so-called "self-excited attractors" has been so far highlighted among different types of attractors by several recently presented fractional-order discrete systems. Quite the opposite, the type of the so-called "hidden attractors", which can be characteristically revealed through exploring the same aforementioned systems, is almost unexplored in the literature. In view of those considerations, the present work proposes a novel 3D chaotic discrete system able to generate hidden attractors for some fractional-order values formulated for difference equations. The map, which is characterized by the absence of fixed points, contains only one nonlinear term in its dynamic equations. An appearance of hidden attractors in their chaotic modes is confirmed through performing some computations related to the 0–1 test, largest Lyapunov exponent, approximate entropy, and the bifurcation diagrams. Finally, a new robust control law of one-dimension is conceived for stabilizing the newly established 3D fractional-order discrete system. [ABSTRACT FROM AUTHOR]