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The Lozi map is well-known and has been studied in various researches. By combining three research trends (discrete map, memristor and fractional calculus) we investigate a fractional memristive Lozi map in this work. Firstly the Grunwald–Letnikov fractional difference operator is used to introduce the new fractional map with no equilibrium point. Then, the coexistence of several chaotic hidden attractors is shown, along with the coexistence of a number of bifurcations, depending on the values of the initial conditions. We found attractive dynamics and characteristics of this fractional Lozi map. The realization with hardware platform illustrates the map's feasibility. [ABSTRACT FROM AUTHOR]

Chaotic dynamics and synchronization in fractional systems described by non-integer order difference operators have attracted increasing attention in recent years. However, very few papers have been published regarding the chaotic behaviors of fractional discrete-time neural networks and their synchronization properties. A novel chaotic h-fractional discrete neural network model is presented in this paper. This model is described using the h-fractional Caputo difference operator of order ν . The paper also presents a novel theorem, which assures that two chaotic fractional discrete neural networks achieve synchronized dynamics using very simple linear control laws. Additionally, the chaotic dynamics of the network are stabilized at the origin via a suitable controller. Finally, simulation results are carried out to demonstrate the feasibility of the theoretical results developed herein. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamics of the biological system modeled by the fractional incommensurate order Van der Pol equations are investigated. The stability of the proposed fractional non-autonomous system is analyzed by varying both the fractional order derivative and system parameters. Moreover, very interesting phenomena such as symmetry, multi-stability and coexistence of attractors are discovered in the considered biological system. Numerical simulations are performed by considering the Caputo fractional derivative and results are reported by means of bifurcation diagrams, computation of the largest Lyapunov exponent, phase portraits in 2D and 3D projections. [ABSTRACT FROM AUTHOR]

Understanding extreme events attracts scientists due to substantial impacts. In this work, we study the emergence of extreme events in a fractional system derived from a Liénard-type oscillator. The effect of fractional-order derivative on the extreme events has been investigated for both commensurate and incommensurate fractional orders. Especially, such a system displays multistability and coexistence of multiple extreme events. [ABSTRACT FROM AUTHOR]

In this paper, we are particularly interested in the fractional form of the Hénon-Lozi type map. Using discrete fractional calculus, we show that the general behavior of the proposed fractional order map depends on the fractional order. The dynamical properties of the new generalized map are investigated by applying numerical tools such as: phase portrait, bifurcation diagram, largest Lyapunov exponent, and 0-1 test. It shows that the fractional order Hénon-Lozi map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. Furthermore, a one-dimensional control law is proposed to stabilize the states of the fractional order map. Numerical results are presented to illustrate the findings. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of a new fractional-order map with no fixed points. Through phase plots, bifurcation diagrams, largest Lyapunov exponent, it is shown that the proposed fractional map exhibit chaotic and periodic behavior. New Hidden chaotic attractors are observed, and transient state is found to exist. Complexity of the new map is also analyzed by employing approximate entropy. Results, show that the fractional map without fixed point have high complexity for certain fractional order. In addition, a control scheme is introduced. The controllers stabilize the states of the fractional map and ensure their convergence to zero asymptotically. Numerical results are used to verify the findings. [ABSTRACT FROM AUTHOR]

Research in the field of dynamic behaviors in neural networks with variable-order differences is currently a thriving area, marked by various significant discoveries. However, when it comes to discrete-time neural networks featuring fractional variable-order nonlocal and nonsingular kernels, there has been limited exploration. This paper stands as one of the initial contributions to this subject, focusing primarily on the topics of stability and synchronization in finite-time within discrete neural networks. The research employs the nabla ABC variable-order difference operator, with a primary approach involving the investigation of a novel Gronwall inequality using the Atangana-Baleanu difference variable-order sum operator. This analysis leads to the development of a uniqueness theorem and a criterion for the stability in finite-time of variable-order discrete neural networks. Furthermore, the requirements stemming from this type of stability and the novel Gronwall inequality serve as the foundation for establishing the conditions necessary for achieving finite-time synchronization in these networks, employing a specific control using state feedback method. Finally, the study utilizes numerical solutions to validate the obtained results. [ABSTRACT FROM AUTHOR]