Fractional-order Memristive System Research Articles

An Encryption Application and FPGA Realization of a Fractional Memristive Chaotic System

The work in this paper extends a memristive chaotic system with transcendental nonlinearities to the fractional-order domain. The extended system’s chaotic properties were validated through bifurcation analysis and spectral entropy. The presented system was employed in the substitution stage of an image encryption algorithm, including a generalized Arnold map for the permutation. The encryption scheme demonstrated its efficiency through statistical tests, key sensitivity analysis and resistance to brute force and differential attacks. The fractional-order memristive system includes a reconfigurable coordinate rotation digital computer (CORDIC) and Grünwald–Letnikov (GL) architectures, which are essential for trigonometric and hyperbolic functions and fractional-order operator implementations, respectively. The proposed system was implemented on the Artix-7 FPGA board, achieving a throughput of 0.396 Gbit/s.

Hidden Dynamics and Hybrid Synchronization of Fractional-Order Memristive Systems

A fractional-order memristive system without equilibrium is addressed. Hidden attractors in the proposed system are discussed and the coexistence of a hidden attractor is found. Via theoretical analysis, the hybrid synchronization of the proposed system with partial controllers is investigated using fractional stability theory. Numerical simulation verifies the validity of the hybrid synchronization scheme.

Fractional-order design of a novel non-autonomous laser chaotic system with compound nonlinearity and its circuit realization

In this paper, a 3-D non-autonomous fractional memristive laser chaotic system which contains a compound nonlinear term multiplied by an absolute value function and a sine function is proposed. The compound nonlinear term and the differential equations of the system are decomposed and solved by the ADM. The chaotic feature of this system is studied by bifurcation diagram and Lyapunov exponential spectrum, and the multiple stability of the system is discovered and analyzed. Besides, the analog simulation circuit and digital DSP corresponding to the new fractional-order memristive system are designed, and this chaotic system is verified by two implementation methods. This research results will provide new research ideas and a theoretical basis for the circuit realization of non-autonomous chaotic systems.

Complex Dynamical Behaviors of a Fractional-Order System Based on a Locally Active Memristor

A fractional-order locally active memristor is proposed in this paper. When driven by a bipolar periodic signal, the generated hysteresis loop with two intersections is pinched at the origin. The area of the hysteresis loop changes with the fractional order. Based on the fractional-order locally active memristor, a fractional-order memristive system is constructed. The stability analysis is carried out and the stability conditions for three equilibria are listed. The expression of the fractional order related to Hopf bifurcation is given. The complex dynamical behaviors of Hopf bifurcation, period-doubling bifurcation, bistability and chaos are shown numerically. Furthermore, the bistability behaviors of the different fractional order are validated by the attraction basins in the initial value plane. As an alternative to validating our results, the fractional-order memristive system is implemented by utilizing Simulink of MATLAB. The research results clarify that the complex dynamical behaviors are attributed to two facts: one is the fractional order that affects the stability of the equilibria, and the other is the local activeness of the fractional-order memristor.

Bifurcation analysis of a fractional order time-delay Chua’s circuit based on coupled memristors

In this paper, a fractional-order (and an integer-order) chaotic system, obtained from Chua’s circuit by substituting Chua’s diode with two active coupled memristors (MRs) characterized by quadratic nonlinearity, is introduced to probe the memristive coupling effect. Two MRs connected in parallel are coupled by the flux. For the integer-order memristive system, the dynamical characteristics depending on the coupling strength coefficient between MRs without changing the circuit parameters are illustrated theoretically and numerically by using phase portraits, time domain diagram, bifurcation diagram and the Lyapunov diagram. Then based on the Adams–Bashforth–Moulton algorithm, the study of dynamic behavior of the fractional-order memristive system containing the time-delay reveals that appropriately setting the coupling strength between MRs generates more interesting attractors that differ from its integer-order counterpart. Besides, the effects of the order and the time-delay on dynamics are discussed briefly. Finally, the simulation results verify the validity of the theoretical analysis.

Characteristic analysis of the fractional-order hyperchaotic memristive circuit based on the Wien bridge oscillator

In this paper, a new hyperchaotic memristive circuit based on the Wien bridge oscillator is built. The numerical solution of the new fractional-order memristive system is calculated by using the Adomian decomposition method. By using the spectral entropy (SE) complexity algorithm and the $ C_0$ complexity algorithm, the dynamic characteristics of the fractional-order system are analyzed. Especially, the fractional-order coexisting attractors are found and the coexisting bifurcation diagrams with different order are presented. With varying the order q , the phenomenon of coexisting evolution is observed. Finally, the practical circuit is realized. The results of the theoretical analysis and the numerical simulation show that the fractional-order Wien bridge hyperchaotic memristive circuit system has very complex dynamical characteristics. It provides a theoretical guidance for the chaotic related field.

Non-fragile state estimation for delayed fractional-order memristive neural networks

The issue of non-fragile estimation for fractional-order memristive system is provided in this paper. By endowing the Lyapunov technique, the corresponding works that ensuring the globally asymptotic stability of the error model are presented, which can be calculated efficiently. In the end, the analytical methods are voiced by two simulations.

On Coexistence of Fractional-Order Hidden Attractors

Abstract This paper proposes new fractional-order (FO) models of seven nonequilibrium and stable equilibrium systems and investigates the existence of chaos and hyperchaos in them. It thereby challenges the conventional generation of chaos that involves starting the orbits from the vicinity of unstable manifold. This is followed by the discovery of coexisting hidden attractors in fractional dynamics. All the seven newly proposed fractional-order chaotic/hyperchaotic systems (FOCSs/FOHSs) ranging from minimum fractional dimension (nf) of 2.76 to 4.95, exhibit multiple hidden attractors, such as periodic orbits, stable foci, and strange attractors, often coexisting together. To the best of the our knowledge, this phenomenon of prevalence of FO coexisting hidden attractors in FOCSs is reported for the first time. These findings have significant practical relevance, because the attractors are discovered in real-life physical systems such as the FO homopolar disc dynamo, FO memristive system, FO model of the modulation instability in a dissipative medium, etc., as analyzed in this work. Numerical simulation results confirm the theoretical analyses and comply with the fact that multistability of hidden attractors does exist in the proposed FO models.