Designing Twin Memristor-Based Multiscroll Systems by Varying the Flux Variable of Memristor

Memristor, as a nonlinear element in nanometer size, is feasible to generate chaotic signals. Especially, it can improve the randomness of the signals and the complexity of chaotic systems. A novel multiscroll hyperchaotic system based on the flux-controlled memristor is designed. Its twin system with a different topological structure is obtained by varying only the flux variable of the memristor, which is a considerable difference from other chaotic systems existing in the literature. Scroll numbers of the proposed system and its twin, especially, are sensitive to the system parameters. To further investigate the characteristics of the system and its twin, their behaviors with respect to the strengths of the memristor, simulation time and coexistence of multiscroll hyperchaotic attractors are analyzed. Moreover, complex dynamical behaviors of the system and its twin are analyzed by phase diagrams, Poincaré cross-sections, Lyapunov indexes, power spectra, 0-1 test diagrams, and time series. Finally, a novel image encryption scheme is proposed based on the system and its twin. The system and its twin have independent key spaces, and the proposed algorithm makes them have an intersection, which greatly improves the key spaces. In addition, the security and reliability of the image encryption algorithm are demonstrated by different security analysis methods, including correlation analysis, robustness analysis and information entropy test. The results of all experiments prove that the proposed image encryption scheme is superior to other existing ones.

Memristor Initial-Offset Boosting in Memristive HR Neuron Model with Hidden Firing Patterns

A new three-dimensional (3D) memristive HR neuron model is presented, which is improved from an existing memristive HR neuron model using a memristor synapse with sine memductance to substitute the original one. The improved memristive HR neuron model has no equilibrium but hidden firing activities can emerge with discrete memristor initial-offset boosting. Treating the neuron model as a two-dimensional (2D) major subsystem controlled by a magnetic flux variable, fold bifurcations for hidden chaotic and periodic firing patterns are elaborated. The coexistence of hidden firing patterns induced by memristor initial boosting is quantitatively analyzed and numerically simulated by bifurcation plots, phase plots, and basins of attraction. The results demonstrate that the improved memristive HR neuron model can exhibit a discrete memristor initial-offset boosting behavior owning infinitely many disconnected basins of attraction and the generating firing patterns can be boosted to different discrete levels by changing the memristor initial value, differing entirely from various boosting behaviors reported previously. Therefore, infinitely many hidden coexisting offset-boosted firing patterns with the same initial-offsets and attractor types are disclosed along the boosting route, which are homogenous with extreme multistability and are perfectly validated by PSIM circuit simulations based on a physically implementation-oriented analog circuit.

A Mixed Discontinuous Galerkin Method for the Helmholtz Equation

In this paper, we introduce and analyze a mixed discontinuous Galerkin method for the Helmholtz equation. The mixed discontinuous Galerkin method is designed by using a discontinuous Pp+1−1−Pp−1 finite element pair for the flux variable and the scattered field with p≥0. We can get optimal order convergence for the flux variable in both Hdiv-like norm and L2 norm and the scattered field in L2 norm numerically. Moreover, we conduct the numerical experiments on the Helmholtz equation with perturbation and the rectangular waveguide, which also demonstrate the good performance of the mixed discontinuous Galerkin method.

Superconvergence of a fully conservative finite difference method on non-uniform staggered grids for simulating wormhole propagation with the Darcy–Brinkman–Forchheimer framework

In this paper, a finite difference scheme on non-uniform staggered grids is proposed for wormhole propagation with the Darcy–Brinkman–Forchheimer framework in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration and auxiliary flux with second-order superconvergence in different discrete norms are established rigorously and carefully on non-uniform grids. We also obtain second-order superconvergence for some terms of the $H^{1}$ norm of the velocity on non-uniform grids. Finally, some numerical experiments are presented to verify the theoretical analysis and effectiveness of the proposed scheme.

Modulated wave formation in myocardial cells under electromagnetic radiation

We exclusively analyze the onset and condition of formation of modulated waves in a diffusive FitzHugh–Nagumo model for myocardial cell excitations. The cells are connected through gap junction coupling. An additive magnetic flux variable is used to describe the effect of electromagnetic induction, while electromagnetic radiation is imposed on the magnetic flux variable as a periodic forcing. We used the discrete multiple scale expansion and obtained, from the model equations, a single differential-difference amplitude nonlinear equation. We performed the linear stability analysis of this equation and found that instability features are importantly influenced by the induced electromagnetic gain. We present the unstable and stable regions of modulational instability (MI). The resulting analytic predictions are confirmed by numerical experiments of the generic equations. The results reveal that due to MI, an initial steady state that consisted of a plane wave with low amplitude evolves into a modulated localized wave patterns, soliton-like in shape, with features of synchronization. Furthermore, the formation of periodic pulse train with breathing motion presents a disappearing pattern in the presence of electromagnetic radiation. This could provide guidance and better understanding of sudden heart failure exposed to heavily electromagnetic radiation.