Finite difference method

Abstract: A finite-difference time-domain modeling of finite-size zero thickness space–time-modulated Huygens’ metasurfaces based on generalized sheet transition conditions is proposed and numerically demonstrated. A typical all-dielectric Huygens’ unit cell is taken as an example and its material permittivity is modulated in both space and time, to emulate a traveling-type spatio-temporal perturbation on the metasurface. By mapping the permittivity variation onto the parameters of the equivalent Lorentzian electric and magnetic susceptibility densities, $\chi _{\text {ee}}$ and $\chi _{\text {mm}}$ , the problem is formulated into a set of second-order differential equations in time with nonconstant coefficients. The resulting field solutions are then conveniently solved using an explicit finite-difference technique and integrated with a Yee-cell-based propagation region to visualize the scattered fields taking into account the various diffractive effects from the metasurface of finite size. Several examples are shown for both linear and space–time varying metasurfaces which are excited with normally incident plane and Gaussian beams, showing detailed scattering field solutions. While the time-modulated metasurface leads to the generation of new collinearly propagating temporal harmonics, these harmonics are angularly separated in space, when an additional space modulation is introduced in the metasurface.

Abstract: In this paper we are interested in the fractional-order form of Chua’s system. A discretization process will be applied to obtain its discrete version. Fixed points and their asymptotic stability are investigated. Chaotic attractor, bifurcation and chaos for different values of the fractional-order parameter are discussed. We show that the proposed discretization method is different from other discretization methods, such as predictor-corrector and Euler methods, in the sense that our method is an approximation for the right-hand side of the system under study.