This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and $4\times 4$-grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms.

In this work we propose an extension to the Standard Model in which we consider a type-III two-Higgs-doublet model (2HDM) plus massive neutrinos and the horizontal flavor symmetry $S_3$ ($\nu 2\mathrm{HDM}\otimes S_3$). In the above framework and with the explicit breaking of flavor symmetry $S_3$, the Yukawa matrices in the flavor-adapted basis are represented by means of a matrix with two texture zeros. Also, the active neutrinos are considered as Majorana particles and their masses are generated through the type-I seesaw mechanism. The unitary matrices that diagonalize the mass matrices, as well as the flavor-mixing matrices, are expressed in terms of fermion mass ratios. Consequently, in the mass basis the entries of the Yukawa matrices naturally acquire the form of the so-called Cheng-Sher ansatz. For the leptonic sector of $\nu 2\mathrm{HDM}\otimes S_3$, we compare, through a ${\chi }^2$ likelihood test, the theoretical expressions of the flavor-mixing angles with the masses and flavor-mixing leptons current experimental data. The results obtained in this ${\chi }^2$ analysis are in very good agreement with the current experimental data. We also obtain allowed value ranges for the “Dirac-like” phase factor, as well as for the two Majorana phase factors. Furthermore, we study the phenomenological implications of these numerical values of the $CP$-violation phases on the neutrinoless double-beta decay, and for long baseline neutrino oscillation experiments such as T2K, $\mathrm{NO}\nu \mathrm{A}$, and DUNE.