Analysis of the basins of attraction generated by a pseudo Newton-Raphson method iterating over complex order derivatives

The aim of this study is to analyze visually the evolution and dynamic behavior of the basins of attraction and polynomiographs generated by a pseudo Newton-Raphson method, where the first order derivative is replaced by fractional derivatives of complex order. The functions of choice are polynomials as they have been studied extensively in the past and the dynamical analysis of iterative methods based on them is well understood. The pseudo Newton-Raphson method iterates over both the analytical form of the derivatives and their approximation from the definition of Grünwald. The generated polynomiographs and basins of attraction are colored according to the required number of iterations for the method to converge and the corresponding convergent roots of the studied polynomials respectively. In the last section, an analysis of polynomiographs generated from imaginary order derivatives is presented.