Quasi-sine Fibonacci M set with perturbation

Abstract

Noise is a kind of random variable, and according to the form it appears in the dynamical equations, it can be divided into additive noise and multiplicative noise. By adding the noise into the map of the quasi-sine Fibonacci M set as a kind of random perturbation, research the effect of the random perturbation. In this paper, we first introduce the perturbations into the quasi-sine Fibonacci Mandelbrot set (short for M set). Through changing the strength of the noised parameters, and using the method of combining the mathematical proof and computer graphics, we find two conclusions: (1) when the parameters are conjugate, the two figures of noised quasi-sine Fibonacci M set are symmetry when flipping along the x-axis. When the parameters are real numbers, the figures are symmetry along the x-axis; (2) under different kinds of parameters, the quasi-sine Fibonacci M set shows a different character and degree and rules evolve on the parameter place, but the procedure is continuous. That is to say different noises make different characteristic and degree shape changes. Also, we conclude the change rules, and do the proofs of the given properties.