Fractional Mandelbrot sets with impulse

Abstract

This paper proposes fractional Mandelbrot sets with impulses based on a fractional difference map, and preliminarily investigates their fractal dynamic characteristics by numerical methods and graphical explorations. The influence of impulse and parameters on the novel fractal sets is performed by dimension analysis. It confirms that the symmetry of the Mandelbrot set can be preserved under piecewise linear random impulse. Several extensive numerical experiments are done to support the main conclusions.