First- and second-order framings of Mandelbrot sets and structure of fixed points of quadratic polynomials

The paper continues the research of R. Kronover and J. Minlor, H.-O. Peitgen and P. H. Richter. Using mathematical methods and computer experiments, the first- and second-order framings of the Mandelbrot sets of three families of the quadratic polynomials of a complex variable are revealed. The connection between the first- and second-order framings of Mandelbrot sets of the functions $f_{2}(z)=z^2+cz$, $f_{3}(z)=z^{2}+z+c$, and $f_{4}(z)=cz^{2}+c$ with remarkable curves (cardioid, lemniscate, and circle) was established. The algorithms of constructing the framings of Mandelbrot sets of considered functions in the mathematical package MathCad and Pascal programs have been worked out. Algorithms for constructing Mandelbrot sets in Pascal programs are developed.