The Hansen-Patrick Root-Finding Method is a one-parameter family of cubically convergent root-finding methods. The parameter is called alpha and can be any complex number. With a few different values of alpha, Hansen-Patrick becomes equivalent to other, more well-known root-finding methods. For example, when alpha equals -1, Hansen-Patrick becomes equivalent to Halley’s Method. There has been previous research into the dynamical systems that arise when varying the initial starting point or varying a family of functions. This paper deals with what happens when the initial point and function are fixed but the root-finding method varies. We are interested in spurious cycles that can attract points which would ideally converge to a root of the function. We vary alpha near a known spurious cycle and track what happens to this cycle as alpha varies. Some results we obtain are the standard bifurcation diagram complete with cycle doubling and chaos, as well as Mandelbrot Sets by varying alpha in the complex direction.
Hardy, Preston, "Bifurcation and Non-Convergence in the Hansen-Patrick Root-Finding Method" (2014). Honors Theses. 37.