We study the dynamics of some Newton-type iterative methods when they are applied of polynomials degrees two and three. The methods are free of high-order derivatives which are the main limitation of the classical high-order iterative schemes. The iterative schemes consist of several steps of damped Newton's method with the same derivative. We introduce a damping factor in order to reduce the bad zones of convergence. The conclusion is that the damped schemes become real alternative to the classical Newton-type method since both chaos and bifurcations of the original schemes are reduced. Therefore, the new schemes can be utilized to obtain good starting points for the original schemes.
"Reducing Chaos and Bifurcations in Newton-Type Methods." Abstr. Appl. Anal. 2013 (SI36) 1 - 10, 2013. https://doi.org/10.1155/2013/726701