Research Article Using the Logistic Map as Compared to the Cubic Map to Show the Convergence and the Relaxation of the Period–1 Fixed Point Patrick Akwasi Anamuah Mensah , 1 William Obeng-Denteh , 2 Ibrahim Issaka, 2 Kwasi Baah Gyamfi, 2 and Joshua Kiddy K. Asamoah 2 1 St. Ambrose College of Education, Dormaa Akwamu, Ghana 2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Correspondence should be addressed to Patrick Akwasi Anamuah Mensah; nanaamonoo12@yahoo.com and William Obeng-Denteh; wobengdenteh@gmail.com Received 19 May 2022; Accepted 20 June 2022; Published 7 July 2022 Academic Editor: Harvinder S. Sidhu Copyright © 2022 Patrick Akwasi Anamuah Mensah et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we employ the logistic map and the cubic map to locate the relaxation and the convergence to the periodic fixed point of a system, specifically, the period—1 fixed point. e study has shown that the period—1 fixed point of a logistic map as a recurrence has its convergence at a transcritical bifurcation having its power-law fit with exponent β − 1 when α  1 and μ  0. e cubic map shows its convergence to the fixed point at a pitchfork bifurcation decaying at a power law with exponent β −(1/2) α  1 and μ  0. However, the system shows their relaxation time at the same power law with exponents and z − 1. 1. Introduction 1.1. Preliminary. In dynamical systems, the most frequent map or function that has been extensively studied is the one- dimensional logistic map [1]. In many studies, the effects of the control parameters when changed shows a change in behavior (asymptotic behavior) of the orbits or trajectories of the map. e transition process of the logistic map shows the cascades of period-doubling bifurcation leading to chaos [2, 3]. Reference [4] asserted that the structural changes in the trajectory or orbit of a given system are termed bifurcation and it was first used in a work by Henri Poincare. In a dynamical system, bifurcation appears when there is a change in parameters that affects the structural system [5]. Bifur- cation is very important in dynamics since the structural change of a system in its behavior or nature is core in its studies. e study of stability (attractors) and instability (repeller) happens when there is bifurcation [6]. In the dy- namical system, the bifurcation diagram helps one to un- derstand the behavior of the system either in its fixed points, stability, periodicity, etc., for instance, when the parameter in the system is varied the system changes affecting its stability. e system may be in equilibrium states when the bifurcation of the system is in one dimension, hence local bifurcation. Local bifurcation as stated by [7, 8] occurs when the points in the neighborhood are in equilibrium, and there are three main types/forms of bifurcation are Saddle-node, pitchfork, and transcritical see [6, 9]. Within the neighborhood [0, 1] of a system, the bifurcation of the map is dependent on the parameter as we keep on varying and iterating it through/ within the neighborhood, see [10, 11]. When the parameter of the system passes its critical value, the type of bifurcation is flipped as a result of a loss of stability of the periodic orbit [10]. e flip bifurcation as cited by [9, 11] is locally su- percritical when the period of the parameter value is double with stable periodic orbits. It is locally subcritical when the periodic orbit is unstable with twice the period of the pa- rameter value as the critical values show a new one. A point exhibits some sort of recurrence behavior when the dynamical system returns the point to itself or to a Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2022, Article ID 1255614, 7 pages https://doi.org/10.1155/2022/1255614