Using the Logistic Map as Compared to the Cubic Map to Show the Convergence and the Relaxation of the Period–1 Fixed Point

Abstract

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. The 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 $\beta =-1$ when $\alpha =1$ and $\mu =0$ . The cubic map shows its convergence to the fixed point at a pitchfork bifurcation decaying at a power law with exponent $\beta =-\left(1/2\right)$ $\alpha =1$ and $\mu =0$ . However, the system shows their relaxation time at the same power law with exponents and $z=-1$ .