Using the logistic map as compared to the cubic map to show the convergence and the relaxation of the period-1 fixed point (Q2081866)

Summary: 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 \(z=-1\).