Logistic map: Stability and entrance to chaos

Abstract

Chaos and nonlinear dynamics have taken a crucial place in the mathematics, physics, and engineering worlds. The main focus of this paper is about one famous map in the dynamical system that has an extreme sensitivity to the initial conditions, the logistic map.We first discuss the behaviours of the logistic map under different µ: convergence to 0 when√μ ∈ (0,1), convergence to 1−1/µwhen µ∈ (1,3), 2-cycle when µ∈ (3,1 + 6), further period doubling and eventual chaos, which is in good accordance with our simulation. In the end, we proved three relevant results: the criteria for stability of cycle, the Coppel Theorem, and the famous slogan "period three implies chaos".