Dynamical systems are complex and constantly changing systems that exhibit predictable and unpredictable behaviour because of their inherent randomness and sensitivity to initial conditions. In the last few years, the dynamics of various fixed-point recursive methods and chaotic maps have received significant attention from the research community. Generally, in dynamical systems, the standard dynamics revolve around the chaotic map λp(1-p), where the growth rate parameter λ ∈ [0, 4]. In this article, a novel Geometric Mean-Based fixed point recursive method is used to examine the dynamical behaviour in the chaotic map λp(1−p) in which the growth rate parameter λ ∈ [0, 4]. approaches a maximum value of 6.7. Furthermore, the mathematical and computational study reveals the efficiency of the proposed approximation method. In this method, the logistic map admits extra freedom in the parameter λ, which gives improved dynamic properties such as fixed point, periodicity, chaos, and Lyapunov exponent. Additionally, it has been noted that better dynamic performance could enhance various applications such as weather forecasting, secure communications, neural networks, cryptography, and discrete traffic flow models, etc.