Stability analysis of a discrete chaotic map in superior orbit

Discrete chaotic maps have found a pivotal place in the study of nonlinear dynamics and chaos theory, due to their significant applications in several disciplines including discrete traffic control, secure communications, cryptography, weather forecasting and the population biology of species. The stability analysis of discrete systems via different iterative orbits also plays a prominent role in studying the dynamical behavior of nonlinear systems. In this article, the stability performance of a discrete map is analyzed using two-step superior (Mann) iterative orbit through various dynamical aspects like fixed points, time-series analysis, period-doubling bifurcations and Lyapunov exponent. It is evident to notice that in the superior orbit, due to the freedom of an additional control parameter \(\eta \) $\eta$ , the discrete map admits an improved stability performance up to higher ranges of the growth-rate parameter \(\rho \) $\rho$ . Numerical simulations along with analytical analysis are used to examine the enhanced stability performance in the proposed discrete system. Further, the comparative bifurcation and Lyapunov diagrams report the superior stable behavior in the proposed system for decreasing values of parameter \(\eta \) $\eta$ and in contrast to the existing discrete systems. Thus, the superior stability performance of the discrete system may be used for improved control-based applications like traffic control and population control in the future.