Chaotic Dynamics, Sequence Density, and Chaos Suppression: A Study in Euclidean and Riemannian Spaces Inspired by Collatz-like Problems

In this paper, we explore the surprising behavior of certain discrete maps, drawing inspiration from Collatz-like problems and advanced techniques in analytic number theory, dynamical systems, and differential geometry. Specifically, we investigate a driven cubic-quintic Duffing equation and predict the number of limit cycles around equilibrium points. Furthermore, we develop a novel theoretical framework for chaos suppression in damped-driven systems, leveraging sequences derived from Collatz-like problems. Our study also focuses on analyzing the density of these sequences in both Euclidean and Riemannian metric spaces, providing a comparative analysis of their distribution properties. We estimate the growth rate of sequence density analytically and numerically, highlighting the sensitivity of the density values to the choice of parameters in both metric spaces. Our results present new insights into the behavior of these sequences and their implications for number theory, chaotic dynamics, and geometric analysis.