Nonlinear Dynamics In Number Theory: Exploring Iterative Functions And Chaos

Abstract

The role of chaos theory in discrete number theoretic iterations has not been studied extensively. It studies the emergence of chaos in modular recurrence functions and develops a rock-solid mathematical framework of their dynamical properties. By employment of the Lyapunov exponent, bifurcation analysis, and fractal dimension calculations, this study provides new links between nonlinear dynamics and number theory. Adapting classical sensitivity to initial conditions, topological mixing, and periodic point density criteria, this defines chaos in discrete number theoretic systems, that is, modular arithmetic functions. Lyapunov exponents for exponential divergence are computed for modular recurrence functions to quantify the rates at which two points diverge, and bifurcation diagrams are generated to visualize stability transitions. Furthermore, fractal analysis is used to analyze the self-similarity and complexity of prime-based sequences. Theoretical conclusions are corroborated using computational simulations. The results show that iterative number theoretic functions have unique chaotic properties. Lyapunov exponent analysis shows that for some modular functions λ>0.8, i.e., sensitive dependence on initial conditions. It is shown that the analysis of bifurcations exhibits transitions from periodic to chaotic behavior in the form of classical period-doubling cascades. Moreover, further computations of fractal dimension confirm that prime-generated modular sequences possess self-similarity with Hausdorff dimension D_H≈1.58. This study furnishes a very rigorous mathematical and computational framework to detect chaos in discrete number theoretic systems, and the implications are in cryptography, pseudo-random number generation, and computational complexity. These results indicate that chaotic modular recurrence functions can be used in the design of secure encryption schemes and efficient randomness sources. Future work should involve higher dimensional modular iterations and chaos-based cryptographic application research.