Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

TL;DR

This paper introduces a novel chaotic model for star positional dynamics using Ricci flow and Perelman's entropy, revealing increased unpredictability over time and advancing celestial chaos understanding.

Contribution

It combines Ricci flow, Perelman's entropy, and chaos theory to model star dynamics, providing new insights into celestial system complexity and unpredictability.

Findings

Entropy increases exponentially, indicating rising unpredictability.

Lyapunov exponents confirm chaotic behavior.

Bifurcation analysis shows parameter sensitivity.

Abstract

This paper explores a novel approach to modeling the positional dynamics of stars using discrete dynamical systems. We define star evolution through discrete-time update rules based on right ascension, declination, and distance, incorporating chaotic behavior via nonlinear functions and external perturbations. By applying Ricci flow and Riemannian metrics, we provide new insights into the positional dynamics of stars. Theoretical computations of Perelman entropy are used to assess system complexity, with high-precision Runge-Kutta methods ensuring accurate solutions for our chaotic model. We quantify chaos using Lyapunov exponents and perform bifurcation analysis to study how parameter variations affect the dynamics. Comparing our model to the Lorenz attractor reveals both similarities and unique characteristics in stellar dynamics. Our results show that entropy increases exponentially, indicating that predicting star positions with precision becomes increasingly challenging over time. This study advances the understanding of chaos in celestial systems and contributes to dynamical systems theory by integrating chaos theory with astronomical modeling.