Paper · Sep 15, 2024

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

Zeraoulia Rafik, Sobhan Sobhan Allah

math.DSarXiv:2410.02457
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As on arXiv

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.

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Submitted
Sep 15, 2024
Subjects
math.DS, astro-ph.SR, math.DG

Comments: 30 pages,8 figures

Ricci flowPerelman entropy

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)3Chan 2021 Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow (Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al., 2022)8Cartas-Fuent 2022 Geometric flows and the Swampland (Davide De Biasio, 2023)7De Biasio 2023 Local smooth convergence of F-limit flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)6Chan 2023 Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear σ model (Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al., 2024)2Kim 2024 The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory (Sergiu I. Vacaru and Elşen V. Veliev, 2025)5Vacaru 2025 Stochastic Thermodynamics on Time-Evolving Curved Spaces (Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al., 2026)1Nagase 2026 On the singularity formation of gauge fields coupled to Ricci flow (Andoni Royo Abrego, 2026)4Abrego 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al.

    Topics Perelman entropy · Shared terms “contributes”, “quantify”

  2. Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear model

    Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al.

    Topics Perelman entropy · Shared terms “utilizing”, “integrating”

  3. A local Sobolev inequality on Ricci flow and its applications

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

  4. On the singularity formation of gauge fields coupled to Ricci flow

    Andoni Royo Abrego

    Topics Perelman entropy

  5. The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

    Sergiu I. Vacaru and Elşen V. Veliev

    Topics Perelman entropy

  6. Local smooth convergence of -limit flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

  7. Geometric flows and the Swampland

    Davide De Biasio

    Topics Perelman entropy · Shared terms “dynamical”

  8. Evolution of Lifshitz metric anisotropies in Einstein-Proca theory under the Ricci-DeTurck flow

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar et al.

    Topics Perelman entropy · Shared terms “increase”, “exponent”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.