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Dynamical systems defined on simplicial complexes: Symmetries, conjugacies, and invariant subspaces
1ICMC São Carlos, Universidade de São Paulo, Av. Trab. São Carlense 400, São Carlos SP 13566-590, Brasil.
Chaos (Woodbury, N.Y.)
|October 1, 2022
Summary
We explore dynamical systems on simplicial complexes, revealing how their symmetries influence system dynamics and create invariant subspaces. This work clarifies the relationship between topological structure and system behavior.
Area of Science:
- Mathematics
- Dynamical Systems Theory
- Topology
Background:
- Dynamical systems are fundamental to understanding complex phenomena.
- Simplicial complexes provide a powerful framework for modeling discrete structures.
- Understanding the interplay between structure and dynamics is crucial.
Purpose of the Study:
- To develop a general model for dynamical systems on simplicial complexes.
- To characterize the conjugacy classes of these systems.
- To investigate how symmetries in the complex affect the system's dynamics.
Main Methods:
- Defining dynamical systems on the vertices and edges of a simplicial complex.
- Analyzing the structure of conjugacy classes for these systems.
- Identifying and characterizing invariant subspaces within the dynamics.
Main Results:
- A comprehensive description of conjugacy classes for dynamical systems on simplicial complexes.
- Demonstration of how symmetries in the simplicial complex directly influence the dynamics.
- Identification of specific invariant subspaces arising from these symmetries.
Conclusions:
- The study provides a theoretical framework for analyzing dynamical systems on structured spaces.
- Symmetries in the underlying topological structure are key determinants of dynamical behavior.
- The findings offer insights into the emergence of order and stability in complex systems.
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