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Sealable Femtoliter Chamber Arrays for Cell-free Biology

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Random templex encodes topological tipping points in noise-driven chaotic dynamics.

Gisela D Charó^1,2,3, Michael Ghil^4,5,6, Denisse Sciamarella^1,3,7

¹Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, C1428EGA, Buenos Aires, Argentina.

Chaos (Woodbury, N.Y.)

|October 30, 2023

Summary

This study introduces random templexes, a new mathematical tool to analyze chaotic systems. Random templexes reveal tipping points in dynamical systems by tracking changes in their structure over time.

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Area of Science:

Dynamical Systems Theory
Chaos Theory
Stochastic Processes

Background:

Random attractors are time-evolving pullback attractors in chaotic and perturbed systems.
Branched Manifold Analysis and templexes (cell complexes with digraphs) describe deterministic chaotic attractors.
Existing methods lack a framework for analyzing the temporal evolution of random attractors.

Purpose of the Study:

Introduce the concept and mathematical framework of random templexes.
Develop a method to analyze the temporal structure of random attractors.
Identify and characterize tipping points in stochastic dynamical systems.

Main Methods:

Define random templexes as a sequence of cell complexes, each representing a snapshot of the random attractor.
Utilize a directed graph (digraph) to connect generators (holes) of successive cell complexes.
Apply the random templex framework to the noise-driven Lorenz system.

Main Results:

Random templexes provide a temporal description of random attractors.
Tipping points manifest as significant changes in the holes of the random templex (birth, splitting, merging, death).
The noise-driven Lorenz system's random attractor was successfully computed using random templexes.

Conclusions:

Random templexes offer a novel approach to understanding the dynamics of random attractors.
This framework allows for the identification of critical transitions (tipping points) in stochastic systems.
The study provides a computational method for analyzing complex, time-varying chaotic behavior.