Contractibility of a persistence map preimage.

Area of Science:

• Dynamical Systems Theory
• Topological Data Analysis
• Differential Equations

Background:

• Dynamical systems are often analyzed using time series data.
• Persistence diagrams, from topological data analysis, capture essential topological features of system snapshots.
• Connecting topological features to the underlying dynamical system's solutions is a key challenge.

Purpose of the Study:

• To investigate what conclusions can be drawn about solutions of a dynamical system from its time series of persistence diagrams.
• To associate persistence diagrams with initial conditions in an N-dimensional ordinary differential equation system.
• To explore the application of this association for identifying fixed points of differential equations.

Main Methods:

• Associating a persistence diagram with each point (initial condition) in the N-dimensional ordinary differential equation system.
• Analyzing the topological properties of the preimage of each associated persistence diagram.
• Developing conditions for inferring the existence of fixed points from multiple time series of persistence diagrams.

Main Results:

• The core finding is that the preimage of every associated persistence diagram is contractible.
• This contractibility property provides a theoretical foundation for relating topological features back to the system's dynamics.
• Demonstrated conditions under which multiple time series of persistence diagrams can reliably indicate the existence of a fixed point.

Conclusions:

• The association of persistence diagrams to dynamical systems provides a powerful tool for data-driven analysis.
• The contractibility of persistence diagram preimages is a significant theoretical result with practical implications.
• This framework enables the detection of fundamental properties, such as fixed points, of differential equations from observed topological data.