Identification of critical points in transient electroconvection dynamics using tensor bundles

Area of Science:

• Physics
• Dynamical Systems
• Nonlinear Dynamics

Background:

• Transient dynamics in complex systems are challenging to analyze.
• Existing methods often require extensive data and complex computations.
• Understanding phase transitions and system behaviors is crucial in fields like fluid dynamics and ecology.

Purpose of the Study:

• To present a novel method for quantitative and qualitative description of transient dynamics from empirical data.
• To simplify Jacobian estimation for analyzing complex systems.
• To apply the method to a novel transient in liquid crystal electroconvection and a predator-prey model.

Main Methods:

• Constructing a tensor bundle around an exemplar transient.
• Creating aligned charts at each step of propagation.
• Reducing Jacobian estimation by one dimension.
• Applying topological analysis to estimated tensor bundles.

Main Results:

• The method reduces data requirements and clarifies results for Jacobian estimation.
• Analysis identified the onset of a boundary crisis in a predator-prey model.
• The tensor bundle from liquid crystal data detailed system behavior during a phase transition.
• A saddle point driving initial dynamics during voltage increase was quantified.

Conclusions:

• The developed method provides a robust framework for analyzing transient dynamics.
• This approach simplifies the study of complex systems, including liquid crystals and ecological models.
• The quantification of a phase space saddle point offers insights into sudden dynamical changes.