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An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

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Two-fold singularities in nonsmooth dynamics-Higher dimensional analogs.

Simon Webber¹, Mike R Jeffrey¹

¹Department of Engineering Mathematics, University of Bristol, Merchant Venturer's Building, Woodland Road, Bristol BS8 1UB, United Kingdom.

Chaos (Woodbury, N.Y.)

|October 2, 2020

Summary

Higher dimensional fold and multi-fold singularities in discontinuous differential equations surprisingly simplify. These complex systems reduce to the classic two-fold singularity, offering new tools for analyzing discontinuous systems.

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Area of Science:

Dynamical Systems Theory
Differential Equations
Singularity Theory

Background:

Discontinuities in ordinary differential equations can lead to fold and two-fold singularities.
The classic two-fold singularity is well-understood but represents only the simplest case.
Higher-dimensional analogs arise from multiple intersecting thresholds and complex sliding flows.

Purpose of the Study:

To investigate the nature of higher-dimensional fold and multi-fold singularities.
To determine if complex higher-dimensional singularities can be simplified.
To provide new analytical tools for discontinuous dynamical systems.

Main Methods:

Analysis of systems of ordinary differential equations with discontinuities.
Geometric investigation of flow behavior along intersecting thresholds.
Mathematical reduction of higher-dimensional singularity equations.

Main Results:

Higher-dimensional analogs of two-fold and multi-fold singularities were identified.
It was surprisingly shown that these complex systems reduce to the classic two-fold singularity equations.
This reduction provides a foundational step for studying higher-dimensional discontinuous systems.

Conclusions:

Higher-dimensional singularities in discontinuous differential equations are not necessarily more complex than the classic two-fold.
The reduction to the classic two-fold provides a powerful new tool for analysis.
This work opens avenues for understanding intricate dynamics in higher-dimensional discontinuous systems.