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A Modeling and Simulation Method for Preliminary Design of an Electro-Variable Displacement Pump

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Assigning probabilities to non-Lipschitz mechanical systems.

Danny E P Vanpoucke¹, Sylvia Wenmackers²

¹Faculty of Sciences, UHasselt, Agoralaan Gebouw D, 3590 Diepenbeek, Belgium.

Chaos (Woodbury, N.Y.)

|January 1, 2022

Summary

This summary is machine-generated.

We developed a novel method to assign probabilities to solutions of initial value problems with Lipschitz singularities, using a deterministic model for indeterministic systems. This approach provides a new way to analyze complex differential equations, including Norton's dome.

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Area of Science:

Mathematics
Physics
Applied Mathematics

Background:

Initial value problems with Lipschitz singularities pose significant analytical challenges.
The physical interpretation of these problems, such as Norton's dome, highlights the need for robust solution methodologies.
Existing methods may struggle with the inherent indeterminism and singularities.

Purpose of the Study:

To introduce a novel method for assigning probabilities to solutions of initial value problems with Lipschitz singularities.
To demonstrate the method's efficacy using a specific physical model (Norton's dome).
To bridge the gap between deterministic modeling and probabilistic outcomes in complex systems.

Main Methods:

Utilizing finite difference equations for a deterministic framework.
Applying techniques from alpha-theory for analyzing infinitesimal perturbations.
Incorporating a uniform prior on the canonical phase space.
Developing a deterministic, hyperfinite grid model.

Main Results:

Successfully assigned probabilities to solutions of the initial value problem.
The method provides a probabilistic interpretation for a deterministic model.
Demonstrated applicability to problems like the Norton's dome example.

Conclusions:

The proposed method offers a robust framework for probabilistic analysis of singular initial value problems.
This approach can handle systems previously considered intractable due to singularities.
It opens new avenues for research in mathematical physics and computational methods.