The Regularized Visible Fold Revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter ϵ → 0 . Alternatively, these singularly perturbed systems can be thought of as r...

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Bibliographic Details
Published in:Journal of nonlinear science 2020-12, Vol.30 (6), p.2463-2511
Main Author: Kristiansen, K. Uldall
Format: Article
Language:English
Subjects:
Analysis
Belt conveyors
Bifurcations
Classical Mechanics
Economic Theory/Quantitative Economics/Mathematical Methods
Mass-spring systems
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Singular perturbation
Theoretical
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Summary:The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter ϵ → 0 . Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit ϵ → 0 , grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law.
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-020-09627-8