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Creation of discontinuities in circle maps
G Derks1, P A Glendinning2, A C Skeldon1
1Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK.
Proceedings. Mathematical, Physical, and Engineering Sciences
|February 24, 2022
Summary
Structural transitions in circle maps, crucial for modeling physical and biological systems, are analyzed. We reveal how discontinuities create singularities and alter bifurcation patterns, particularly in threshold systems and Cherry flows.
Area of Science:
- Dynamical systems theory
- Mathematical modeling in physics and biology
Background:
- Circle maps are fundamental in modeling diverse phenomena, including neuronal activity, cardiac arrhythmias, and sleep/wake cycles.
- Understanding structural transitions and discontinuities in these maps is key to accurately representing complex systems.
Purpose of the Study:
- To investigate how structural transitions occur in circle maps, focusing on the creation of discontinuities.
- To analyze the impact of these discontinuities on the mathematical properties and bifurcations of the maps.
Main Methods:
- Analysis of circle maps near discontinuity creation.
- Investigating singularity formation in map derivatives.
- Examining generic properties of maps with gaps, including border collisions and saddle-node bifurcations.
Main Results:
- Discontinuities in threshold systems and Cherry flows naturally lead to singularities in the map's derivative.
- Threshold systems exhibit square root singularities, altering Arnold tongue structures.
- Loss of injectivity results in multiple gaps and a novel codimension-two bifurcation.
Conclusions:
- The study provides a detailed analysis of circle map behavior near discontinuities.
- Findings amend existing bifurcation theory, particularly the Arnold tongue picture, by incorporating map gaps.
- A new codimension-two bifurcation arising from loss of injectivity is identified.
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