Global Hopf bifurcation for differential equations with multiple threshold‐type state‐dependent delays

• Published in: Mathematical methods in the applied sciences Vol. 47; no. 2; pp. 1065 - 1094
• Main Authors: Efendiyev, Messoud, Hu, Qingwen
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30-01-2024
• Subjects: Bifurcation theory; Carbon dioxide; Carbon dioxide concentration; Differential equations; global Hopf bifurcation; Hopf bifurcation; Mathematical models; multiple threshold‐type state‐dependent delays; Oscillations; Respiratory system; S1$$ {S}^1 $$‐equivariant degree
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9700

Using the
S1$$ {S}^1 $$‐equivariant degree, we develop a global Hopf bifurcation theory for system of differential equations with multiple threshold‐type state‐dependent delays whose prototype is the human respiratory system with multiple blood transport time delays. To establish a theoretic framework for modeling practices of periodic breathing, we further investigate the periodic oscillations of carbon dioxide concentrations in the lung, brain, and tissue compartments and conduct a local and global Hopf bifurcation analysis for the model when varying the commensurate scale of the multiple delays in a transformed system. Such a global Hopf bifurcation will indicate the onset and persistence of the periodic oscillations.