AIMR Math Group Seminar2025/7/7
Date: July. 7th (Mon) 2025 10:00-12:00
Venue: AIMR, 2F Seminar room
1.
10:00 -- 10:50
Speaker: Pallab Navojit Dhali (Mathematical Institute, Tohoku University)
Title: Canard Dynamics and Synchronization in the Network of Three Time Scale Systems
Abstract:
In this study, three time scale systems are analyzed to explore complex oscillatory dynamics, with a focus on canard dynamics and synchronization in heterogeneous oscillator networks. Geometric singular perturbation theory is employed to investigate the slow manifolds, while the blow-up method is used to reveal dynamics at non-hyperbolic points. A synchronization condition is derived that tunes the coupling strength by leveraging canard-induced bursting delays, ensuring that fast variables remain synchronized within a specified tolerance. The study applies this analysis to the Pancreatic β-cell model, where numerical simulations reveal synchronization thresholds for bursting initiation, crucial for insulin secretion. The framework is useful for research fields such as neuroscience, chemistry, and ecology, offering insights into multiscale systems.
2.
11:00 -- 11:50
Speaker: Zhou Changyu (Mathematical Institute, Tohoku University)
Title: Quasi-Homogeneous Integrable Systems: Free Parameters, Kovalevskaya Exponents, and the Painlevé Property
Abstract:
This work explores quasi-homogeneous integrable systems, focusing on their resonance condition, Frobenius manifold structure, and Hamiltonian nature. Motivated by Kovalevskaya exponents in four-dimensional Painlevé equations, we derive an arithmetic resonance condition governing the emergence of fractional powers and the breakdown of the Painlevé property. We construct a Frobenius manifold structure on the parameter space, which becomes conformal when the weights coincide. Furthermore, we show that the induced flow preserves a symplectic form and deduce the pairing property of the Kovalevskaya exponents. Additionally, we clarify the interaction between the deformation and original flow exponents via the parameter space, revealing a structural link between their singularity behaviors. These results advance the understanding of deformation theory and singularity structures in integrable systems.
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