Existence and stability of sawtooth periodic solutions in a state-dependent switching system

Jianshe Yu; Chunming Ju; Yufeng Wang; Kai Wang; Huaiping Zhu

doi:10.3934/dcds.2025173

Article Contents

2026, Volume 49: 381-402. Doi: 10.3934/dcds.2025173

This volume Previous Article Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy Next Article Ruelle's inequality and Pesin's formula for Anosov geodesic flows in non-compact manifolds

Existence and stability of sawtooth periodic solutions in a state-dependent switching system

1.
Guangzhou Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China
2.
LAMPS and Center of Diseases Modeling CDM, Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada

^*Corresponding author: Kai Wang
^*Corresponding author: Kai Wang

Received: June 2025

Revised: October 2025

Early access: October 30, 2025

Published online: October 30, 2025

The research is supported by the National Natural Science Foundation of China (12331017, 12401633) and the China Postdoctoral Science Foundation (2024T170191).

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Abstract

In this paper, we investigate the existence and stability of sawtooth periodic solutions of a new scalar switching system with state dependence. We find two threshold regions for the switch, denoted as $ D_M^* $ and $ D_M^{**} $, such that every solution will eventually converge to a constant if the threshold is located in $ D_M^* $, but there exists a unique globally asymptotically stable sawtooth periodic solution if it is located in $ D_M^{**} $ under suitable conditions with the aid of the Poincaré mapping method. Furthermore, the non-existence of sawtooth periodic solutions is investigated through utilizing an ingenious contradiction argument which seems to be the first attempt. As an example, we apply our theory to show the existence of novel forms of sawtooth periodic solutions in a mosquito population suppression model.

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Mathematics Subject Classification: Primary: 34A38, 34C25; Secondary: 34D05, 34D23.

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Figure 1. Schematic diagram of sequence $ \beta_{v}^{ip} $

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Figure 2. Schematic diagram of function $ \mathcal{B}_v(s;u_{0}) $

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Figure 3. Depiction of $ \mathcal{B}_{v}(vp;{u}_{0}^{f}) $ within one period

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Figure 4. Stability of $ w_2^\mathcal{R} $ in Theorem 4.1 (iii) and (v)

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Figure 5. Sawtooth periodic solutions in Theorem 4.1 (vi-1) and (vi-2)

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Figure 6. Sawtooth periodic solutions in Theorem 4.1 (vii-1) and (vii-2)

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