[MS01] Delay differential equations in mathematical biology and life sciences

[MS01-06] Resonance in Delay Differential Equations with Exact Solutions

*Toru Ohira¹ (1. Graduate School of Mathematics, Nagoya University (Japan))

Keywords:

Delay,Resonance,Exact Solutions,Non-autonomous Delay Differential Equations,W function

Resonance is a fundamental phenomenon, particularly in physics, occurring when an external perturbation matches the inherent oscillation frequency of a system. Meanwhile, the presence of delay, especially in feedback mechanisms, is well known for inducing complex behaviors, including oscillations. Understanding the effects of delays has become a significant issue across various fields, such as biology, mathematics, economics, and engineering. Even in simple systems, delays in feedback or interactions can lead to oscillatory and intricate dynamics. Delay Differential Equations (DDEs) serve as essential tools for analyzing such phenomena.Building on this background, we investigate resonance induced by delay. We introduce two non-autonomous DDEs that exhibit frequency and amplitude resonances, both of which can be solved analytically. In the first case, solutions are constructed as a sum of Gaussian dynamics via Fourier transforms, while in the second, solutions are obtained using Lambert’s W function.A key feature of these models is that, unlike typical resonance phenomena, they do not require an oscillating external force. Consequently, the resulting expressions are remarkably simple, making these mechanisms among the simplest for generating resonance. These examples highlight the rich dynamics introduced by the presence of delay.