Hopf Bifurcation of the Ring Unidirectionally Coupled Toda Oscillators with Distributed Delay

This paper investigates the bifurcation problem associated with a ring-shaped one-way coupled Toda oscillator incorporating distributed time delays. We define precise criteria to assess the stability and bifurcation conditions of the positive equilibrium within the model incorporating distributed time delays, applying the Routh–Hurwitz criterion. Our study enriches the characterization of Toda oscillators to better capture their dynamic properties. This improvement takes into account the infinite memory feature associated with distributed time delays, thereby refining our understanding of their behavior. To substantiate our theoretical analysis, we conduct comprehensive numerical simulations.