Simple Time-Periodic Delay Can Support Complex Dynamics

In this paper, we investigate the complex dynamics of a mapping derived from a differential equation with simple time-periodic delay. Firstly, we calculate the truncated normal form of 1:1 resonance of the mapping at a degenerate fixed point and obtain an approximating system of the mapping by using Picard iteration. By analyzing the approximate system, we find that the mapping will undergo a 1:1 resonance at the degenerate fixed point. Secondly, the qualitative property and the stability of the degenerate fixed point are determined, which provide a new view to understand the dynamic of differential equation with simple time-periodic delay. However, the approximate system does not have the versal unfolding of the Bogdanov–Takens singularity of codimension 2. These phenomena show that simple time-periodic delay can support complex dynamics. Finally, a numerical simulation is carried out to verify the analytic results.