Bifurcation Analysis and Coexisting Attractors in a Hénon Map with State-Dependent Step Delay

We study a two-parameter family of planar difference equations obtained from the classical Hénon map by letting the delay of the quadratic term depend on the sign of the current state: the term 𝑥2 𝑛 is used when 𝑥𝑛 ≥0 and 𝑥2 𝑛−1 when 𝑥𝑛 <0. The resulting map is discontinuous on the switching line 𝑥𝑛 =0 and non-invertible. We prove that the two Hénon fixed points persist but that the saddle one becomes a repeller, that the stable 2-cycle of the Hénon map persists exactly up to 𝑎∗ =(1−𝑏)2, and that no 2-cycle exists beyond this value. Numerical experiments for 𝑏 =0.3 show that, beyond 𝑎∗, the attractor is a sequence of stable odd-period cycles whose period decreases as 𝑎 grows, followed by chaotic regimes, and that several attractors (for instance a 5-cycle and a 10-cycle at 𝑎 =0.9) coexist with finely interwoven basins of attraction. For 𝑏 =0.3 we prove, with explicit tail estimates, that stable cycles of every odd period 𝑁 ≥5 exist on intervals of 𝑎 that accumulate at 𝑎∗ geometrically.