We investigate the dynamical consequences of distributed delay in a coupled Higgs-type scalar field model describing nucleon-meson interactions. Starting from a nonlinear partial differential equation with convolution memory, we derive a reduced four-dimensional autonomous system through a traveling-wave reduction that preserves the delay-induced nonlocal structure. A rigorous bifurcation analysis reveals the existence of a degenerate codimension-two Bautin bifurcation at the trivial equilibrium when the effective cubic nonlinearity vanishes ( \(n=0\) ), identifying a critical organizing center for delay-driven oscillatory instabilities. Beyond this degeneracy, the system exhibits persistent hyperchaotic dynamics characterized by multiple positive Lyapunov exponents (confirming at least two expanding directions in phase space), broadband frequency spectra obtained from Fast Fourier Transform analysis, and fractal attractor geometry. Our results demonstrate that distributed temporal memory acts as a destabilizing mechanism, driving symmetry-breaking transitions and complex scalar field oscillations. To assess the predictability of such delay-induced hyperchaos, we construct data-driven forecasting models using NARX and LSTM architectures. The comparative analysis reveals a crossover in performance: NARX achieves superior accuracy for short-term predictions (up to \(0.10\tau _L\) ), while LSTM maintains better long-term coherence and outperforms NARX at \(h=50\) steps ( \(0.24\tau _L\) ). The study establishes a direct connection between delay-induced bifurcation degeneracy and hyperchaotic scalar field dynamics, offering new insights into instability mechanisms in nonlinear Higgs-type field models.
