Minimal Cyclic Delay Systems: Symmetry Constraints and the Emergence of Dynamical Complexity

We investigate a minimal delayed dynamical system consisting of a three-node cyclic network.We show that this topology is the smallest directed system whose interaction operator admitsa complex conjugate eigenvalue pair, enabling intrinsic rotational dynamics.Analytical results establish the existence of a delay-induced Hopf bifurcation. Near thebifurcation point, the dynamics reduce to a Stuart–Landau normal form, and numerical evidenceindicates a supercritical transition leading to a stable limit cycle.We demonstrate that cyclic symmetry constrains the dynamics to a low-dimensional phaselockedmanifold: increasing delay deforms the geometry of trajectories but does not increasedynamical dimension, and no quasi-periodic behavior is observed.High-dimensional dynamics arise only when symmetry is broken through modified coupling.These results identify symmetry as a fundamental structural constraint on dynamical complexity,and provide a structural foundation for cyclic delay systems such as the LOGOS framework.