Theory for “gluing” cyclic unions in multi‑gait networks

Develop theoretical results that characterize fixed point supports, survival conditions, and attractor structure for networks constructed by gluing multiple cyclic unions (rather than forming a single cyclic union), as in the 24‑node five‑gait quadruped network. Provide criteria analogous to the cyclic union theorem to predict FP(G), basin structures, and coexistence of gait limit cycles without parameter changes.

Background

Cyclic unions admit a complete characterization of FP(G) and sequential attractors. The five‑gait quadruped network is built by ‘gluing’ several cyclic‑union‑based gait modules that share leg nodes, producing coexistence of diverse limit cycles without parameter changes.

The authors note that this glued architecture is not a pure cyclic union and that available theorems do not directly apply. A general theory of ‘gluing’ cyclic unions would extend existing CTLN structural results and explain the observed robustness and coexistence of attractors.