Higher-Order Interactions as Activators of Latent Bifurcation Modes: A Formal Mapping Between Polyadic Coupling and the Landau Free Energy Expansion

Higher-order interactions—simultaneous couplings among three or more dynamical
units—produce collective phenomena such as explosive synchronization, multistability,
and hysteresis that are absent in pairwise-coupled networks. We prove that these phe-
nomena arise from a formal mathematical correspondence between higher-order network
dynamics and classical Landau theory of phase transitions. Specifically, we show that
the Ott-Antonsen reduced equilibrium equation for the higher-order Kuramoto model is
structurally identical to the critical point condition of a Landau free energy functional, with
the effective order of each interaction term mapping to the degree of the corresponding
Landau expansion term. On this basis, we establish two main results. First, an interaction
of effective order 𝑝 generates a term of degree 2𝑝 in the Landau free energy (Theorem 1).
Second, higher-order interactions of effective order 𝑝 ≥ 2 are necessary and sufficient to ac-
tivate subcritical (discontinuous, first-order) bifurcations; pairwise-only coupling is confined
to supercritical (continuous, second-order) transitions (Theorem 2). We derive a general
formula for the maximum number of coexisting stable states as a function of interaction
order and show that the correspondence is preserved through the entire phase reduction
chain from Hopf bifurcation to the macroscopic order parameter. These results reframe the
diverse phenomenology of higher-order network dynamics as the progressive activation of
latent bifurcation modes within the Landau-Stuart framework, with implications for social,
biological, and ecological systems where polyadic interactions dominate.