Energy landscapes and synergetic state transitions in frustrated Stuart-Landau oscillator networks: a homotopy continuation study.

Introduction: Energy landscapes provide a useful lens for understanding multistability and state transitions in self-organizing systems, but systematic characterization of the energy landscapes for coupled oscillator networks remains less well developed. Here, we study a frustrated Stuart-Landau oscillator network on a 2D toroidal lattice with competing local coupling and global frustration and characterize how its macroscopic states and noise-driven transitions reorganize as the frustration strength is varied. Method: We combine a mode-based, phase-decoupled approximation with homotopy continuation and track representative families of equilibria from the approximation to the full model. In singular cases where the leading XY-Hamiltonian phase interaction yields non-isolated critical points, we impose a second-order phase equilibrium condition and solve it in a constrained-solvability sense to ensure that continuation is well posed. Results: Numerical continuation shows partial selectivity of this homotopy, in which lower-energy initializations tend to continue to similarly lower-energy, lower-index equilibria in the full system. Across frustration regimes, continuation shows how approximate equilibria split and reorder by energy and index; at intermediate frustration, extensive runs reveal multistability between an isolated global minimum and multiple 1D troughs contained within the edges of a multigraph. Guided by the resulting landscape-level picture, simulations demonstrate distinct noise-dependent synergetic phenomena, including noise-induced synchronization, noise-induced desynchronization, and persistent switching among coexisting macrostates. Stationary transition kinetics exhibit both Arrhenius-Kramers-like (activated) and diffusion-limited scaling with respect to noise. Discussion: These results support a coupled-oscillator-based framework for analyzing the energy landscape of multivariate time series, with potential applications in neuroscience, physiology, and beyond.