A nonlinear FBSDE framework for weakly coupled stochastic Rabinovich–Fabrikant attractors under common noise

Abstract This paper develops a nonlinear forward–backward stochastic differential equation (FBSDE) framework for a pair of weakly coupled Rabinovich–Fabrikant (RF) oscillators under common stochastic excitation. The study is motivated by the fact that RF dynamics are much less explored in stochastic control and backward stochastic analysis than more classical chaotic models, even though they possess highly intricate nonlinear structures and rich attractor morphology. A six-dimensional controlled stochastic model is first constructed by combining diffusive coupling, common Brownian forcing, and a synchronization-oriented control channel. A quadratic performance criterion is then introduced to balance synchronization quality and control expenditure. On this basis, the stochastic maximum principle yields a fully coupled nonlinear FBSDE in which the forward equation is governed by the coupled RF drift, while the backward adjoint equation is driven by the transpose Jacobian of the chaotic vector field. This adjoint structure transmits the geometric sensitivity of the attractor into the backward dynamics and makes the resulting system a demanding benchmark for stochastic–chaotic control theory. The paper further derives the Hamiltonian representation, the optimal feedback characterization, the decoupling relation, and the corresponding dynamic programming interpretation. The formulation clarifies the interaction among weak deterministic coupling, common noise-induced coherence, and energy-limited control action. It also provides a mathematically consistent route for studying controlled deformation of RF attractors in stochastic environments. The resulting framework is expected to be useful for future work on synchronization transitions, common noise effects, hidden attractor-related regulation, and numerical FBSDE analysis in strongly nonlinear systems.