A fully coupled FBSDE framework for optimal orbit steering near an Arneodo-Coullet-Tresser spiral attractor

This paper develops a fully coupled forward-backward stochastic differential equation (FBSDE) framework for optimal orbit steering near a spiral attractor generated by an Arneodo-Coullet-Tresser-type chaotic oscillator. The central objective is not to suppress chaos by point stabilization, but to preserve the intrinsic orbital geometry of the attractor while reducing stochastic deviations from a prescribed reference orbit segment. To achieve this goal, a stochastic tracking problem is formulated for a noisy controlled ACT system, and the corresponding stochastic maximum principle yields a nonlinear fully coupled FBSDE in which the third backward component enters the active chaotic channel through the optimal feedback law. The resulting model establishes a direct link between spiral chaotic dynamics, stochastic optimal control, and adjoint backward propagation. A compact tensor formulation is provided, together with a variational derivation, a local solvability discussion under structured assumptions, and a decoupling-field interpretation. The framework offers a mathematically coherent and comparatively underexplored route for investigating stochastic orbit steering, random attractor neighborhoods, and control-energy allocation in a spiral chaotic environment.