Uncontrolled manifold analysis provides the synergy index, which quantifies how well a neuro-musculo-skeletal system of an animal, characterized by redundant degrees of freedom with respect to dynamic tasks, is coordinated to maintain a task performance under the influence of perturbations such as noise. The synergy index has been estimated and analyzed for several systems from observation data, while the relation between the underlying dynamics of motions and the synergies, which possibly leads to give new insights to the control theory and the physiology, remains unexplored theoretically. Hence, in this study, we develop a method to obtain the synergy index in stable rhythmic movements subjected to weak white Gaussian noise from a model equation using the Floquet theory. The fundamental property of the Floquet vector, which is covariant with the dynamics, is utilized to uncouple the modes in the complex nonlinear dynamics of body movements. These uncoupled modes provide a semi-analytical expression of the synergy index.
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