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A. The Tulczyjew's triplet is a geometric framework that makes the Legendre transformation available even for the singular systems. In this work we present the trivialization, in both the horizontal and the vertical terms, and the reduction of the classical triplet under a symmetry group action in the presence of an Ehresmann connection. We thus establish a geometric pathway for the Legendre transformations of singular dynamical systems after reduction.

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Phase synchronization of weakly coupled limit cycle oscillators are related to the stability of the zero solution of the reduced-order dynamics of phase differences, represented by a systems of differential equations on a hypertorus. Using Rantzer's density function, a dual form of Lyapunov function, we propose a method to certify almost global stability of an equilibrium on a hypertorus. We show that the proposed method can certify robustness of phase synchronization of all-to-all and weakly coupled limit cycle oscillators with respect to disturbances in phases. The method leverages sum of squares polynomial optimization to construct the certification function.