Geometric decomposition and potential-based representation of nonlinear systems

This paper considers the problem of representing a sufficiently smooth nonlinear dynamical as a structured potential-driven system. The proposed approach is based on a decomposition of a differential one-form that encodes the divergence of the given vector fields into its exact and anti-exact components, and into its co-exact and anti-coexact components. The decomposition method, based on the Hodge decomposition theorem, is rendered constructive by introducing a dual operator to the standard homotopy operator. The dual operator inverts locally the co-differential operator, and is used in the present paper to identify the structure of the dynamics. Applications of the proposed approach to gradient systems, Hamiltonian systems, and generalized Hamiltonian systems are given to illustrate the proposed approach.