Dual Variational Problems and Action Principles for Chen–Lee and Hopf–Langford Systems

ABSTRACT We describe the construction of dual variational principles and action functionals for nonlinear dynamical systems using a methodology based on the dual Lagrange multiplier formalism and a convex optimization approach, to derive families of dual actions that correspond to the given nonlinear ordinary differential system. The method involves a mapping between the original or primal variables and their duals, with the primal variables being expressible in terms of the dual variables and their derivatives and will be referred to as the dual‐to‐primary (or primal) (DtP) map. We demonstrate the application of this technique to the three‐dimensional, autonomous Chen–Lee system, known for its chaotic behavior, and the Hopf–Langford system, which arises in the context of turbulence studies. The derived dual action principles provide an alternative variational description of these nonlinear systems in terms of the Euler–Lagrange equations.