Approximate First Integrals for a System of Two Coupled Van Der Pol Oscillators with Linear Diffusive Coupling

The approximate partial Noether operators for a system of two coupled van der Pol oscillators with linear diffusive coupling are presented via a partial Lagrangian approach. The underlying system of two equations, in general, do not admit a standard Lagrangian. However, the approximate first integrals are constructed by utilization of the partial Noether’s theorem with the help of approximate partial Noether operators associated with a partial Lagrangian. These approximate partial Noether operators are not approximate symmetries of the system under study and they do not form an approximate Lie algebra. Moreover, we show how approximate first integrals can be constructed for perturbed ordinary differential equations (ODEs) without making use of a standard Lagrangian.