  The auxiliary function method allows computation of extremal long-time
averages of functions of dynamical variables in autonomous nonlinear ordinary
differential equations via convex optimization. For dynamical systems defined
by autonomous polynomial vector fields, it is operationally realized as a
semidefinite program utilizing sum of squares technology. In this contribution
we review the method and extend it for application to periodically driven
non-autonomous nonlinear vector fields involving trigonometric functions of the
dynamical variables. The damped driven Duffing oscillator and periodically
driven pendulum are presented as examples to illustrate the auxiliary function
method's utility.
