  We introduce and analyze an $hp$-version $C^1$-continuous Petrov-Galerkin
(CPG) method for nonlinear initial value problems of second-order ordinary
differential equations. We derive a-priori error estimates in the $L^2$-,
$L^\infty$-, $H^1$- and $H^2$-norms that are completely explicit in the local
time steps and local approximation degrees. Moreover, we show that the
$hp$-version $C^1$-CPG method superconverges at the nodal points of the time
partition with regard to the time steps and approximation degrees. As an
application, we apply the $hp$-version $C^1$-CPG method to time discretization
of nonlinear wave equations. Several numerical examples are presented to verify
the theoretical results.
