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We employ nonsmooth transformations of the independent coordinate to analytically construct families of strongly nonlinear periodic solutions of the harmonically forced nonlinear pendulum. Each family is parametrized by the period of oscillation, and the solutions are based on piecewise constant generating solutions. By examining the behavior of the constructed solutions for large periods, we find that the periodic orbits develop sensitive dependence on initial conditions. As a result, for small perturbations of the initial conditions the response of the system can ‘jump’ from one periodic orbit to another and the dynamics become unpredictable. An analytical procedure is described which permits the study of the generation of periodic orbits as the period increases. The periodic solutions constructed in this work provide insight into the sensitive dependence on initial conditions of chaotic trajectories close to transverse intersections of invariant manifolds of saddle orbits of forced nonlinear oscillators.