At 1125 Mountain time, the special light-sport airplane sustained substantial damage following a loss of control during landing. The flight instructor and student sport pilot were not injured. Visual conditions prevailed. The airplane touched down on its main wheels, then the nose wheel was lowered, slightly right of the runway centerline. The instructor asked the student to apply light rudder pressure during the roll-out to realign the aircraft with the runway centerline. Ample left rudder pressure was applied and countered with opposite right rudder pressure, resulting in pilot-induced oscillations. The airplane came to a stop in a drainage ditch. Winds were reported as calm during the accident. Examination revealed the nose gear was bent aft and the firewall was buckled.