I work in early intervention with children with autism. I'm an occupational therapist. Children who want to bounce due so for a reason. They are seeking propioceptive input. When they bounce, the nerves in their muscles and joints are stimulated and give them a feeling of being alive. it's similar to the good feeling we adults can get when we run or take a jazz aerobic class and we feel vital and alert.
Rather than begin the impossible task of getting him to stop, it's better to honor his need and serve it in different ways. One way is to serve this need in a game form between the parent and child. Since children with ASD (autistic syndrome disorder) have difficulty with social interaction, be part of your child's bounce. Be the one who gives him that joy. Sit him on your knees facing you and play "pony boy" where you bounce him and sing the pony boy song "Pony boy, pony boy, won;t you be my pony boy. Don't say no, here we go. Ride away with me". Bounce him up and down doing the first verse. Side to side during the second and, if he likes it, let him fall backwards while you are holding his hands, on the last.
You can also play a similar game using a large gym ball where he is sitting on the ball facing you and you are holding his hips.As you bounce him, sing whatever song comes to mind. Have fun with him.
Other less bouncy ways that serve his need is to wrap him up tight in a small blanket and then unroll it letting him roll out.
You can also buy tight velcro vest or even weighted blankets he can wrap himself up in. All these way give him the feedback that he is seeking because his sensory system in not integrated.
Don't worry, he won't want to bounce forever. His system will get integrated especially if he get good early intervention
Check out the book Out-of Sync child by Carol Kranowitz for more ideas or email me. I'll be happy to help you.
Barbara.sher@gmail.com