Chaotic dynamical systems provide a rich source of fascinating examples for the mathematical community. However, much of what is believed to be true about such systems has not been rigorously verified, only inferred from numerical experimentation. My work, largely joint with Hüseyin Koçak and Kenneth J. Palmer, bridges the gap between numerical experimentation and rigorous mathematical analysis by a technique called shadowing. Using this technique we have been successful in computing uniform estimates of solutions of chaotic dynamical systems over long time intervals and we have been able to prove the existence and accurately describe periodic orbits of dynamical systems, some with surprisingly long periods. We have now turned our attention to obtaining new classes of results via shadowing.We have found several orbits homoclinic to periodic orbits of the Cremona map. Click on the following links to see animations of orbits for the stated parameter values. Username and password available on request.