Mathematical Sciences: A study of chaos: theoretical and numerical investigations of transversal homoclinic points

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One of the most important situations where chaotic behavior occurs is in systems with transversal homoclinic points. Moreover, this is almost the only instance for which the presence of chaotic dynamics can be rigorously established. In this project Professor Palmer and Kocak will 1) Enlarge the class of equations where the existence of transversal homoclinic points can be proved rigorously; 2) Develop numerical methods for the detection of transversal homoclinic points in specific equations; 3) Analyze the accuracy of numerically computed orbits of dynamical systems, and investigate the role of hyperbolicity therein. The mathematical development will provide insight into the behavior of physical and biological systems in which seemingly random dynamics are actually governed by deterministic natural laws.