Mathematical Sciences: Computational Algorithms in Chaotic Dynamical Systems and Homoclinic Phenomena

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The investigators introduce new mathematically rigorous experiments to compute dynamical entities of chaotic systems, and they consider the rigorous reliability of commonly used algorithms. In particular, they construct suitable algorithms to compute orbits of chaotic ordinary differential equations, such as the Lorenz equations, and derive rigorous estimates of the shadowing error in such numerically computed orbits; determine general conditions under which an attractor of a differential equation is well-approximated by an attractor of the discretized system resulting from a one-step method; and devise a rigorous numerical algorithm to test for hyperbolicity in specific dynamical systems. They also study homoclinic phenomena, which play an essential role in the analysis of chaotic behavior. More specifically, they estimate the intervals of the parameter on a dense subset of which a one-parameter family of dissipative planar diffeomorphisms has homoclinic tangencies, and establish the presence of transversal homoclinic points in slowly varying systems. Some of the more noteworthy discoveries in dynamical systems have largely been based on numerical experiments. It is important to devise such experiments whose reliability can be rigorously verified. Most of this project is concerned with chaotic systems. In recent years it has been realized that much of nature behaves chaotically. For example, small changes on a given day in weather parameters such as temperature and barometric pressure can lead to large fluctuations in weather patterns a few days later. A mathematical model that describes a chaotic system in nature will almost inevitably behave chaotically also. Because these mathematical models can rarely be solved exactly, knowledge of their behavior relies greatly on computer simulations. The main object of this project is to develop tests that will tell how close a computer simulation of a chaotic system is to a true behavior of the system.