UMD Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/3

New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.

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    Dynamics of Some Fermi Acceleration Models
    (2020) Zhou, Jing; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    In the thesis we describe the dynamics of two variations of the Fermi acceleration models. The first model consists of a rectangular billiard with two periodically vertically oscillating slits. A point particle bounces elastically against the billiard table and the slits. We assume that the horizontal motion of the particle is in resonance with those of the slits. In this case, we have found a mechanism of trapping regions which provides the exponential acceleration for almost all initial conditions with sufficiently high initial energy. Under an additional hyperbolicity assumption on the parameters of the system, we estimate the waiting time after which most high-energy orbits start to gain energy exponentially fast. The second model depicts a point particle bouncing elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion is piecewise smooth with one singularity. If the second derivative of the platform motion behaves well, i.e. it is either always positive or always less than the negative of the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits coexist with bounded orbits at arbitrarily high energy levels.
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    Abundance of escaping orbitsin a family of anti-integrable limitsof the standard map
    (2009) De Simoi, Jacopo; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    We give quantitative results about the abundance of escaping orbits in a family of exact twist maps preserving Lebesgue measure on the cylinder T × R; geometrical features of maps of this family are quite similar to those of the well-known Chirikov-Taylor standard map, and in fact we believe that the techniques presented in this work can be further improved and eventually applied to studying ergodic properties of the standard map itself. We state conditions which assure that escaping orbits exist and form a full Hausdorff dimension set. Moreover, under stronger conditions we can prove that such orbits are not charged by the invariant measure. We also obtain prove that, generically, the system presents elliptic islands at arbitrarily high values of the action variable and provide estimates for their total measure.