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.

More information is available at Theses and Dissertations at University of Maryland Libraries.

Browse

Subject: search.filters.subject.dynamical systems

Search Results

Now showing 1 - 5 of 5
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps
    (2014) Contreras, Fabian Elias; Dolgopyat, Dmitry; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    This dissertation consists of two parts. In the first part, we consider a piecewise expanding unimodal map (PEUM) $f:[0,1] \to [0,1]$ with $\mu=\rho dx$ the (unique) SRB measure associated to it and we show that $\rho$ has a Taylor expansion in the Whitney sense. Moreover, we prove that the set of points where $\rho$ is not differentiable is uncountable and has Hausdorff dimension equal to zero. In the second part, we consider a family $f_t:[0,1] \to [0,1]$ of PEUMs with $\mu_t$ the correspoding SRB measure and we present a new proof of \cite{BS1} when considering the observables in $C^1[0,1]$ . That is, $\Gamma(t)=\int \phi d\mu_t$ is differentiable at $t=0$, with $\phi \in C^1[0,1]$, when assuming $J(c)=\sum_{k=0}^{\infty} \frac{v(f^k(c))}{Df^k(f(c))}$ is zero. Furthermore, we show that in fact $\Gamma(t)$ is never differentiable when $J(c)$ is not zero and we give the exact modulus of continuity.
  • Thumbnail Image
    Item
    ROBUSTNESS OF ATTRACTING ORBITS
    (2014) Joglekar, Madhura R.; Yorke, James A; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
    Understanding the transition to turbulence is a long-lasting problem in fluid dynamics, particularly in the case of simple flows in which the base laminar flow does not become linearly unstable. For flows at a low Reynolds number, all initial conditions decay to the laminar profile. At higher Reynolds numbers, above a critical value, turbulence is observed, often in the form of a chaotic saddle. The magnitude of the perturbation that disrupts the laminar flow into the turbulent region depends on the Reynolds number and on the direction of the perturbation. In Chapter 2, we investigate the robustness of the laminar attractor to perturbations in a 9-dimensional sinusoidal shear flow model. We examine the geometry of the `edge of chaos', where the edge denotes the boundary of the chaotic saddle, which is embedded in the basin of attraction of the laminar state, and is accessible from that state. For a smooth dynamical system $x_{n+1} = F(C, x_n)$ (depending on a parameter C), there may be infinitely many periodic windows, that is, intervals in C having a region of stable periodic behavior. However, the smaller of these windows are easily destroyed with tiny perturbations, so that only finitely many of the windows can be detected for a given level of noise. For a fixed perturbation size $\epsilon$, we consider the system behavior in the presence of noise. In this Chapter, we look at the ``$\epsilon$-robust windows'', that is, those periodic windows such that for the superstable parameter value C in that window, the general periodic behavior persists despite noise of amplitude $\le \epsilon$. We focus on the quadratic map, and numerically compute the number of periodic windows that are $\epsilon$-robust. In Chapter 3, we obtain a robustness-exponent $\alpha \approx .51 \pm .03$, which characterizes the robustness of periodic windows in the presence of noise. The character of the time-asymptotic evolution of physical systems can have complex, singular behavior with variation of a system parameter, particularly when chaos is involved. A perturbation of the parameter by a small amount $\epsilon$ can convert an attractor from chaotic to non-chaotic or vice-versa. We call a parameter value where this can happen $\epsilon$-uncertain. The probability that a random choice of the parameter is $\epsilon$-uncertain commonly scales like a power law in $\epsilon$. Surprisingly, two seemingly similar ways of defining this scaling, both of physical interest, yield different numerical values for the scaling exponent. In Chapter 4, we show why this happens and present a quantitative analysis of this phenomenon. Many dynamical systems reach a level of maximum topological entropy as the system parameter is increased followed by a decrease to zero entropy. In Chapter 5, we give an example such that the number of cascades continues to increase for arbitrarily large values of the parameter. We investigate the map $S_{\mu}:[0,1] \rightarrow [0,1)$ defined by $S_{\mu}(x) := \mu \sin(2\pi x) \bmod{1}.$ For this map, the entropy increases without bound as $\mu \rightarrow \infty$, and the system has an ever-increasing number of solitary cascades for $\mu \in [0,m]$ as $m$ is increased to higher and higher integer values. Specifically, we calculate the number of period-$k$ cascades of the map, for $k>1$, for positive integer values of $\mu \in [0,m]$, where $m \in \mathbb{N}$.
  • Thumbnail Image
    Item
    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.
  • Thumbnail Image
    Item
    A Dynamical Systems Approach to Estimating the Sequences of Repeat Regions in the Genome
    (2004-05-11) Meloney, Kathleen Ann; Yorke, James A; Mathematics
    In 1982, Fred Sanger introduced a cloning technique on which shotgun sequencing is based. Shotgun sequencing is a method for determining the sequence of bases (or letters) in the genome and since its introduction, many groups have used this technique to sequence the genomes of various organisms. The shotgun technique involves breaking the DNA into a large number of small pieces, each of whose sequence of letters is determined experimentally. Current technology limits the length of the sequenced pieces to approximately 500 letters. Then, like a puzzle, the pieces are assembled using computer algorithms to produce the complete sequence. The greatest difficulty with the shotgun technique is the presence of subsequences longer than 500 letters that occur multiple times in the genome with minor variations. We present a dynamical systems approach to estimating the sequence of letters of these long, highly repetitive subsequences in the genome. Our results suggest that this approach produces good representatives of the long repetitive subsequences in a genome. We also present potential applications of this method to genome assembly.