Browsing by Subject "Differential equations"
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Item Applications of moving frames to group foliation of differential equations(2013-10) Thompson, RobertThe classical group foliation algorithm uses the continuous symmetries of a differential equation to aid in its integration. This is accomplished by transforming the differential equation into two alternative systems, called the resolving and automorphic systems. Incorporating the theory of equivariant moving frames for Lie pseudogroups, a completely symbolic and systematic version of the group foliation algorithm is introduced. In this version of the algorithm, the resolving system is derived using only knowledge of the structure of the differential invariant algebra, requiring no explicit formulae for differential invariants. Additionally, the automorphic system is replaced by an equivalent reconstruction system, again requiring only symbolic computation. The efficacy of this approach is illustrated through several examples. Further applications of aspects of group foliation are given, including the construction of Backlund transformations using resolving systems and a reconstruction process for an invariant submanifold flow corresponding to a given invariant signature evolution.Item Chaos, attractors and the Lorenz conjecture: Noninvertible transitive maps of ivariant sets are sensitive.(2010-06) Taft, Garrett ThomasIn 1989, Edward Lorenz published a paper entitled, “Computational chaos- a prelude to computational instability” [L]. His paper looked at Euler approximations to differential equations. If the time increment of the approximating function was increased, he found that computational chaos set in. Since the numerics suggested transitivity and noninvertibility, he conjectured that transitive, noninvertible maps of an attractor were chaotic. To set the stage for investigating this conjecture, this thesis looked to examine the relationships between some of the standard definitions of chaos and attractor used throughout the literature. In addition to offering a proof of the Lorenz conjecture, a review of a number of related results was conducted. A side product of the work done was a partial result that tried to address whether topological transitivity carries sensitivity at a point to sensitivity on a set.Item Differential Independence of Meromorphic Functions(University of Minnesota. School of Mathematics, 2003-01) Markus, LawrenceItem Dynamics of a Singularly Perturbed Quadratic Family(2017) Wang, LeshengSome of the dynamics of the family x→x^2+c+β/x^d are described. Different behaviors occur as the parameter β and c are varied. These transitions are called bifurcations. This singular perturbed quadratic family is treated both as a real system. This paper is based on the various trials done in the Mathematica 10. The main focus of this paper is done for the parameter β and c. Specifi cally, the summary is done to present the general behaviors of the real case. Meanwhile, this paper creates some general behaviors of the system with different parameter d. The dynamics of one dimensional quadratic maps is present.Item Periodic brake orbits in the N-body problem(2014-08) Chen, Nai-ChiaThe thesis is devoted to finding periodic brake orbits in the N-body problem. We consider certain subsystems of the N-body problem that have two degrees of freedom, including the isosceles three-body problem and other highly symmetric sub-problems. We prove the existence of several families of symmetric periodic orbits, including ``Schubart-like" orbits and brake orbits, by using topological shooting arguments.