275 research outputs found
Classical quasi-particle dynamics in trapped Bose condensates
The dynamics of quasi-particles in repulsive Bose condensates in a harmonic
trap is studied in the classical limit. In isotropic traps the classical motion
is integrable and separable in spherical coordinates. In anisotropic traps the
classical dynamics is found, in general, to be nonintegrable. For
quasi-particle energies E much smaller than thechemical potential, besides the
conserved quasi-particle energy, we identify two additional nearly conserved
phase-space functions. These render the dynamics inside the condensate
(collective dynamics) integrable asymptotically for E/chemical potential very
small. However, there coexists at the same energy a dynamics confined to the
surface of the condensate, which is governed by a classical Hartree-Fock
Hamiltonian. We find that also this dynamics becomes integrable for E/chemical
potential very small, because of the appearance of an adiabatic invariant. For
E/chemical potential of order 1 a large portion of the phase-space supports
chaotic motion, both, for the Bogoliubov Hamiltonian and its Hartree-Fock
approximant. To exemplify this we exhibit Poincar\'e surface of sections for
harmonic traps with the cylindrical symmetry and anisotropy found in TOP traps.
For E/chemical potential very large the dynamics is again governed by the
Hartree-Fock Hamiltonian. In the case with cylindrical symmetry it becomes
quasi-integrable because the remaining small chaotic components in phase space
are tightly confined by tori.Comment: 13 pages Latex, 6 eps.gz-figure
A striking correspondence between the dynamics generated by the vector fields and by the scalar parabolic equations
The purpose of this paper is to enhance a correspondence between the dynamics
of the differential equations on and those
of the parabolic equations on a bounded
domain . We give details on the similarities of these dynamics in the
cases , and and in the corresponding cases ,
and dim() respectively. In addition to
the beauty of such a correspondence, this could serve as a guideline for future
research on the dynamics of parabolic equations
Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps
We study dynamics and bifurcations of two-dimensional reversible
maps having non-transversal heteroclinic cycles containing symmetric saddle fixed
points. We consider one-parameter families of reversible maps unfolding the
initial heteroclinic tangency and prove the existence of infinitely many sequences
(cascades) of bifurcations and birth of asymptotically stable, unstable and elliptic
periodic orbitsPeer ReviewedPostprint (published version
B-Spline based uncertainty quantification for stochastic analysis
The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification.
At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails.
Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it.
Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided
Iterative solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order method
We consider the solution to the biharmonic equation in mixed form discretized
by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic
problems can be decoupled via the introduction of a new unknown, corresponding
to the boundary value of the solution of the first Laplacian problem. This
technique yields a global linear problem that can be solved iteratively via a
Krylov-type method. More precisely, at each iteration of the scheme, two
second-order elliptic problems have to be solved, and a normal derivative on
the boundary has to be computed. In this work, we specialize this scheme for
the HHO discretization. To this aim, an explicit technique to compute the
discrete normal derivative of an HHO solution of a Laplacian problem is
proposed. Moreover, we show that the resulting discrete scheme is well-posed.
Finally, a new preconditioner is designed to speed up the convergence of the
Krylov method. Numerical experiments assessing the performance of the proposed
iterative algorithm on both two- and three-dimensional test cases are
presented
Asymptotic techniques in the analysis of invariant manifolds of dynamical systems
Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 14-06-2018Esta tesis tiene embargado el acceso al texto completo hasta el 14-12-201
- …