60,825 research outputs found
Periodic Homogenization for Inertial Particles
We study the problem of homogenization for inertial particles moving in a
periodic velocity field, and subject to molecular diffusion. We show that,
under appropriate assumptions on the velocity field, the large scale, long time
behavior of the inertial particles is governed by an effective diffusion
equation for the position variable alone. To achieve this we use a formal
multiple scale expansion in the scale parameter. This expansion relies on the
hypo-ellipticity of the underlying diffusion. An expression for the diffusivity
tensor is found and various of its properties studied. In particular, an
expansion in terms of the non-dimensional particle relaxation time (the
Stokes number) is shown to co-incide with the known result for passive
(non-inertial) tracers in the singular limit . This requires the
solution of a singular perturbation problem, achieved by means of a formal
multiple scales expansion in Incompressible and potential fields are
studied, as well as fields which are neither, and theoretical findings are
supported by numerical simulations.Comment: 31 pages, 7 figures, accepted for publication in Physica D. Typos
corrected. One reference adde
The Hess-Appelrot system and its nonholonomic analogs
This paper is concerned with the nonholonomic Suslov problem and its
generalization proposed by Chaplygin. The issue of the existence of an
invariant measure with singular density (having singularities at some points of
phase space) is discussed
Topological fluid mechanics of point vortex motions
Topological techniques are used to study the motions of systems of point
vortices in the infinite plane, in singly-periodic arrays, and in
doubly-periodic lattices. The reduction of each system using its symmetries is
described in detail. Restricting to three vortices with zero net circulation,
each reduced system is described by a one degree of freedom Hamiltonian. The
phase portrait of this reduced system is subdivided into regimes using the
separatrix motions, and a braid representing the topology of all vortex motions
in each regime is computed. This braid also describes the isotopy class of the
advection homeomorphism induced by the vortex motion. The Thurston-Nielsen
theory is then used to analyse these isotopy classes, and in certain cases
strong conclusions about the dynamics of the advection can be made
Chaos in Anisotropic Pre-Inflationary Universes
We study the dynamics of anisotropic Bianchi type-IX models with matter and
cosmological constant. The models can be thought as describing the role of
anisotropy in the early stages of inflation. The concurrence of the
cosmological constant and anisotropy are sufficient to produce a chaotic
dynamics in the gravitational degrees of freedom, connected to the presence of
a critical point of saddle-center type in the phase space of the system. The
invariant character of chaos is guaranteed by the topology of the cylinders
emanating from unstable periodic orbits in the neighborhood of the
saddle-center. We discuss a possible mechanism for amplification of specific
wavelengths of inhomogeneous fluctuations in the models. A geometrical
interpretation is given for Wald's inequality in terms of invariant tori and
their destruction by increasing values of the cosmological constant.Comment: 14 pages, figures available under request. submitted to Physical
Review
Cone-like Invariant Manifolds for Nonsmooth Systems
This thesis deals with rigorous mathematical techniques for higher-dimensional nonsmooth systems and their applications. The dynamical behaviour of these systems is a nonlocal problem due to the lack of smoothness.
Motivated by various examples of nonsmooth systems in applications, we propose to explore the concept of invariant surfaces in the phase space which is separated by a discontinuity hypersurface. For such systems the corresponding Poincaré map can be determined; it turns out that under suitable conditions an invariant cone occurs which is characterized by a fixed point of the Poincaré map. The invariant cone seems to serve in a similar way as a generalisation of the classical center manifold for smooth differential systems. Hence, the stability of the whole system can be reduced to investigate the stability on the two-dimensional surface of the cone.
Motivated to study the generation of invariant cones out of smooth systems, a numerical procedure to establish invariant cones and their stability is presented. It has been found that the flat degenerate cone in a smooth system develops under nonsmooth perturbations into a cone-like configuration. Also a simple example is used to explain a paradoxical situation concerning stability. Theoretical results concerning the existence of invariant cones and possible mechanisms responsible for the observed behavior for general three dimensional nonsmooth systems are discussed. These investigations reveal that the system possesses a rich dynamic behavior and new phenomena such as, for instance, the existence of multiple invariant cones for such system.
Our approach is developed to include the case when sliding motion takes place on the manifold. Sliding dynamical equations are formulated by using Filippov's method. Existence of invariant cones containing a segment of sliding orbits are given as well as stability on these cones. Different sliding bifurcation scenarios are treated by theoretical analysis and simulation.
As an application we have investigated the dynamics of an automotive brake system model under the excitation of dry friction force which has served as a motivating example to develop our concepts. This model belongs to the class of nonsmooth systems of Filippov type which is investigated from direct crossing and a sliding motion point of view. Existence of invariant cones and different types of bifurcation phenomena such as sliding periodic doubling and multiple periodic orbits are observed.
Finally, extensions to nonlinear perturbations of nonsmooth linear systems have been obtained by using the nonsmooth linear system as basic system. If the basic system possesses an attractive invariant cone without sliding motion, we have shown that locally the Poincaré map contains the necessary information with regard to attractivity of the invariant cone. The existence of a generalized center manifold reduction of nonlinear system has been proven by using Hadamard graph transformation approach. A class of nonlinear systems having a cone-like invariant "manifold" is presented to illustrate the center manifold reduction and associated bifurcation. The scientific contributions of parts of this thesis are presented in [32,39,66]
Measuring the mixing efficiency in a simple model of stirring:some analytical results and a quantitative study via Frequency Map Analysis
We prove the existence of invariant curves for a --periodic Hamiltonian
system which models a fluid stirring in a cylindrical tank, when is small
and the assigned stirring protocol is piecewise constant. Furthermore, using
the Numerical Analysis of the Fundamental Frequency of Laskar, we investigate
numerically the break down of invariant curves as increases and we give a
quantitative estimate of the efficiency of the mixing.Comment: 10 figure
Periodic homogenization with an interface
We consider a diffusion process with coefficients that are periodic outside
of an 'interface region' of finite thickness. The question investigated in the
articles [1,2] is the limiting long time / large scale behaviour of such a
process under diffusive rescaling. It is clear that outside of the interface,
the limiting process must behave like Brownian motion, with diffusion matrices
given by the standard theory of homogenization. The interesting behaviour
therefore occurs on the interface. Our main result is that the limiting process
is a semimartingale whose bounded variation part is proportional to the local
time spent on the interface. We also exhibit an explicit way of identifying its
parameters in terms of the coefficients of the original diffusion.
Our method of proof relies on the framework provided by Freidlin and Wentzell
for diffusion processes on a graph in order to identify the generator of the
limiting process.Comment: ISAAC 09 conference proceeding
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