8,122 research outputs found
Realizability of the normal form for the triple-zero nilpotency in a class of delayed nonlinear oscillators
The effects of delayed feedback terms on nonlinear oscillators has been
extensively studied, and have important applications in many areas of science
and engineering. We study a particular class of second-order delay-differential
equations near a point of triple-zero nilpotent bifurcation. Using center
manifold and normal form reduction, we show that the three-dimensional
nonlinear normal form for the triple-zero bifurcation can be fully realized at
any given order for appropriate choices of nonlinearities in the original
delay-differential equation.Comment: arXiv admin note: text overlap with arXiv:math/050539
Symmetric Regularization, Reduction and Blow-Up of the Planar Three-Body Problem
We carry out a sequence of coordinate changes for the planar three-body
problem which successively eliminate the translation and rotation symmetries,
regularize all three double collision singularities and blow-up the triple
collision. Parametrizing the configurations by the three relative position
vectors maintains the symmetry among the masses and simplifies the
regularization of binary collisions. Using size and shape coordinates
facilitates the reduction by rotations and the blow-up of triple collision
while emphasizing the role of the shape sphere. By using homogeneous
coordinates to describe Hamiltonian systems whose configurations spaces are
spheres or projective spaces, we are able to take a modern, global approach to
these familiar problems. We also show how to obtain the reduced and regularized
differential equations in several convenient local coordinates systems.Comment: 51 pages, 4 figure
Higher Order Approximation to the Hill Problem Dynamics about the Libration Points
An analytical solution to the Hill problem Hamiltonian expanded about the
libration points has been obtained by means of perturbation techniques. In
order to compute the higher orders of the perturbation solution that are needed
to capture all the relevant periodic orbits originated from the libration
points within a reasonable accuracy, the normalization is approached in complex
variables. The validity of the solution extends to energy values considerably
far away from that of the libration points and, therefore, can be used in the
computation of Halo orbits as an alternative to the classical
Lindstedt-Poincar\'e approach. Furthermore, the theory correctly predicts the
existence of the two-lane bridge of periodic orbits linking the families of
planar and vertical Lyapunov orbits.Comment: 28 pages, 8 figure
Differential-Flatness and Control of Quadrotor(s) with a Payload Suspended through Flexible Cable(s)
We present the coordinate-free dynamics of three different quadrotor systems
: (a) single quadrotor with a point-mass payload suspended through a flexible
cable; (b) multiple quadrotors with a shared point-mass payload suspended
through flexible cables; and (c) multiple quadrotors with a shared rigid-body
payload suspended through flexible cables. We model the flexible cable(s) as a
finite series of links with spherical joints with mass concentrated at the end
of each link. The resulting systems are thus high-dimensional with high
degree-of-underactuation. For each of these systems, we show that the dynamics
are differentially-flat, enabling planning of dynamically feasible
trajectories. For the single quadrotor with a point-mass payload suspended
through a flexible cable with five links (16 degrees-of-freedom and 12
degrees-of-underactuation), we use the coordinate-free dynamics to develop a
geometric variation-based linearized equations of motion about a desired
trajectory. We show that a finite-horizon linear quadratic regulator can be
used to track a desired trajectory with a relatively large region of
attraction
Cubic Differentials in the Differential Geometry of Surfaces
We discuss the local differential geometry of convex affine spheres in
\re^3 and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In
each case, there is a natural metric and cubic differential holomorphic with
respect to the induced conformal structure: these data come from the Blaschke
metric and Pick form for the affine spheres and from the induced metric and
second fundamental form for the minimal Lagrangian surfaces. The local
geometry, at least for main cases of interest, induces a natural frame whose
structure equations arise from the affine Toda system for . We also discuss the global theory and applications to
representations of surface groups and to mirror symmetry.Comment: corrected published editio
Large isoperimetric surfaces in initial data sets
We study the isoperimetric structure of asymptotically flat Riemannian
3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0.
Refining an argument due to H. Bray we obtain an effective volume comparison
theorem in Schwarzschild. We use it to show that isoperimetric regions exist in
(M, g) for all sufficiently large volumes, and that they are close to centered
coordinate spheres. This implies that the volume-preserving stable constant
mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye
as perturbations of large centered coordinate spheres minimize area among all
competing surfaces that enclose the same volume. This confirms a conjecture of
H. Bray. Our results are consistent with the uniqueness results for
volume-preserving stable constant mean curvature surfaces in initial data sets
obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian.
The additional hypotheses that the surfaces be spherical and far out in the
asymptotic region in their results are not necessary in our work.Comment: 29 pages. All comments welcome! This is the final version to appear
in J. Differential Geo
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