337 research outputs found
Normal Forms for Symplectic Maps with Twist Singularities
We derive a normal form for a near-integrable, four-dimensional symplectic
map with a fold or cusp singularity in its frequency mapping. The normal form
is obtained for when the frequency is near a resonance and the mapping is
approximately given by the time- mapping of a two-degree-of freedom
Hamiltonian flow. Consequently there is an energy-like invariant. The fold
Hamiltonian is similar to the well-studied, one-degree-of freedom case but is
essentially nonintegrable when the direction of the singular curve in action
does not coincide with curves of the resonance module. We show that many
familiar features, such as multiple island chains and reconnecting invariant
manifolds, are retained even in this case. The cusp Hamiltonian has an
essential coupling between its two degrees of freedom even when the singular
set is aligned with the resonance module. Using averaging, we approximately
reduced this case to one degree of freedom as well. The resulting Hamiltonian
and its perturbation with small cusp-angle is analyzed in detail.Comment: LaTex, 27 pages, 21 figure
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
We employ KAM theory to rigorously investigate quasiperiodic dynamics in
cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and
superlattices. Toward this end, we apply a coherent structure ansatz to the
Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation
describing the spatial dynamics of the condensate. For shallow-well,
intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather
sets to prove that one obtains mostly quasiperiodic dynamics for condensate
wave functions of sufficiently large amplitude, where the minimal amplitude
depends on the experimentally adjustable BEC parameters. We show that this
threshold scales with the square root of the inverse of the two-body scattering
length, whereas the rotation number of tori above this threshold is
proportional to the amplitude. As a consequence, one obtains the same dynamical
picture for lattices of all depths, as an increase in depth essentially only
affects scaling in phase space. Our approach is applicable to periodic
superlattices with an arbitrary number of rationally dependent wave numbers.Comment: 29 pages, 6 figures (several with multiple parts; higher-quality
versions of some of them available at
http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of
Nonlinear Scienc
Perturbed Three Vortex Dynamics
It is well known that the dynamics of three point vortices moving in an ideal
fluid in the plane can be expressed in Hamiltonian form, where the resulting
equations of motion are completely integrable in the sense of Liouville and
Arnold. The focus of this investigation is on the persistence of regular
behavior (especially periodic motion) associated to completely integrable
systems for certain (admissible) kinds of Hamiltonian perturbations of the
three vortex system in a plane. After a brief survey of the dynamics of the
integrable planar three vortex system, it is shown that the admissible class of
perturbed systems is broad enough to include three vortices in a half-plane,
three coaxial slender vortex rings in three-space, and `restricted' four vortex
dynamics in a plane. Included are two basic categories of results for
admissible perturbations: (i) general theorems for the persistence of invariant
tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff
type arguments; and (ii) more specific and quantitative conclusions of a
classical perturbation theory nature guaranteeing the existence of periodic
orbits of the perturbed system close to cycles of the unperturbed system, which
occur in abundance near centers. In addition, several numerical simulations are
provided to illustrate the validity of the theorems as well as indicating their
limitations as manifested by transitions to chaotic dynamics.Comment: 26 pages, 9 figures, submitted to the Journal of Mathematical Physic
Symmetry and resonance in periodic FPU chains
The symmetry and resonance properties of the Fermi Pasta Ulam chain with
periodic boundary conditions are exploited to construct a near-identity
transformation bringing this Hamiltonian system into a particularly simple
form. This `Birkhoff-Gustavson normal form' retains the symmetries of the
original system and we show that in most cases this allows us to view the
periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville
integrable Hamiltonian. According to the KAM theorem this proves the existence
of many invariant tori on which motion is quasiperiodic. Experiments confirm
this qualitative behaviour. We note that one can not expect it in lower-order
resonant Hamiltonian systems. So the FPU chain is an exception and its special
features are caused by a combination of special resonances and symmetries.Comment: 21 page
Greene's Residue Criterion for the Breakup of Invariant Tori of Volume-Preserving Maps
Invariant tori play a fundamental role in the dynamics of symplectic and
volume-preserving maps. Codimension-one tori are particularly important as they
form barriers to transport. Such tori foliate the phase space of integrable,
volume-preserving maps with one action and angles. For the area-preserving
case, Greene's residue criterion is often used to predict the destruction of
tori from the properties of nearby periodic orbits. Even though KAM theory
applies to the three-dimensional case, the robustness of tori in such systems
is still poorly understood. We study a three-dimensional, reversible,
volume-preserving analogue of Chirikov's standard map with one action and two
angles. We investigate the preservation and destruction of tori under
perturbation by computing the "residue" of nearby periodic orbits. We find tori
with Diophantine rotation vectors in the "spiral mean" cubic algebraic field.
The residue is used to generate the critical function of the map and find a
candidate for the most robust torus.Comment: laTeX, 40 pages, 26 figure
Scaling law in the Standard Map critical function. Interpolating hamiltonian and frequency map analysis
We study the behaviour of the Standard map critical function in a
neighbourhood of a fixed resonance, that is the scaling law at the fixed
resonance. We prove that for the fundamental resonance the scaling law is
linear. We show numerical evidence that for the other resonances , , and and relatively prime, the scaling law follows a
power--law with exponent .Comment: AMS-LaTeX2e, 29 pages with 8 figures, submitted to Nonlinearit
Invariant Sets in Quasiperiodically Forced Dynamical Systems
This paper addresses structures of state space in quasiperiodically forced
dynamical systems. We develop a theory of ergodic partition of state space in a
class of measure-preserving and dissipative flows, which is a natural extension
of the existing theory for measure-preserving maps. The ergodic partition
result is based on eigenspace at eigenvalue 0 of the associated Koopman
operator, which is realized via time-averages of observables, and provides a
constructive way to visualize a low-dimensional slice through a
high-dimensional invariant set. We apply the result to the systems with a
finite number of attractors and show that the time-average of a continuous
observable is well-defined and reveals the invariant sets, namely, a finite
number of basins of attraction. We provide a characterization of invariant sets
in the quasiperiodically forced systems. A theoretical result on uniform
boundedness of the invariant sets is presented. The series of theoretical
results enables numerical analysis of invariant sets in the quasiperiodically
forced systems based on the ergodic partition and time-averages. Using this, we
analyze a nonlinear model of complex power grids that represents the short-term
swing instability, named the coherent swing instability. We show that our
theoretical results can be used to understand stability regions in such complex
systems.Comment: 23 pages, 4 figure
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