10,062 research outputs found
A note on the convergence of parametrised non-resonant invariant manifolds
Truncated Taylor series representations of invariant manifolds are abundant
in numerical computations. We present an aposteriori method to compute the
convergence radii and error estimates of analytic parametrisations of
non-resonant local invariant manifolds of a saddle of an analytic vector field,
from such a truncated series. This enables us to obtain local enclosures, as
well as existence results, for the invariant manifolds
Wigner's Dynamical Transition State Theory in Phase Space: Classical and Quantum
A quantum version of transition state theory based on a quantum normal form
(QNF) expansion about a saddle-centre-...-centre equilibrium point is
presented. A general algorithm is provided which allows one to explictly
compute QNF to any desired order. This leads to an efficient procedure to
compute quantum reaction rates and the associated Gamov-Siegert resonances. In
the classical limit the QNF reduces to the classical normal form which leads to
the recently developed phase space realisation of Wigner's transition state
theory. It is shown that the phase space structures that govern the classical
reaction d ynamicsform a skeleton for the quantum scattering and resonance
wavefunctions which can also be computed from the QNF. Several examples are
worked out explicitly to illustrate the efficiency of the procedure presented.Comment: 132 pages, 31 figures, corrected version, Nonlinearity, 21 (2008)
R1-R11
Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations
We study the dynamics of the five-parameter quadratic family of
volume-preserving diffeomorphisms of R^3. This family is the unfolded normal
form for a bifurcation of a fixed point with a triple-one multiplier and also
is the general form of a quadratic three-dimensional map with a quadratic
inverse. Much of the nontrivial dynamics of this map occurs when its two fixed
points are saddle-foci with intersecting two-dimensional stable and unstable
manifolds that bound a spherical ``vortex-bubble''. We show that this occurs
near a saddle-center-Neimark-Sacker (SCNS) bifurcation that also creates, at
least in its normal form, an elliptic invariant circle. We develop a simple
algorithm to accurately compute these elliptic invariant circles and their
longitudinal and transverse rotation numbers and use it to study their
bifurcations, classifying them by the resonances between the rotation numbers.
In particular, rational values of the longitudinal rotation number are shown to
give rise to a string of pearls that creates multiple copies of the original
spherical structure for an iterate of the map.Comment: 53 pages, 29 figure
Quantum Theory of Reactive Scattering in Phase Space
We review recent results on quantum reactive scattering from a phase space
perspective. The approach uses classical and quantum versions of normal form
theory and the perspective of dynamical systems theory. Over the past ten years
the classical normal form theory has provided a method for realizing the phase
space structures that are responsible for determining reactions in high
dimensional Hamiltonian systems. This has led to the understanding that a new
(to reaction dynamics) type of phase space structure, a {\em normally
hyperbolic invariant manifold} (or, NHIM) is the "anchor" on which the phase
space structures governing reaction dynamics are built. The quantum normal form
theory provides a method for quantizing these phase space structures through
the use of the Weyl quantization procedure. We show that this approach provides
a solution of the time-independent Schr\"odinger equation leading to a (local)
S-matrix in a neighborhood of the saddle point governing the reaction. It
follows easily that the quantization of the directional flux through the
dividing surface with the properties noted above is a flux operator that can be
expressed in a "closed form". Moreover, from the local S-matrix we easily
obtain an expression for the cumulative reactio probability (CRP).
Significantly, the expression for the CRP can be evaluated without the need to
compute classical trajectories. The quantization of the NHIM is shown to lead
to the activated complex, and the lifetimes of quantum states initialized on
the NHIM correspond to the Gamov-Siegert resonances. We apply these results to
the collinear nitrogen exchange reaction and a three degree-of-freedom system
corresponding to an Eckart barrier coupled to two Morse oscillators.Comment: 59 pages, 13 figure
Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the
smoothest invariant manifolds tangent to linear modal subspaces of an
equilibrium. Amplitude-frequency plots of the dynamics on SSMs provide the
classic backbone curves sought in experimental nonlinear model identification.
We develop here a methodology to compute analytically both the shape of SSMs
and their corresponding backbone curves from a data-assimilating model fitted
to experimental vibration signals. Using examples of both synthetic and real
experimental data, we demonstrate that this approach reproduces backbone curves
with high accuracy.Comment: 32 pages, 4 figure
Analytical description of the structure of chaos
We consider analytical formulae that describe the chaotic regions around the
main periodic orbit of the H\'{e}non map. Following our previous
paper (Efthymiopoulos, Contopoulos, Katsanikas ) we introduce new
variables in which the product (constant) gives
hyperbolic invariant curves. These hyperbolae are mapped by a canonical
transformation to the plane , giving "Moser invariant curves". We
find that the series are convergent up to a maximum value of
. We give estimates of the errors due to the finite truncation of
the series and discuss how these errors affect the applicability of analytical
computations. For values of the basic parameter of the H\'{e}non map
smaller than a critical value, there is an island of stability, around a stable
periodic orbit , containing KAM invariant curves. The Moser curves for are completely outside the last KAM curve around , the curves
with intersect the last KAM curve and the curves with are completely inside the last KAM curve. All orbits in
the chaotic region around the periodic orbit , although they seem
random, belong to Moser invariant curves, which, therefore define a "structure
of chaos". Orbits starting close and outside the last KAM curve remain close to
it for a stickiness time that is estimated analytically using the series
. We finally calculate the periodic orbits that accumulate close to the
homoclinic points, i.e. the points of intersection of the asymptotic curves
from , exploiting a method based on the self-intersections of the
invariant Moser curves. We find that all the computed periodic orbits are
generated from the stable orbit for smaller values of the H\'{e}non
parameter , i.e. they are all regular periodic orbits.Comment: 22 pages, 9 figure
Two-dimensional global manifolds of vector fields
We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in zeta(3)-model converging to an attracting limit cycle
Embeddings of 3-manifolds in S^4 from the point of view of the 11-tetrahedron census
This is a collection of notes on embedding problems for 3-manifolds. The main
question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The
terrain of exploration is the Burton/Martelli/Matveev/Petronio census of
triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There
are 13766 manifolds in the census, of which 13400 are orientable. Of the 13400
orientable manifolds, only 149 of them have hyperbolic torsion linking forms
and are thus candidates for embedability in the 4-sphere. The majority of this
paper is devoted to the embedding problem for these 149 manifolds. At present
41 are known to embed. Among the remaining manifolds, embeddings into homotopy
4-spheres are constructed for 4. 67 manifolds are known to not embed in the
4-sphere. This leaves 37 unresolved cases, of which only 3 are geometric
manifolds i.e. having a trivial JSJ-decomposition.Comment: 58 pages, 80+ figures. V6: Included references to libraries valid in
Regina 5.0+. Incorporated changes suggested by Ahmed Issa, following from his
techniques developed with McCoy. Included a few recent references. To appear
in Experimental Mathematic
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