3,260 research outputs found
A Computational Procedure to Detect a New Type of High Dimensional Chaotic Saddle and its Application to the 3-D Hill's Problem
A computational procedure that allows the detection of a new type of
high-dimensional chaotic saddle in Hamiltonian systems with three degrees of
freedom is presented. The chaotic saddle is associated with a so-called
normally hyperbolic invariant manifold (NHIM). The procedure allows to compute
appropriate homoclinic orbits to the NHIM from which we can infer the existence
a chaotic saddle. NHIMs control the phase space transport across an equilibrium
point of saddle-centre-...-centre stability type, which is a fundamental
mechanism for chemical reactions, capture and escape, scattering, and, more
generally, ``transformation'' in many different areas of physics. Consequently,
the presented methods and results are of broad interest. The procedure is
illustrated for the spatial Hill's problem which is a well known model in
celestial mechanics and which gained much interest e.g. in the study of the
formation of binaries in the Kuiper belt.Comment: 12 pages, 6 figures, pdflatex, submitted to JPhys
Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed
Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed
Direct construction of a dividing surface of minimal flux for multi-degree-of-freedom systems that cannot be recrossed
A formula to compute the microcanonical volume of reactive initial conditions in transition state theory
We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or ‘react’ from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure
A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill’s problem
Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
Hamiltonian dynamical systems possessing equilibria of stability type display \emph{reaction-type
dynamics} for energies close to the energy of such equilibria; entrance and
exit from certain regions of the phase space is only possible via narrow
\emph{bottlenecks} created by the influence of the equilibrium points. In this
paper we provide a thorough pedagogical description of the phase space
structures that are responsible for controlling transport in these problems. Of
central importance is the existence of a \emph{Normally Hyperbolic Invariant
Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient
dimensionality to act as separatrices, partitioning energy surfaces into
regions of qualitatively distinct behavior. This NHIM forms the natural
(dynamical) equator of a (spherical) \emph{dividing surface} which locally
divides an energy surface into two components (`reactants' and `products'), one
on either side of the bottleneck. This dividing surface has all the desired
properties sought for in \emph{transition state theory} where reaction rates
are computed from the flux through a dividing surface. In fact, the dividing
surface that we construct is crossed exactly once by reactive trajectories, and
not crossed by nonreactive trajectories, and related to these properties,
minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space
structures contained in it for 2-degree-of-freedom (DoF) systems in the
threedimensional space , and two schematic models which capture many of
the essential features of the dynamics for -DoF systems. In addition, we
elucidate the structure of the NHIM.Comment: 44 pages, 38 figures, PDFLaTe
Control of Integrable Hamiltonian Systems and Degenerate Bifurcations
We discuss control of low-dimensional systems which, when uncontrolled, are
integrable in the Hamiltonian sense. The controller targets an exact solution
of the system in a region where the uncontrolled dynamics has invariant tori.
Both dissipative and conservative controllers are considered. We show that the
shear flow structure of the undriven system causes a Takens-Bogdanov
birfurcation to occur when control is applied. This implies extreme noise
sensitivity. We then consider an example of these results using the driven
nonlinear Schrodinger equation.Comment: 25 pages, 11 figures, resubmitted to Physical Review E March 2004
(originally submitted June 2003), added content and reference
The role of body rotation in bacterial flagellar bundling
In bacterial chemotaxis, E. coli cells drift up chemical gradients by a
series of runs and tumbles. Runs are periods of directed swimming, and tumbles
are abrupt changes in swimming direction. Near the beginning of each run, the
rotating helical flagellar filaments which propel the cell form a bundle. Using
resistive-force theory, we show that the counter-rotation of the cell body
necessary for torque balance is sufficient to wrap the filaments into a bundle,
even in the absence of the swirling flows produced by each individual filament
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