2,155 research outputs found
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
Isomerization dynamics of a buckled nanobeam
We analyze the dynamics of a model of a nanobeam under compression. The model
is a two mode truncation of the Euler-Bernoulli beam equation subject to
compressive stress. We consider parameter regimes where the first mode is
unstable and the second mode can be either stable or unstable, and the
remaining modes (neglected) are always stable. Material parameters used
correspond to silicon. The two mode model Hamiltonian is the sum of a
(diagonal) kinetic energy term and a potential energy term. The form of the
potential energy function suggests an analogy with isomerisation reactions in
chemistry. We therefore study the dynamics of the buckled beam using the
conceptual framework established for the theory of isomerisation reactions.
When the second mode is stable the potential energy surface has an index one
saddle and when the second mode is unstable the potential energy surface has an
index two saddle and two index one saddles. Symmetry of the system allows us to
construct a phase space dividing surface between the two "isomers" (buckled
states). The energy range is sufficiently wide that we can treat the effects of
the index one and index two saddles in a unified fashion. We have computed
reactive fluxes, mean gap times and reactant phase space volumes for three
stress values at several different energies. In all cases the phase space
volume swept out by isomerizing trajectories is considerably less than the
reactant density of states, proving that the dynamics is highly nonergodic. The
associated gap time distributions consist of one or more `pulses' of
trajectories. Computation of the reactive flux correlation function shows no
sign of a plateau region; rather, the flux exhibits oscillatory decay,
indicating that, for the 2-mode model in the physical regime considered, a rate
constant for isomerization does not exist.Comment: 42 pages, 6 figure
Noise and Correlations in a Spatial Population Model with Cyclic Competition
Noise and spatial degrees of freedom characterize most ecosystems. Some
aspects of their influence on the coevolution of populations with cyclic
interspecies competition have been demonstrated in recent experiments [e.g. B.
Kerr et al., Nature {\bf 418}, 171 (2002)]. To reach a better theoretical
understanding of these phenomena, we consider a paradigmatic spatial model
where three species exhibit cyclic dominance. Using an individual-based
description, as well as stochastic partial differential and deterministic
reaction-diffusion equations, we account for stochastic fluctuations and
spatial diffusion at different levels, and show how fascinating patterns of
entangled spirals emerge. We rationalize our analysis by computing the
spatio-temporal correlation functions and provide analytical expressions for
the front velocity and the wavelength of the propagating spiral waves.Comment: 4 pages of main text, 3 color figures + 2 pages of supplementary
material (EPAPS Document). Final version for Physical Review Letter
Resonance Zones and Lobe Volumes for Volume-Preserving Maps
We study exact, volume-preserving diffeomorphisms that have heteroclinic
connections between a pair of normally hyperbolic invariant manifolds. We
develop a general theory of lobes, showing that the lobe volume is given by an
integral of a generating form over the primary intersection, a subset of the
heteroclinic orbits. Our definition reproduces the classical action formula in
the planar, twist map case. For perturbations from a heteroclinic connection,
the lobe volume is shown to reduce, to lowest order, to a suitable integral of
a Melnikov function.Comment: ams laTeX, 8 figure
In Situ Characterisation of Permanent Magnetic Quadrupoles for focussing proton beams
High intensity laser driven proton beams are at present receiving much
attention. The reasons for this are many but high on the list is the potential
to produce compact accelerators. However two of the limitations of this
technology is that unlike conventional nuclear RF accelerators lasers produce
diverging beams with an exponential energy distribution. A number of different
approaches have been attempted to monochromise these beams but it has become
obvious that magnetic spectrometer technology developed over many years by
nuclear physicists to transport and focus proton beams could play an important
role for this purpose. This paper deals with the design and characterisation of
a magnetic quadrupole system which will attempt to focus and transport
laser-accelerated proton beams.Comment: 20 pages, 42 figure
Canonical Melnikov theory for diffeomorphisms
We study perturbations of diffeomorphisms that have a saddle connection
between a pair of normally hyperbolic invariant manifolds. We develop a
first-order deformation calculus for invariant manifolds and show that a
generalized Melnikov function or Melnikov displacement can be written in a
canonical way. This function is defined to be a section of the normal bundle of
the saddle connection.
We show how our definition reproduces the classical methods of Poincar\'{e}
and Melnikov and specializes to methods previously used for exact symplectic
and volume-preserving maps. We use the method to detect the transverse
intersection of stable and unstable manifolds and relate this intersection to
the set of zeros of the Melnikov displacement.Comment: laTeX, 31 pages, 3 figure
Heteroclinic intersections between Invariant Circles of Volume-Preserving Maps
We develop a Melnikov method for volume-preserving maps with codimension one
invariant manifolds. The Melnikov function is shown to be related to the flux
of the perturbation through the unperturbed invariant surface. As an example,
we compute the Melnikov function for a perturbation of a three-dimensional map
that has a heteroclinic connection between a pair of invariant circles. The
intersection curves of the manifolds are shown to undergo bifurcations in
homologyComment: LaTex with 10 eps figure
Elasticity of semiflexible polymers in two dimensions
We study theoretically the entropic elasticity of a semi-flexible polymer,
such as DNA, confined to two dimensions. Using the worm-like-chain model we
obtain an exact analytical expression for the partition function of the polymer
pulled at one end with a constant force. The force-extension relation for the
polymer is computed in the long chain limit in terms of Mathieu characteristic
functions. We also present applications to the interaction between a
semi-flexible polymer and a nematic field, and derive the nematic order
parameter and average extension of the polymer in a strong field.Comment: 16 pages, 3 figure
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