
WITH INTERDISCIPLINARY APPLICATIONS
LOCATION: MAIN ("K") BUILDING OF THE TECHNICAL UNIVERSITY,
FACULTY CLUB (I.66)
Organizers:
Gabor
Domokos and Tim
Healey
(Budapest)
(Cornell)
Hosts:
Department of Applied
Mechanics,
Technical University
of Budapest.
chair:Gabor
Stepan
Engineering
Department of the
Hungarian
Academy of Sciences
The goal of the workshop is to provide opportunity
for the nonlinear mechanics/applied mathematics
community to exchange ideas on the subjects given
in the title.
Financial support is provided by the USHungarian
Science and Technology Joint Fund and the Hungarian
Scientific Research Fund.
Abstracts
List of Speakers
Logistics
Schedule
Events
Pictures
by Andy
The list of speakers is now final.
Whenever available, a link to the personal homepages is
provided.
Please check the links for schedule, events and logistics.


**
Map of the inner city of Budapest. Guesthouse marked with BIG arrow.
Technical
University campus indicated by handdrawn circle.
**
Detailed map of the TU campus, Guesthouse marked as #3, conference
site
(location of talks) indicated by handdrawn arrow.
**How to get from the airport to the Guesthouse or the Gellert Hotel
Cabs at the airport are by default
rather expensive (~USD 20) and tend to overcharge, sometimes badly.
If you decide to take a cab, make sure that you agree
on the rate with the driver in advance. They can charge
ONLY FLAT RATES, which depend on the district of Budapest
where you drive.
I recommend the Airport Minibus Service
(~USD 7), especially for single travellers. This service is available
immediately after you emerge from the customs area. Normally
you have to wait for 1520 minutes, the minibuses
take a couple of travellers and drop them in sequence.
In both cases (cab or minibus) a tip 1015%
to the driver is recommended.
The EXACT address of the guesthouse is in
Hungarian:
BME Professzori Vendégház
Stoczek utca 57 (Martos kollégium)
1111 Budapest
If you print this out and show it to the receptionist
of the Minibus service or the cab driver, they will know.
Both of them would know Hotel Gellert without any further
specifications.

The `Indian rope trick' for a continuously flexible rod;
linear and nonlinear analysis.
Alan R. Champneys
Applied Nonlinear Mathematics Group
Dept of Engineering Maths
email: a.r.champneys@bristol.ac.uk
University of Bristol
phone: (+44) (0)117 928 7510
Bristol BS8 1TR UK
fax: (+44) (0)117 925 1154
Acheson and Mullin have demonstrated both experimentally and
theoretically that it is possible to balance a system of N coupled
rigid pendulums upside down by oscillating its support. In the limit
as
N tends to infinity the theoretical region of stability in frequency
vs. amplitude disappears. Nevertheless experiments of Mullin on a
continuously flexible rod (a piece of curtain wire just long enough
to
fall over under its own weight) suggest it can be stabilised using
the
right frequency and amplitude of excitation.
We propose a theory for this. First, a dimensionless PDE is derived
from rod theory that is fourthorder in arclength and secondorder
in
time. The stability of the vertical solution subject to parametric
excitation is then studied using Floquet theory, which is the
infinitedimensional analogue of the equivalent analysis for the
Mathieu equation. The results of doublescale asymptotics and numerics
agree and lead to an upside down stability region, albeit one which
is
punctured by thin Arnold tongues corresponding to higherorder
resonances.
The asymptotic analysis is extended to include geometric
nonlinearities and viscous damping. The stability of various
bifurcating states at the simplest resonances is calculated. Emphasis
is placed on the distinction between planar and circular motions which
bifurcate into different symmetry subgroups of the problem. A
qualitative agreement is found with the experimental data on the
observed stability boundaries being caused by an interaction between
a
fundamental and subharmonic instability.
Back to List of Speakers
On solutions of the selfcontact problem for elastic rods and applications
to DNA supercoiling
Bernard D. Coleman, Rutgers University, Department of Mechanics and
Materials Science, Piscataway, NJ 088548058, USA
A plasmid is a DNA molecule in which the two strands of the WatsonCrick
double helix (duplex) structure are intertwined closed curves.
There are
circumstances under which it is useful to model a segment of DNA as
an
inextensible, homogeneous, naturally straight, elastic rod of circular
crosssection. (This can be called the idealized rod model for DNA.)
Research of David Swigon, Irwin Tobias, and the speaker has yielded
exact
analytical solutions of Kirchhoff's equations of mechanical equilibrium
for
knotted and unknotted plasmids, with the effects of impenetrability
and
selfcontact forces taken into account. Such explicit solutions
are now
available for cases in which self contact occurs both at isolated points
and along curves. To be presented in this talk are examples of
configurations and results about the stability of equilibrium
configurations. The emphasis will be laid on bifurcation diagrams in
which
the bifurcation parameter is the excess link, a topological parameter
that
is closely related to the Gauss linking number for the two DNA strands.
General rules will be presented concerning (i) the relation of the
symmetry
of equilibrium solutions to their location in bifurcation diagrams
and (ii)
the graphtheory structure of the set of points of selfcontact.
A
discussion will be given of the way in which the analytical solutions
may
be employed to calculate bounds on the activation energy for transitions
between distinguishable metastable equilibrium configurations with
the same
topological parameters.
Discontinuous Piecewise Polynomial Collocation Methods for Elliptic PDEs
Eusebius Doedel
Applied Math 21750
California Institute of Technology
Pasadena CA 91125
(on leave from Concordia University, Montreal)
Piecewise polynomial collocation methods, as implemented in software
such as COLSYS and AUTO, have been useful in studying the behavior
of dynamical systems represented by ODEs. For example, the behavior
and bifurcations of periodic solutions as a parameter varies
can be
tracked very effectively by using collocation coupled to numerical
continuation and bifurcation techniques. Adaptive mesh selection
allows
the computation of "difficult" orbits, e.g., nearhomoclinic
periodic
orbits. Although collocation methods have been extended to PDEs,
their
use in studying bifurcation phenomena in such equations has been.
In this
talk I introduce a class of highorder accurate, globally
discontinous piecewise polynomial collocation methods for solving
nonlinear
elliptic PDEs, using adaptive triangulations of the problem domain.
The nested dissection technique used in solving the linearized
systems
that arise in Newton's method will be briefly described. Numerical
results that illustrate the effectiveness of the methods will
be given.
It is also indicated how problems with certain symmetries and
invariances
can be treated.limited.
Back to List of Speakers
G. Domokos
Technical University of Budapest
H1521 Budapest, Hungary
It is widely known that discretization can yield
spurious solutions not only in Initial Value Problems (IVPs)
but also in Boundary Value Problems (BVPs). However,
it is still believed by many engineers that "sufficiently
smooth/continuous" solutions of discretized models
are valid approximations of the solutions in the
continuous model.
This talk addresses the above question; the goal
is to show that there exist arbitrarily smooth/continuous
("ghost") solutions of the discretized model which are bounded away
from any of the continuous solutions. We will
illutrate this on Euler's elastica problem
and indicate the class of problems in which
similar phenomena can be expected. We also
discuss the physical relevance of ghosts.
Special shapes of bent and twisted rods
Zsolt Gaspar and Robert Nemeth
Budapest University of Technology and Economics
Dept. of Structural Mechanics
H1521 Budapest Mûegyetem rkp. 3. Hungary
An inextensible, long, straight elastic rod of circular
cross
section is considered. First it is bent into a torus form, then a
relative rotation between the end cross sections will be produced.
Beyond
the critical value of this relative rotation the axis of the bar will
be a spatial curve, and there will be selfcontact points.
We are dealing with the special interval of the relative rotation
when the deformed rod has three symmetry axes, and there is also an
interval of the contact points. For these cases the differential
equations will be derived, the shape and the stresses of the rod will
be showed.
Homoclinic collapse in the statics and dynamics of biological filaments
by A.Goriely, Dept. of Mathematics, University of Arizona.
In this talk, I will consider models for long thin intrinsically curved
filaments. These models are used in a variety of biological systems
ranging from bacterial flagella to climbing plants. As
tension is
decreased or intrinsic curvature is increased, these filaments bifurcate
to solutions connecting asymptotically helices with opposite
handedness.
The statics of these heteroslinic structures is studied within the
framework of the Kirchhoff theory with linear and nonlinear elasticity.
A center manifold reduction and a normal form transformation
for a
triple zero eigenvalue reduce the dynamics to a third order reversible
dynamical system. The analysis of this reduced system reveals that
the
heteroclinic connection representing the physical solution results
from
the collapse of pairs of symmetric homoclinic orbits.
The dynamics of moving fronts and the possibilty of multiheteroclinic
orbits
will be discussed. Applications to realistic biological systems
such as the
flagellar bistability in Salmonella and the motion of spirochetes will
also
be considered.
Nonlinear dynamics and stability of spherical membranes
Oded Gottlieb
Faculty of Mechanical Engineering
TechnionIsrael Institute of Technology
Haifa, Israel.
This paper focuses on the nonlinear dynamic inflation process of spherical
membranes. We consider isotropic hyperelastic constitutive laws portrayed
by a classical MooneyRivlin material and by a family of compressible
BlatzKo materials where the effect of material compressibility is
characterized by the material Poisson function.
A bifurcation analyses of the integrable problem reveals conditions
controlling local stability of periodic solutions and existence of
limiting homoclinic solutions. Perturbation of the integrable structure
with (weak) dissipation and a time dependent inflation process renders
the
problem nonintegrable. Approximate conditions for aperiodic dynamics
are
determined using Melnikov's method complemented by numerical simulations.
The results shed light on the nonlinear (initial) inflation dynamics
of
spherical membranes and reveal the influence of several controlling
parameters governing nonlinear compressible and incompressible
viscoelastic materials utilized in large (e.g. space) and small (e.g.
medical applications) structures and biological tissue materials.
Coherent structures and mixing in twodimensional turbulence
G. Haller, Division of Applied Mathematics, Brown University
It is well known that in twodimensional turbulent fluid flows
coherent structures tend to emerge. These structures have primarily
been described based on instantenous Eulerian quantities, such as
energy, vorticity, or enstrophy, all giving different answers for
the exact location and properties of coherent structures. At the same
time,
in many industrial and geophysical flow problems the exact location
of
the boundaries of these structures, as well as their impact on mixing,
is
of fundamental importance. In this talk we describe new dynamical
systemsbased
methods that do detect exact coherent structure boundaries in the Lagrangian
sense.
We show how such boundaries can be extracted from numerical simulations
and
experimental data.
Multiple Parameter Continuation
Michael E. Henderson
IBM Research
P.O. Box 218, Yorktown Heights, NY 10598
A new continuation method for computing implicitly defined manifolds
of
arbitrary dimension will be described.
The method is based on a representation of the manifold as the projection
of a set of overlapping spherical balls,
with new points being added by choosing a point on the boundary of
the
collection of balls. This will be shown
to be equivalent to computing the vertices of a certain set of
convex
polyhedra, which form a polyhedral decomposition
of the manifold, and which are a weighted Voronoi diagram for the centers
of the balls. This decomposition
can be used to generate a triangulation of the manifold similar to
the
Delaunay triangulation.
Several applications will be described, including the computation
of a
manifold of periodic motions of a pair
of coupled pendula. The representation may also be used for manifolds
defined in other ways, provided that a method
can be found to project from the tangent space onto the manifold. Methods
for invariant manifolds of mappings and flows
will be presented.
Back to List of Speakers
Hidden Symmetry of Global Solutions in Twisted Elastic Rings
Tim Healey, Gabor Domokos
Cornell University
Department of Theoretical and Applied Mechanics
Ithaca, NY 148531503 USA
We investigate global equilibria of twisted, isotropic
elastic rings. The high degree of symmetry in the problem
leads to nonisolated solutions. We prove that
{\em all} solutions are flipsymmetric, and from this
we can globally isolate all solution branches. We
derive possible other choices for boundary conditions
systematically and show that other conditions necessarily
lead to {\em spurious} solutions. We show global computations
performed by the Parallel Simplex Algorithm.
Back to List of Speakers
Unconstrained Euler buckling in a potential field.
P. Holmes, G. Domokos and G. Hek
We consider elastic buckling of an inextensible rod with free ends,
confined to the plane, and in the presence of distributed body forces
derived from a potential. We formulate the geometrically nonlinear
(Euler) problem with nonzero preferred curvature, and show that it
may
be written as a threedegreeoffreedom Hamiltonian system. We focus
on
the special case of an initially straight rod subject to body forces
derived from a quadratic potential uniform in one direction; in this
case the system reduces to two degrees of freedom. We find two classes
of trivial (straight) solutions and study the primary nontrivial
branches bifurcating from one of these classes, as a load parameter,
or
the rod's length, increases. We show that the primary branches may
be
followed to large loads (lengths) and that segments derived from
primary solutions may be concatenated to create secondary solutions,
including closed loops, implying the existence of disconnected
branches. At large loads all finite energy solutions approach
homoclinic and heteroclinic orbits to the other class of straight
states, and we prove the existence of an infinite set of such
`spatially chaotic' solutions, corresponding to arbitrary
concatenations of `simple' homoclinic and heteroclinic orbits. We
illustrate our results with numerically computed equilibria and global
bifurcation diagrams.
Back to List of Speakers
Maxwell
Lower Bounds for Bifurcation at Infinite Load
G.W. Hunt
Department of Mechanical Engineering and Centre for Nonlinear
Mechanics
University of Bath, UK
There are a number of problems of structural stability, oneway
buckling of
(heavy) railway tracks and pipelines from impenetrable beds for example,
or
the wellknown failure mechanism for fibrous and layered media known
as
kinkbanding, which carry quite pathological stability characteristics.
A realistic modelling process leads in each case to a symmetrical
(prebuckling) equilibrium state that is apparently a local energy
minimum
for all values of applied compressive load. Not only does this suggest,
erroneously, that the system can maintain stability up to infinite
load, but
the usual practices of linearization can now play no useful purpose.
Such
systems are clearly sensitive to external disturbances and imperfections,
but
assessing the level of this sensitivity is difficult.
Instabilities of this kind are often associated with a form of localization
in
which one part of a structure unloads elastically into a localizing
region.
One measure of sensitivity then lies in the ratio of the lengths of
the
unloading and localizing regions: the greater this value, the less
the
energy per unit length, averaged over the structure as a whole, necessary
to
trigger the instability. Such considerations lead naturally to lower
bound
descriptions for critical loads or critical applied displacements in
terms
of the Maxwell criterion, where stability rests only with global, as
opposed
to local, energy minima. The paper will review the accuracy of
the criterion
for simple one and two degree of freedom systems, modelled as dynamical
systems
and including effects of stochastic noise.
Back to List of Speakers
Fractality, chaos, and reactions in imperfectly mixed
open hydrodynamical flows
György Károlyi, Tamás Tél
TU Budapest , Eötvös University
Recent developments in the field of chaotic advection in
hydrodynamical or environmental flows encourage us to revisit the
chemical and biological dynamics of active tracers in open aquatic
systems.
Passively advected tracers in open hydrodynamical flows
are known to trace out permanent fractal patterns.
If these advected tracers undergo certain chemical or biological
interactions, the activity takes place on a fractal set.
This observation leads to a novel form of a reaction
equation containing essential chaos parameters.
In biological respects,
while in homogeneous, well mixed environments only
one species could survive a competition for a common limiting resource,
coexistence of competitors is typical in our hydrodynamical system.
Curvature and parallel transport in averaging
Mark Levi
Stabilization of the inverted pendulum by vibration of its
suspension point is a well known and surprising phenomenon. A deeper
study
of this effect leads to a new observation: there is some hidden geometry
geometry behind various mechanical phenomena associated with highfrequency
vibrations. One of these is a relationship of normal form (averaging
theory)
and parallel transport.
Symmetry Breaking and Stability in DNA minicircles
John H. Maddocks
Abstract
DNA minicircles are formed when the double helix of a short DNA
molecule bonds with its own tail to form a closed loop. Minicircles
of
200 or so basepairs are a very convenient experimental motif for
chemists. In the first part of this talk I will describe some old work
in which my collaborators and I used computation of the bifurcation
diagram for an idealized, highly symmetric, elastic rod model, combined
with symmetry breaking techniques, and stability analysis to model
various experimental data for minicircles. The computations and
analysis make extensive use of the appropriate Hamiltonian form of
the
equilibrium conditions and a new theory of isoperimetric conjugate
points for constrained variational principles. In the second part of
the
talk I will describe more recent work that addresses the issue of
identifying perturbations for which the symmetry breaking is not
generic. This analysis explains recent Monte Carlo simulations of
Katritch and Vologodski in which multipeaked Link distributions have
been observed, and offers the promise of designing particular basepair
sequences that should exhibit atypical behaviour in minicircle
experiments.
Nonstandard Reduction of Noisy Mechanical Systems
N. Sri Namachchivaya and Richard Sowers
Department of Aeronautical and Astronautical Engineering
University of Illinois at UrbanaChampaign
Urbana, IL 61801, USA
The purpose of this paper is to formulate and develop
methods to analyze certain behaviors of {\em mechanical systems}.
Small random excitations affect many mechanical systems, and
the presence of random vibrations can lead to a host of undesirable
effects: reduction of the dynamic strength via fatigue crack growth
or other modes of failure, stochastic instabilities, etc.
An understanding of their dynamics necessitates a study of the
complex interactions between {\em noise},
{\em symmetries}, and {\em nonlinearities}.
Our approach will consist of the application of some recent
abstract theories of stochastic dimensional reduction to some relevant
mechanical models.
Our main analytical tool is a certain method of {\em dimensional reduction}
of nonlinear systems with symmetries and small noise.
As the noise becomes asymptotically small, one can exploit symmetries
and a
separation of scales to use wellknown methods (viz. stochastic averaging)
to
find an appropriate lowerdimensional description of the system.
The interest of this paper is when classical methods fail because the
original Hamiltonian system has bifurcations, which give rise
to singularities in the lowerdimensional description. Such singularities
complicate the analysis of standard designrelated statistical measures
of
response and stability (e.g., mean exit times and stationary measures);
our
methods are uniquely adapted to study exactly such problems.
The focus of this work is the development
of general techniques of stochastic averaging of randomlyperturbed
fourdimensional integrable Hamiltonian systems.
The integrable system here has certain nontrivial (yet generic) types
of fixed points.
Stochastic averaging makes use of these integrable structures to
identify a reduced diffusive model on a space which
encodes the structure of the fixed points and can have dimensional
singularities.
At these singularities, {\em glueing conditions} will be derived,
these glueing conditions completing the specification of the
dynamics of the reduced model.
Qualitative dependence of statistical measures of the reduced system
upon
various coefficients can be studied by extensions of known techniques
such as stochastic bifurcation theory.
The symmetry of some of the simple nonholonomic examples precludes assymtotic stability.
Andy Ruina
Cornell University
Department of Theoretical and Applied Mechanics
ruina@cornell.edu
Conservative nonholonomic mechanical systems are not Hamiltonian
in general and thus are not constrained to obey the theorems
applicable to Hamiltonion systems. In particular Hamiltonian
systems cannot have assymptotic stability and nonholonomic
systems can.
But this fact has been often missed by the nonexperts. Perhaps
this is because many of the common examples of nonholonomic
systems have symmetries that preclude stability. In particular,
a system that has a reflection symmetry that correponds to
moving backwards cannot be so stable. Such systems include
a rolling hoop, ball, or symmetric top while rolling on
a flat plane, a surface of revolution or a prismatic channel.
The Chapligin sleigh is also an example of such a system if
the center of mass is on the line normal to the skate.
Nonholonomic systems that violate this symmetry, and thus
better reveal the stability possibilities of nonholonomic
systems, include a general Chaplygin Sleigh, MonteHubbards
skateboard, and a bicycle.
KEY WORDS: nonholonomic, symmetry, stability
Back to List of Speakers
Dynamic Stability and Performance of Systems of Nonlinear Vibration Absorbers
Steven W. Shaw
Department of Mechanical Engineering
Michigan State University
East Lansing, MI 488241226 USA
Centrifugal pendulum vibration absorbers are devices that are used to
attenuate torsional vibrations in rotating machinery. They consist
of a
set of masses that are mechanically suspended from a rotating shaft
and are
free to move relative to it in a plane perpendicular to the axis of
rotation. The paths along which these masses move are designed
so that
their motions dynamically cancel the effects of external fluctuating
torques applied to the shaft. Some key features of these passive
absorbers
include the following: they are linearly tuned to match the frequency
of
the applied torque over all operating speeds; they operate most effectively
when lightly damped; one can design the paths of the absorber masses
to
achieve desired performance; the absorbers are typically identical
and
coupled to one another indirectly through the (relatively large) inertia
of
the rotor. It is shown that the equations of motion of such a
system
possess SN symmetry, and the fully symmetric response is desired in
applications. Scaling of the system parameters allows one to
express the
equations of motion in a form that is amenable to asymptotic analysis.
Application of the method of averaging shows that as system parameters
are
varied, the desired response can exhibit bifurcations that maintain
the
full symmetry as well as those that result in a lower order of symmetry.
It is also shown that imperfections among the absorbers can lead to
localized responses in which a small subset of absorbers undergo large
amplitude motions. The effects of system and excitation parameters
on
these undesirable behaviors are examined, with the goal of minimizing
torsional vibration levels of the rotor. It is shown that by
proper
selection of the absorber paths, both the instabilities and the
localization can be avoided, thereby providing good performance over
a wide
range of operating conditions.
Back to List of Speakers
Stability and bifurcations in dynamical systems
with time delay and parametric excitation
G. Stépán and T. Insperger
Department of Applied Mechanics
Budapest University of Technology and Economics
Budapest H1521 Hungary
email: stepan@mm.bme.hu, inspi@m.bme.hu
The stability analysis of linear autonomous ordinary differential equations can based on their characteristic roots. The Routh–Hurwitz criterion (back to 1875) supports to check whether all the characteristic roots are located in the left half of the complex plane. In case of single degree of freedom mechanical systems, the necessary and sufficient condition of asymptotic stability is the presence of positive stiffness and damping.
When the parameters of the ordinary differential equation depend on the time periodically, that is in case of parametric excitation, the linearization of the system still describes the stability of the trivial solution. However, the socalled Floquet theory (back to 1883) does not provide closed form results even for the simplest single degree of freedom mechanical system with time periodic stiffness and zero damping. The stability chart of the corresponding Mathieu equation was constructed via power series by Incze and Strutt in the last century.
When time delay appears in the autonomous systems, the differential equation is not ordinary anymore. It belongs to the theory of functional differential equations which involves differentialdifference (in other words delaydifferential) equations. The linear stability criterion is based on characteristic roots again. This time, infinitely many characteristic roots are to be checked whether they have negative real parts. In some cases, closed form results can be given. For example, the undamped, single degree of freedom mechanical system with delay in its stiffness was fully analyzed by Hsu and Bhatt in 1966.
In this paper, the combination of the two cases above, that is, the
stability investigation of the timeperiodic delaydifferential equation
is presented. As a mechanical application, the stability and
bifurcation analysis of the milling process is considered.
Transition from spherical circle packing to covering
Tibor Tarnai
Budapest University of Technology and Economics
Department of Structural Mechanics
Budapest, Muegyetem rkp. 3., H1521 Hungary
email: tarnai@epmech.me.bme.hu
How must n equal circles (spherical caps) of given angular
radius r be arranged on the surface of a sphere so that the
area covered by the circles will be as large as possible?
In this paper, conjectural solutions of this problem for
n=5,7,8,9,10,11,13 are given when r varies from the maximum
packing radius to the minimum covering radius. It is apparent
that this variation of r provides a transition from the
maximum circle packing to the minimum circle covering on
a sphere. To a circle arrangement, there is associated a contact
polyhedron whose vertices are the centres of the spherical
circles and whose edges appear between the centres of
two circles having at least one point in common. During this
configuration transition, the symmetry and the number of edges of
the contact polyhedra change. The change is in general continuous,
but sometimes at certain points, discontinuous as in the case
n=7,11,13. For n=2,3,4,6,12 we obtain the regular polyhedra
as solutions for any r, that means that the configurations
are constant during the packing to covering transition. The case
n=13 introduces several features not present for lower numbers
of circles: large numbers of symmetry and edge transitions in
contact
polyhedra, crowding of transitions within small ranges of r and
three separate intervals where the position of one circle has
a limited freedom to "rattle" without changing the fractional
covering of the sphere. The relation of the results of this
problem to the extremal configurations of n mutually repulsive equal
point charges on the surface of a sphere, and to chemical isomerization
processes will be discussed. The role of symmetry will be also analyzed.
Application of Normal Form theory in an optimal controlled production
model
Alois Steindl, Gustav Feichtinger
Institute of Mechanics II and Institute of Econometrics, Operations
Research
and System Theory, Vienna University of Technology
We investigate the creation of periodic solutions in a continuous production
model with nonconvex production cost and constant demand.
For vanishing interest rate a ``Hamiltonian Hopf bifurcation'' occurs,
leading to a family of periodic solutions bifurcating from a steady
state. Normal Form Theory introduces a mathematical phase shift
symmetry and leads to a first integral, admitting the determination
of
the periodic solutions. If the interest rate is assumed small, but
different from zero, almost all periodic
solutions vanish; only an hyperbolic unique branch of periodic solutions
is
obtained.
Back to List of Speakers
Elastic stability of knotted DNA plasmids
David Swigon, Rutgers University, Piscataway, New Jersey 088548058 USA
In research of Bernard Coleman, Irwin Tobias, and the speaker on the
theory
of the idealized elastic rod model for DNA exact analytical representations
have been obtained for equilibrium configurations of DNA segments showing
both isolated points and intervals of selfcontact. Criteria
have been
derived for determining whether an equilibrium configuration is stable
in
the sense that it gives a strict local minimum to elastic energy.
Here
this theory will be applied to knotted DNA plasmids with emphasis placed
on
the dependence on excess link of minimum energy configurations of DNA
trefoil knots. Among the results to be presented is the following:
the
minimum bending energy configuration a (2,q) torus knot shows selfcontact
along a circle. This result remains valid even when the integer q is
even,
i.e., when the "knot" reduces to a link of 2 unknots.
The chaotic instability of a slowly spinning asymmetric top
J.M.T. Thompson and
Gert Van der Heijden
Centre for Nonlinear Dynamics,
University College London
We consider a top that is spinning in an upright state at an angular
velocity
that is less than the critical value predicted by classical bifurcation
theory. The top is then in a state of unstable relative equilibrium,
and
given an infinitesimal perturbation it will fall away from the upright
configuration. In the absence of dissipation, it will however return,
again
and again, very close to the upright configuration. In technical terms,
its
motion will lie very close to a homoclinic orbit that, as time passes
from
minus infinity to plus infinity, departs from the upright state and
returns
to it. For a top spinning about an axis of symmetry, the repeated returns
will be regular and predictable. However, for a nonsymmetric top there
are
an infinite number of different homoclinic orbits, implying that the
repeated
returns are chaotic and unpredictable. We use homoclinic pathfollowing
techniques to explore the structure of these homoclinic orbits, and
illustrate the physical motions involved.
Stability Analysis of Relative Equilibria of Tethered Satellite Systems
Martin Krupa, Martin Schagerl, Alois Steindl, and
Hans Troger,
Vienna University of Technology, A1040 Vienna, Austria
After shortly explaining the concept of tethered satellite
systems and giving some practical applications of this concept, we
present two
different mathematical models of such systems.
The first is a finite dimensional model consisting of a rigid
rod and two point masses. The second more complicated model is
infinite dimensional and
consists of two masses, which are connected by a massive perfectly
flexible and extensible string.
For both systems relative equilibrium positions for a circular motion
around
the Earth exist, due to the $O(3)$symmetry of the system. Their
stability
will be investigated by means of the reduced energy momentum method
since by symmetry, besides energy, also the angular momentum
is conserved.
For the continuous system we distinguish between two different
types of relative equilibria.
One are the socalled trivial relative equilibria, for example,
the practically important radial configuration.
The second are nontrivial (wavy) tether configurations which
require active
control of the motion of at least one of the satellites.
For the solution of the nontrivial
relative equilibrium problem we formulate a bifurcation problem, where
we vary the position of one of the satellites relative to the other.
Numerical methods have to be applied, first, in the calculation of
the nontrivial relative equilibria by Finite Differences and,
further, in the evaluation
of the stability condition following from the reduced energy
momentum method.
Back to List of Speakers
Passive Nonlinear Energy Pumping due to Symmetry Breaking in Coupled
Oscillators
Alexander F. Vakakis
Mechanical and Industrial Engineering
Theoretical and Applied Mechanics
University of Illinois
Oleg Gendelman
Institute of Chemical Physics
Russian Academy of Sciences
Igor Rozhkov
Theoretical and Applied Mechanics
University of Illinois
We present numerical evidence of nonlinear energy pumping from a linear
system
of coupled oscillators to a symmetrybreaking essentially nonlinear
element. By
energy pumping we mean the oneway, irreversible, transient \221channeling\222
of
externally induced energy from the linear to the nonlinear part of
the system.
This phenomenon occurs only when the energy is above a critical level.
To
analytically study energy pumping we first restrict our attention
to a two
degreeoffreedom system, transform the equations of motion into actionangle
form, and apply twofrequency averaging. We show that energy pumping
is due to
resonance capture on a 1:1 resonance manifold of the system, and perform
a
perturbation analysis in an O() neighborhood of this manifold to study
the
timevarying attracting region responsible for this phenomenon. An
alternative
analytical approach based on the assumption of 1:1 internal resonance
in the
fast dynamics of the system, complexifies and averages the fast dynamics.
This
leads to satisfactory analytical approximations of the nonlinear transient
responses of the system in the energy pumping regime.
In the next step of our step we consider energy pumping in a periodic
semiinfinite linear periodic system of coupled oscillators, attached
to an
essentially nonlinear oscillator. We study in detail the interaction
of the
linear chain with the essentially nonlinear element in order to identify
the
harmonic components of the interaction that are responsible for the
energy
pumping phenomenon. An analytical technique is then employed to analytically
study the transient response of the system in the energy pumping regime.
The
practical applications of the nonlinear energy pumping phenomenon to
the
active/passive control of largescale mechanical systems are discussed.
Back to List of Speakers