0.3truecm

**A Fredholm Determinant for Semi-classical Quantization
0.01truecm**

We investigate a new type of approximation to quantum determinants,
the ``Quantum Fredholm Determinant", and test numerically the conjecture
that for Axiom A
hyperbolic flows such determinants have a larger domain of analyticity
and better convergence than the quantum Selberg products derived from the
Gutzwiller trace formula. The conjecture is supported by numerical
investigations of the
3-disk repeller, a normal-form model of a flow, and a model 2- map
[][].
-0.1truecm *P. Cvitanovic, P.E. Rosenqvist, G. Vattay, and H.H. Rugh
0.3truecm*

**Entire Fredholm Determinants for Evaluation of
Semi-classical and Thermodynamical Spectra
0.01truecm**

Proofs that Fredholm determinants of transfer operators for hyperbolic
flows
are entire can be extended to a large new class of multiplicative
evolution
operators. We construct such operators both for the Gutzwiller
semi-classical
quantum mechanics and for classical thermodynamic formalism, and
introduce a
new functional determinant which is expected to be entire for Axiom A
flows,
and whose zeros coincide with the zeros of the Gutzwiler-Voros zeta
function.
[].
-0.1truecm *P. Cvitanovic and G. Vattay
0.3truecm*

**Creeping Orbits in Semi-classical Quantization
0.01truecm**

We try to incorporate the diffraction phenomena in the semi-classical
description of classically chaotic systems.
This leads us to introduction of periodic creeping orbits which are created by
diffraction and thus have no classical particle trajectory analogue.
We are studying their effects in
simple 2-disk and 3-disk scattering systems, and are
comparing our results to exact quantum mechanical results.
-0.1truecm *P. Cvitanovic, P.E. Rosenqvist, G. Vattay, and (A. Wirzba)
0.3truecm*

**Trace Formula for Dynamic Systems in the Presence of Noise
0.01truecm**

Evolution operator methods for evaluation of averages in chaotical
dynamical systems require arbitrarily fine symbolic dynamics partition of
the phase space; at the same time,
the attainable accuracy in physical dynamical systems is always
limited by external noise. We use external noise as a parameter which
defines the smallest phase-space partitions that need to be taken into
account in a given computation, and generate finite Markov diagram
approximation to the dynamics.
-0.1truecm *P. Cvitanovic and (R. Mainieri)
0.3truecm*

**Advection of Vector Fields by Chaotic Flows
0.01truecm**

We have introduced a new transfer operator for chaotic flows whose leading
eigenvalue yields the dynamo rate of the fast kinematic dynamo, and applied
cycle expansion of the Fredholm determinant of the new operator to evaluation
of its spectrum. The theory has been tested on a normal form model of the
vector advecting dynamical flow. If the model is a simple map with constant
time between two iterations, the dynamo rate is the same as the escape rate
of scalar quantities. However, a spread in Poincaré section return times
lifts the degeneracy of the vector and scalar advection rates, and leads to
dynamo rates that dominate over the scalar advection rates. For sufficiently
large time spreads we have even found repellers for which the magnetic field
grows exponentially, in contrast to the scalar densities which decay
exponentially
[].
-0.1truecm *(N.J. Balmforth), P. Cvitanovic, (G.R. Ierley), (E.A. Spiegel), and G. Vattay
0.3truecm*

**Periodic Orbit Expansions for Power Spectra of Chaotic Systems
0.01truecm**

We develop the periodic orbit theory of power
spectra for chaotic dynamical systems. The theory is tested
numerically on several 1-dimensional mappings.
-0.1truecm *P. Cvitanovic, (M.J. Feigenbaum), and (A.S. Pikovsky)
0.3truecm*

**Symbolic Dynamics Description of Chaotic Systems
0.01truecm**

We study a number of different chaotic systems using the theory of
symbolic dynamics. We show how symbolic dynamics can be used to
describe -modal one-dimensional maps and two-dimensional maps of
the Hénon type. Different billiard systems, such as the three disk
system, hyperbola billiard, stadium billiard and the wedge billiard
are described and we show how to construct a well ordered symbolic
plane and find pruning fronts for these systems. These results will be
applied to describe semiclassically quantum systems.
-0.1truecm *P. Cvitanovic and K.T. Hansen
0.3truecm*

**Sound Waves in Crystals
0.01truecm**

The experimental study of eigenmodes of microwave cavities has
received considerable credit in recent years. For an essentially
two dimensional cavity, the corresponding electrodynamical wave
equation coincides mathematically with the quantum mechanical
Schrödinger equation. The cavities have to be cooled to a
temperature of about 2K to achieve a high resolution in the spectra.
It is investigated whether a comparably high resolution can be achieved
by exciting the eigenmodes of a crystal with sound waves. The
study of three dimensional geometries should also be possible.
-0.1truecm *C. Ellegaard, T. Guhr, and M. Oxborrow
0.3truecm*

**Correlation Holes in Microwave Spectra
0.01truecm**

In order to achieve a high resolution in microwave cavity spectra,
it is necessary to cool the cavity down to a temperature of about
2K. The correlation hole technique, frequently used in molecular
physics, decouples to some extent the statistical influences of the
widths and the positions of the resonances. We study if this method
works for cavity spectra as well, to become less dependent on high
resolution and cooling.
-0.1truecm *T. Guhr, (R. Hofferbert), and (A. Richter)
0.3truecm*

**Fourier Bessel Analysis in Graded Matrix Spaces
0.01truecm**

The generating function of the correlation functions in theories
involving random matrices can often be mapped onto theories,
that show a so called hopping coupling and that are
formulated in spaces of graded matrices. To evaluate the integrals
an expansion is employed similar to the well known plane wave expansion.
The angular degrees of freedom appear in generalized spherical harmonics
and can thus be integrated by using results from representation theory.
-0.1truecm *T. Guhr
0.3truecm*

**Mixing of a Fixed Matrix to a Gaussian Unitary Ensemble
0.01truecm**

A given matrix is, with a strength parameter, mixed to a Gaussian Unitary
Ensemble. The correlation functions for arbitrary level number are evaluated
by using the graded eigenvalue method. The results involve the graded
symmetric functions which are related to the representations of supergroups.
-0.1truecm *T. Guhr
0.3truecm*

**Thermoelectricity in Bimetallic Film
0.01truecm**

In 1821, Thomas Seebeck discovered that an D.C. electromotive
force can be generated by heating a junction between two different
conductors. This phenomenon, the so-called Seebeck effect, is
reversible; it can be used to convert thermal energy into
electrical energy with Carnot efficiency. Because irreversible
phenomena (Joule heating and thermal conduction) inevitably
accompany the Seebeck effect, however, no device based on the
Seebeck effect, that can generate a *sizeable* amount of
electricity at a *reasonable* efficiency, has ever been
built.
In 1993, I am attempting to generate an A.C.thermoelectric
current, within a sputtered Al-Ni bilayer that lies on a
piezoelectric PVDF substrate, by applying an ultrasonic (2MHz)
wave normal to the substrate's surface. So far, I have generated
only heat.
-0.1truecm *M. Oxborrow
0.3truecm*

**Embedded Gaussian Unitary Ensemble
0.01truecm**

The Embedded Gaussian Ensembles model the forces in physical systems
more realistically than the plain Gaussian Ensembles. The graded
eigenvalue method and the graded Fourier Bessel Analysis are applied
in order to study the Embedded Gaussian Unitary Ensemble.
-0.1truecm *T. Guhr
0.3truecm*

**Four Spheres Scattering Problem
0.01truecm**

The scattering of a particle in a geometry of three discs in two dimensions
was studied classically and quantum mechanically by several authors.
Particularly, it was one of the first problems investigated with the
periodic orbit theory. We study the analogous problem in three
dimensions, the tetrahedron of four spheres. We expect new information
about the dependence of (semi)classical methods on the dimensionality.
The model is also more realistic for applications in condensed
matter physics.
-0.1truecm *(A. Wirzba) and T. Guhr
0.3truecm*

**Periodic Orbits of Nonscaling Hamiltonian Systems from Quantum
Mechanics.
0.01truecm**

The spectral formfactor is utilized to define a quantum equivalent of
the classical -plot of the periodic orbits of a Hamiltonian
system.
For scaling systems where the classical dynamics is independent of the
energy (except for a trivial scaling) the quantal -plot may
be reduced to the Fourier transform of the energy spectrum.
For nonscaling systems on the other hand, the quantal -plot
reveals the richness of the classical dynamics with its bifurcations of
periodic orbits.
The analysis may be extended into the regime where the diagonal form of
the semiclassical formfactor is not expected to hold and where the
connection to Random Matrix Theories is largely unknown.
-0.1truecm *(M. Baranger), (M. Haggarty), B. Lauritzen, (D. Meredith), and (D. Provost)
0.3truecm*

**Bound States in Twisting Tubes
0.01truecm**

It is known that an infinite tube with constant cross-section has
propagating modes if the frequency is above a certain cutoff frequency.
What is not so widely known is that if the tube is bent there can also
be localized states around the bend. These states exist as solutions
of both Schrodinger's equation and also of Maxwell's equations when the tube is
very narrow in one dimension so as to be quasi two-dimensional. This
contrasts with the corresponding classical motion which is not
localized. I am interested in understanding the connection between
the classical and quantum behaviour by using semiclassical
approximations to the quantum Green function. Hopefully this project
will result in predictions for microwave experiments.
-0.1truecm *N.D. Whelan
0.3truecm*

**Classical Transport in Near-Integrable Chaotic Systems
0.01truecm**

The KAM theorem of classical mechanics asserts that if an integrable
Hamiltonian is subject to a small perturbation then only a small fraction of
the tori are destroyed. However, this theorem has certain smoothness
conditions for the perturbation. These conditions are violated for a number
of systems of common interest such as the Bunimovich stadium, the Sinai
billiard and the wedge in a constant gravitational field. Because of this
violation these three problems
share the property of being integrable for one choice of parameter and
completely chaotic for arbitrarily close parameter values. This non-KAM
behaviour is not very well understood.
I am interested in understanding
the nature of the classical transport of these chaotic systems when their
hamiltonians are close to the integrable limit. Results obtained so far
indicate that a trajectory remains trapped for a long time in a small region
of phase space before making a sudden jump to a different region. It would be
good to understand the mechanism for this behaviour and the extent to which
understanding the nearby integrable Hamiltonian helps in understanding the
chaotic Hamiltonian. In the longer term, it would also be interesting to see
how this is reflected in the corresponding quantum dynamics.
-0.1truecm *N.D. Whelan
0.3truecm*

**Semiclassical Analysis of a Hyperbolic System
0.01truecm**

Much is now understood about the connection between classical periodic
orbits and the quantum Green function. We want to use this
understanding to study an apparently simple problem. We consider free motion
between confocal hyperbolae. This problem is integrable both
classically and quantum mechanically. The quantum eigenvalue problem can be
solved in terms of Mathieu functions and the classical problem has two
independent integrals of motion in involution. What makes this problem
interesting is that there is only one classical periodic orbit, which is
unstable. This should make the connection between the quantum and
classical problems particularly clear. We hope to be able to describe
scattering resonances in terms of the one unstable periodic orbit.
-0.1truecm *G. Vattay and N.D. Whelan
0.3truecm*

**Determination of the Noise Level of Chaotic Time Series
0.01truecm**

We analyze analytically the influence of gaussian measurement noise on the
shape of the estimated corelation integral of an otherwise low dimensional
signal. If the data is embedded in a space of higher dimension than strictly
required to reconstruct the dynamics, the extra dimensions are dominated by the
noise which results in a certain shape of the correlation integral. For the
case that only Gaussian noise is present, this shape can be calculated
analytically as a function of the noise level. This leads to a method to
determine the noise level present in the data. Furthermore, the analytical
result shows that a noise level of more than 2% will obscure any possible
scaling of the correlation integral and thus makes it impossible to estimate
the correlation dimension.
-0.1truecm *T. Schreiber
0.3truecm*

**Tunnelling in Multidimensional Integrable Systems
0.01truecm**

We examine tunnelling from a point of view that makes it possible
to treat intrinsically multidimensional systems. The semiclassical theory
of tunnelling in dimension is very well understood, but much less
so in more dimensions. The -dimensional theory is interpreted in
terms of the construction of wavefunctions from complex Lagrangian
manifolds in complexified phase space. With this interpretation it is
possible to see how to generalize the calculation to more dimensions
in cases where a quantum state can be identified with a Lagrangian
manifold. The most obvious case is that of classically integrable systems
where the quantum states are derived from invariant tori. We illustrate
the theory by calculating the splittings in a -dimensional double
well potential for which the Hamiltonian is integrable, but
nonseparable. Because the system is nonseparable, it is not possible
to reduce it to -dimensional problems - which has been the only
way to solve tunnelling in integrable systems before.
-0.1truecm *S. Creagh
0.3truecm*

**Anomalous Tunnelling Splittings in Near-integrable Systems
0.01truecm**

We compute the tunnelling rates in a near-integrable system
under the assumption that the states can be associated with a complex
Lagrangian manifold. The energy levels, without tunnelling corrections,
have been well predicted by applying EBK quantization to KAM tori.
The tunnelling splittings are computed using the technique previously
used for exactly integrable systems. The splittings so obtained are found
to be significantly smaller that the actual splittings. This indicates
that the mechanism for tunnelling in even near-integrable systems is
significantly different from that in exactly integrable systems. This
is in line with previous calculations that have shown that the introduction
of nonintegrability significantly enhances tunnelling rates.
-0.1truecm *S. Creagh
0.3truecm*

**Long Time Behaviour in Quantum Maps
0.01truecm**

I have been using quantum maps to investigate the
long-time behaviour of semiclassical approximations.
The recent advances of reorganization of the trace formula
into Riemmann-zeta-function-like determinants has the potential
problem that they require pushing the semiclassical approximations
to times that are very large on classical scales, and it is not
obvious that the approximations are always valid for such long times.
I investigated the validity of the formalism by testing it for
quantizations of a perturbed Arnol'd cat map. For sufficiently
small perturbations, the map is hyperbolic and semiclassical
approximations hold well for long times. It is found that the
zeta-function formulation gives accurate results in this case,
including an explicit and simple realization of the `resurgence'
predicted by Berry, (*Les Houches Lecture Series 52*
(North-Holland, 1990)). As the perturbation
is increased, nonhyperbolic structures appear and the twists and folds
associated with these should make the semiclassical behaviour much worse.
It is indeed found that the zeta function deteriorates after the
transition to nonhyperbolic behaviour.
-0.1truecm *S. Creagh
0.3truecm*

**Chaos in Periodically Driven Gunn Diodes
0.01truecm**

We have applied numerical simulation to
explore the highly nonlinear dynamic phenomena that can arise in
Gunn diodes by interaction between the internally generated
domain mode and an external microwave signal. By adjusting the
time of domain formation and the speed of propagation, the
internal oscillation entrains with the external signal. This
produces a devil's staircase of frequency-locked solutions. At
higher microwave amplitudes, period-doubling and other forms of
mode-converting bifurcations can be seen. In this interval the
diode also exhibits spatio-temporal chaos. At still higher
microwave amplitudes, transitions to delayed, quenched and
limited space-charge accumulation modes take place.
-0.1truecm *R. Feldberg, C. Knudsen, E. Mosekilde, and J.S. Thomsen
0.3truecm*

**Bifurcation Analyses for a Forced Chemical Oscillator
0.01truecm**

We have performed a detailed numerical
bifurcation analysis of the forced Brusselator model, exposing
local and global bifurcation curves that constitute the internal
structure of the dominant Arnol'd tongues. The results are
presented as phase diagrams and one-parameter bifurcation
diagrams. Two theorems concerning the existence of global
bifurcations near generic codimension-2 bifurcation points are
stated and proved. It is argued that the results are generic to
a class of periodically forced self-oscillating systems.
-0.1truecm *C. Knudsen, (J. Sturis), and J. S. Thomsen
0.3truecm*

**Duffing's equation for a forced nonlinear oscillator
0.01truecm**

In collaboration with Y. Ueda, University of Kyoto, we have
investigated the solutions to Duffing's equation for a forced
nonlinear oscillator. This equation, which in many ways is
prototypical to modern nonlinear dynamics, is known to exhibit a
wealth of complex phenomena, including deterministic chaos,
coexisting regular and chaotic solutions, fractal basin
boundaries, cascades of period-doubling bifurcations, as well as
blue sky catastrophes. We have determined the phase diagram for
forcing amplitudes exceeding those hitherto studied. New regions
of chaotic behavior have been found. The basins of attraction
for two coexisting chaotic attractors have been determined, and
the chaotic solutions have been characterized in terms of their
Lyapunov exponents and fractal dimensions. We have also studied
an extension to Duffing's equation to three variables with
external forcing. Starting from a phase-space conserving chaos,
three prototypes of chaotic attractors with a dimension larger
than three have been obtained.
-0.1truecm *E. Mosekilde and J.S. Thomsen
0.3truecm*

**Phase Locking in Long Josephson Diodes
0.01truecm**

Phase-locking of fluxon motion to an external signal has been extensively studied using a simple particle model that captures many of the characteristics observed experimentally in long Josephson junctions. When the influence of the external ac drive manifests itself only through the boundary conditions, and when there is exactly one fluxon present in the junction at all times, the simple model reduces the perturbed sine-Gordon equation into an implicit two-dimensional map. Hence, it is possible to obtain analytical expressions for the existence of fixed points and their stability. When the fluxon is in a phase-locked state, it has been observed that the stability of the fixed point changes under parameter variations. This can result in period-doubling bifurcations and chaotic oscillations in the soliton propagation.

We have investigated the effect of thermal noise on the phase-locked state and shown that the noise measurements can be used to detect internal bifurcations in the system. This is particularly relevant to the prediction of the line width of the emitted signal from a phase-locked Josephson junction, since the line width increases with the variance of the time of flight for the fluxon.

The project has also led to the development of a numerical
scheme for solving a nonlinear partial integro-differential
equation with nonlocal time dependence. In particular we have
considered the dynamics of long Josephson diodes modeled by use
of the microscopic theory for tunneling between superconductors.
It is demonstrated that the detailed behavior of a solitonic
mode is different from the results of the conventional
sine-Gordon equation.
-0.1truecm *M. Samuelsen, (N. Grønbech-Jensen), and (Y.S. Kivskar)
0.3truecm*

**Deterministic Approach to Die Tossing
0.01truecm**

Usually, the motion of a die is so rapid that the human eye is unable to follow the trajectory in detail. Combined with the actual complexity of the motion, which appears to have defied mathematical treatment for centuries, this may have contributed to the widespread notion that a die toss is random. However, by virtue of the dissipation of energy, a die must always come to rest within a finite time. Therefore, a die toss cannot show the sensitive dependence on initial conditions found in chaotic systems or in systems with fractal basin boundaries. Nonetheless, die tossing has certain properties in common with these systems. In particular, the basins of attraction in the space of initial conditions show a certain self-similarity in structure.

In the present project we have investigated the structure of
these basins of attraction by means of a recursive
transformation involving the construction of flight and
collision maps. We have proposed quantitative measures for the
average size of the basins of attraction and for the mixing of
these basins, and we have applied these measures to determine
how the sensitivity to the initial conditions varies with the
energy and orientation of the die.
-0.1truecm *R. Feldberg, C. Knudsen, and E. Mosekilde
0.3truecm*

**Vehicle Dynamics
0.01truecm**

The dynamical behavior of railway vehicles is controlled by two nonlinear influences arising from the specific form of the wheel and rail profiles and from the dry friction in the contact between wheel and rail, respectively. Because of these nonlinearities, bifurcations, self-sustained oscillations and various forms of deterministic chaos may arise in the wheel/rail interaction. These phenomena can be observed in theoretical investigations and in experiments under laboratory conditions. In practical railway operation the deterministic response tends to be swamped by stochastic phenomena arising from various irregularities. However, the deterministic response may influence the wear of the wheels and the suspense system.

In the present project we have studied a dicone moving on a pair
of cylindrical rails. Accounting for nonlinear friction, the
equations of motion for this mechanical system consist of a set
of nonlinear differential-algebraic equations. We have
investigated the nonlinear dynamic behavior arising from these
equations in detail with the aid of special in-house software
and path-following algorithms.
-0.1truecm *C. Knudsen, R. Feldberg, (E. Slivsgaard), (H. True), and (M. Rose)
0.3truecm*

**Chaos in Economic Systems
0.01truecm**

In collaboration with the Sloan School of Management, MIT, we have performed a series of experiments with human decision making behavior in simulated corporate environments. Participants were asked to operate a simplified production-distribution chain to minimize costs. Performance was systematically suboptimal, however, and in many cases the subjects were unable to secure the stable operation of the system. As a result, large-scale oscillations and various forms of highly nonlinear dynamic phenomena were observed.

A model of the applied ordering policy has been proposed. Econometric estimates show that the model is an excellent representation of the actual decisions. With different parameters, computer simulations of the estimated order policy produce a great variety of complex dynamic behaviors. Analyses of the parameter space reveal an extremely complex structure having a fractal boundary between the stable and unstable solutions, and with fingers of periodic solutions penetrating deeply intro regions representing quasiperiodic and chaotic solutions. In certain parts of the parameter space, any neighborhood of a given solution contains a qualitatively different solution. Thus, changes on the margin can produce a completely different system behavior.

Our results provide direct experimental evidence that chaos can be produced by the decision making behavior of real people in simple managerial systems. The consequent implications for the ability of human subjects to cope with complex dynamical systems are explored.

We have also studied how mode-locking and other nonlinear
dynamic phenomena arise through the interaction of two
capital-producing sectors in a disaggregated economic long-wave
model. One sector might represent the construction of buildings
and infrastructure capital with long lifetimes while the other
represents production of machinery, transport means, etc., with
much shorter lifetimes.
-0.1truecm *(E.R. Larsen), Chr. Haxholdt, Chr. Kampmann, E. Mosekilde, and J.S. Thomsen
0.3truecm*

**Experimental Observation of the Lorentz Group in Nonlinear Circuits
0.01truecm**

We have measured Berry's geometrical phase of the Lorentz group in a pair
of connected driven dissipative electronic oscillators in the vicinity
of a period-doubling bifurcation. Surprisingly, under certain
conditions, a small signal transforms just as the Lorentz transformation
of a spinor, apart from some factors. By measuring along what
corresponds to a closed loop on a hyperboloid (the invariant surface of the
Lorentz group), we can determine Berry's phase of the Lorentz group.
-0.1truecm *H. Svensmark and P. Dimon
0.3truecm*

**Recursive definition of global cellular-automata mappings
0.01truecm**

We have proposed a recursive definition of the global mappings for
deterministic cellular automatas.
The method is based on a graphical representation of global
cellular-automata mappings.
The recursive definition defines the change of the global mapping
as the size of the automata is increased.
A proof of lattice size invariance of the global
mappings is derived from an approximation to the global mappings.
The recursive definitions are applied to calculate the
fractal dimension of the set of
reachable states and of the set of fixed points of cellular automata on an
infinite lattice.
-0.1truecm *R. Feldberg, C. Knudsen, and (S. Rasmussen)
*

*
*

Mon Feb 6 10:24:27 MET 1995