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Steady Fast Kinematic Dynamos: The Topological Entropy Bound
and the Effect of Magnetic Field Diffusion
truecm**

We utilize recently developed periodic orbit techniques
to establish that the topological entropy bounds the dynamo growth rate,
estimate the magnetic field diffusion induced supressesion of
the high frequency eigenmodes,
and give periodic orbit formulas for the cancellation
exponent and the generalized dimension , quantities
characterizing the singular nature of dynamo eigenfuctions.
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P. Cvitanovic and (E. Ott)
truecm*

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Sum Rules for Periodic Orbit Theory
truecm**

A family of exact sum rules for stabilities of
periodic points of **d-dimensional polynomial mappings is
derived and checked numerically. Associated Fredholm
determinants are of particularly simple
polynomial form. Possible applications to periodic orbit calculations
for continuous time flows are discussed.
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P. Cvitanovic, K.T. Hansen and G. Vattay
truecm*

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Trace Formula for Dynamic Systems in the Presence of Noise
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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.
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P. Cvitanovic, (R. Mainieri) and G. Vattay
truecm*

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Symbolic Dynamics and Markov Partitions for the Stadium Billiard
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We investigate the Bunimovich stadium dynamics and show that in the
limit of infinitely long stadium the symbolic dynamics is a subshift
of finite type. For a stadium of finite length the Markov partitions
are infinite, but the inadmissible symbol sequences can be determined
exactly by means of the appropriate pruning front. We outline a
construction of a sequence of finite Markov graph approximations by
means of approximate pruning fronts with finite numbers of steps.
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P. Cvitanovic and (K.T. Hansen)
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Topology and Bifurcation Structure of Maps of Hénon type
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We construct a series of n-modal approximations to
maps of the Hénon type and
apply the corresponding symbolic dynamics to
description of the admissible orbits and
possible bifurcation structures for such maps.
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P. Cvitanovic and (K.T. Hansen)
truecm*

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

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Deterministic Approach to Die Tossing
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By means of a recursive transformation involving the
construction of flight and collision maps we have investigated
the structure of the basins of attraction for a die toss. 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. A similar analysis has been performed for the Galton
Apparatus (or so-called Probability machine). We have shown that
the outcome distribution for this machine in general is not
binominal. It may be approximated by a Gaussian distribution but
the standard deviation in this distribution will depend on the
coefficient of restitution in the ball-pin collisions. For
certain parameters the outcome distribution may be double humped.
truecm
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M. Christensen, L.-U. W. Hansen and E. Mosekilde
truecm*

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Bifurcation Analyses
truecm**

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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. We have also investigated the
solutions to Duffing's equation for a forced nonlinear
oscillator. This equation 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. Finally, we have
studied an extension of 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.
truecm
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C. Knudsen, E. Mosekilde, (J. Sturis) and J.S. Thomsen
truecm*

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Phase Locking in Long Josephson Diodes
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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. 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.
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(N. Grønbech-Jensen), (Y.S. Kivskar) and M. Samuelsen
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Vehicle Dynamics
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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. Poincare sections and return maps
have been applied to describe the structure of the periodic,
quasiperiodic and chaotic motions. The analyses show that part
of the chaotic behavior can be fully understood as a nonlinear
iterative process.
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R. Feldberg, C. Knudsen and (H. True)
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Fri Mar 29 00:13:44 MET 1996