The
paper shows that only for the first few time steps, the trajectory of
the most likely state stays near the images of the most likely initial
state. Then the trajectory jumps to a new dynamics which originates at
one of the singular points. Several such jumps can occur before the
trajectory reaches the limiting state. The paper presents the first
evidence for these jumps and shows, that they can be predicted. This
makes it possible to predict the behavior of the chaotic system for
much longer time spans, than with any other known prediction
method. The paper also suggests that the trajectory of the most likely
state can be much more complex and qualitatively different from the
dynamics of the images of the most likely initial state.
Current and future work: Hübler's
group tries to observe the jumps in the trajectory experimentally. A
mechanical system with chaotic motion has been completed and
systematic studies will start this summer. Future work will include
prediction of highdimensional time continuous systems with
lowdimensional time discrete models. For instance, if the equation of
motion for the center of mass of a physical pendulum is discretized
with Euler's method, the dynamics can be chaotic due to numerical
instabilities. Preliminary studies suggest that this type of chaos can
be found in an experiment, if the time step of the discretization
matches the period of the leading vibrational mode of the object. Thus,
the discrete version of the equation of motion for the center of mass
describes more features of the dynamics than the timecontinuous
version. We will study the role of noise in iterated functions and
cellular automata, in particular noiseinduced phase transitions.
Funding: NSF DMS 0325939 ITR, $3,991,798,
200308, Materials Computation Center, A. Hübler coPI
2. Control of chaos
Context: Hübler was the first to recognize that seemingly
erratic, random motions associated with deterministic chaos could, in
fact, be controlled, and that `chaotic' systems could be steered with
less effort than systems undergoing more regular motion
(A. Hübler, Adaptive Control of Chaotic Systems, Helv. Phys. Acta
62, 343 (1989)). His work has inspired a large number of papers which
expanded his methodology or present alternative approaches.
Most significant recent work: P. Melby, J. Kaidel, N. Weber, A.
Hübler, Adaptation to the Edge of Chaos in the SelfAdjusting
Logistic Map, Phys. Rev. Lett. 84, 5991(2000).
This paper explores systems, where chaos is suppressed due to a
lowpass filtered feedback. If a dynamical system has a range of
parameter values with periodic dynamics and a range of parameter values
with mostly chaotic dynamics, a selfadjusting system will evolve
toward a narrow parameter range near the boundary between the two
regimes. The paper offers the first undisputed explanation for a
phenomenon `adaptation to the edge of chaos', which suggest that
adaptive systems are much more likely to be found in a state with
complex periodic dynamics or weakly chaotic motion than in a highly
chaotic state or a simple periodic state. The predictions of the paper
were confirmed experimentally (Melby, Weber & Hübler, Chaos
15, 033902 (2005)) and seem to have a wide range of applicability
(Melby, Weber & Hübler, Phys. Fluct. and Noise Lett. 2,
L285 (2002)). More recently it has been shown that there is a conserved
quantity which helps to simplify models of the adaptation process
(Baym & Hübler, accepted by Phys. Rev. E).
Current and future work: The pattern of capillary waves on the
surface of a vibrated water droplet depends on the shape of the
droplet. Preliminary experiments show that if the droplet has an
initial shape where the wave pattern is chaotic, the shape changes
until the wave dynamics becomes periodic. This suggests that adaptation
to the edge of chaos can be observed in systems with wave chaos and
quantum chaos. Future studies will include quantitative models for
adaptation to the edge of chaos, in spatially extended systems,
nonstationary systems, and highdimensional systems.
Funding: NSF 0022948, $167,908, 200205, Experimental Study of
Adaptation to the Edge of Chaos and Critical Scaling in the
Selfadjusting PeroxidaseOxidase Reaction, A. Hübler PI.
3. Resonance spectroscopy with chaotic forcing
functions
Context: Another groundbreaking discovery was Hübler's
observation that nonlinear dynamical systems react most sensitive to a
forcing function that complements it's natural motion. In a sequence
of papers he showed that such aperiodic forcing functions have a
perfect impedance match. He developed a theory of nonlinear resonance
spectroscopy based on aperiodic forcing functions. This may lead to a
new generation of spectroscopic instruments with an unusually large
signal to noise ratio.
Most significant recent work: G. Foster, A. Hübler,
K. Dahmen, Resonance Spectroscopy with Chaotic Forcing Functions,
submitted to Phys. Rev. Lett. 2006.
This paper explores the final response of multidimensional chaotic
map dynamics to additive aperiodic forcing functions with equal
variance. The forcing function, which produces the largest response is
called resonant forcing function. It is shown that the product of the
resonant forcing function and the displacement dynamics of neighboring
trajectories is a conserved quantity at each time step. Hence, the
optimal forcing function complements the system dynamics. If the
optimal forcing function is computed with a set of models, the system
response reaches an absolute maximum if the model is correct. This can
be used for system identification. The resonance curve of the chaotic
map dynamics has an absolute maximum, if the model parameters are
correct.
Current and future work: The same approach will be used to
determine resonant forcing function of time continuous systems, such as
the Lorenz attractor. For physical systems, such as a chaotic coupled
pendulum dynamics, we anticipate that the conserved quantity has a
physical meaning. It is probably equal or related to the reaction
power. In addition we will explore of coupled chaotic oscillators and
resonances of real nonlinear oscillators with a bidirectional
instantaneous coupling with a real time implementation of a model
system on a computer.
Funding: NSF DMS 0325939 ITR, $3,991,798, 200308, Materials
Computation Center, A. Hübler coPI
4. Modeling and prediction of the growth of
fractal networks with graph theoretical methods
Context: Hübler carried out the first systematic
experimental studies on the growth and the dynamics of fractal
particle agglomerates (Dueweke, Dierker & Hübler,
Phys. Rev. E 54, 496 (1996)). His graph theoretical network analysis
illustrates deficiencies in earlier fractal growth models and this may
lead to the first detailed quantitative models of fractal growth.
Most significant recent work: Joseph K. Jun, A. Hübler,
Formation and structure of ramified charge transportation networks in
an electromechanical system, PNAS 102, 536 (2005).
This is a systematic study of the growth of fractal particle
agglomerates in an electric field. The study suggest that some
observables, such as the number of endpoints, the number of branching
points, the limiting resistance, and the fractal dimension are highly
reproducible, whereas other quantities, such as the number of trees
depend sensitive on the initial condition.
Current and future work: Currently there is no model known
that describes the dynamics of the observables during the growth of
the fractal network. However some preliminary studies suggest that a
minimum spanning tree growth model might be able to reproduce the
dynamics of all observables. In addition to such graph theoretical
models the underlying physical equations will be used to describe
certain aspects of the pattern formation, such as the opening of closed
loops or the formation of linear strands at the initial stages of the
growth process. Eventually we may be able to merge the graph
theoretical and the physical model. This system has a large number of
stable attractors. We will explore regularities in the attractor
structure. In addition we will try to carry out the experiment on a
microscopic scale, and explore hardware implementations of neural nets
on an atomic scale (Sperl, Chang, Weber & Hübler,
Phys. Rev. E 59, 3165 (1999)).
Funding: NSF DMS 0325939 ITR, $3,991,798, 200308, Materials
Computation Center, A. Hübler coPI
5. Resonances in the studentcomputer
interaction
Context: Based on the idea that the student's attention to a
virtual tutor would be particularly high if the response time and the
preferences of the virtual tutor match those of the student,
Hübler designed webbased tutoring software with this specific
goal in mind (Hübler & Assad, CyberProf: An Intelligent
HumanComputer Interface for Asynchronous Wide Area Training and
Teaching, World Wide Web Journal 4, 231(1996); Raineri, Hübler
& Mertens, CyberprofTM: An Intelligent HumanComputer Interface
for Interactive Instruction on the World Wide Web, JALN 1, 20 (1997)).
Most significant recent work: Hübler & Martinez. A
complex systems perspective to computerassisted learning. In TW
Chan, A. Collins and J. Lin (Eds.), Global Education on the Net
(Springer: New York, NY, 1998), vol. 1, pp 512.
Web technology has made it possible for the Internet to penetrate a
large proportion of American society. The educational sector has lagged
behind the progress curve, however. This lag in the application of
technology in education is due to not only financial reasons but also
because education is more than information delivery and simulation. To
teach a student via the Web, at least two components are required: a
human computer interface that resonates with the student and artificial
intelligence that understands specific areas of knowledge and applies
appropriate responses. It is in this content that we study an
asynchronous learning environment, CyberprofTM, where the authors apply
several fundamental complex systems paradigms to human computer
interaction: nonlinear resonances applied in a Digital Mirror,
dynamical reconstruction applied in a FieldSpecific Tutor and control
of chaos applied in a Five Senses Human Computer Interface. To measure
the effectiveness of the new learning environment the authors suggest
using a Differential Concept Analysis (DCA). In DCA the authors propose
to monitor student's actions step by step, determine which concepts the
students are using, in which sequence the concepts are used, and if the
students are able to substitute and alter the sequence of the applied
concepts.
Current and future work: Hübler's group tries to collect
data for Differential Concept Analysis from a large student population
in introductory University courses. Another goal is to introduce an
English language parser so that the virtual tutor can evaluate student
input to free response questions and answer questions by the students.
In collaboration with the UIUC College of Education the effectiveness
of resonances in the studentcomputer interaction will be tested.
Funding: NSF DGE 0338215, $1,836,231, 200409, TRACK 2, GK12:
EdGrid Graduate Fellows Program, A. Hübler coPI
6. Computer Assisted Reasoning Software
Context: Hübler was one of the first researchers to
specify a precise definition of a concept (Durak & Hübler,
Scaling of Knowledge in Random Conceptual Networks, Lecture Notes in
Computer Science 2074, 976985 (2001)). He used this definition to
conceptualize courseware systematically in Science and Mathematics.
Once the courseware is conceptualized the courseware and student input
can be parsed by educational software. This allows computerassisted
tutoring on a high level of abstraction.
Most significant recent work: Hübler, Vlasic, Stiegler,
Bievenue, & Raineri. Interactive middle school courseware on
abstract reasoning skills. In C. Crawford et al. (Eds.), Proceedings of
Society for Information Technology and Teacher Education International
Conference 2006 (AACE: Chesapeake, VA, 2006) pp 389394.
Quantitative reasoning skills are a fundamental tool in many fields,
ranging from Mathematics to Engineering and from Business to Rhetoric.
However, quantitative reasoning is almost never taught as a course,
except in the context of other disciplines, such as Mathematics or
Physics. This paper introduces basic elements of a course in reasoning,
such as the definition of a concept and the definition of a strategy
and studies the response of the students. These definitions are applied
to algebraic proofing. Algebraic concepts are conceptualized, i.e. each
concept is named, its range of applicability is specified and each
concept is illustrated with typical examples. Strategies for proofing
are conceptualized as well. The paper indicates that a diverse
population of female middle school students readily accepts this
approach and achieves proofing skills on a level which is comparable to
university freshmen.
Current and future work: The courseware covered by the
computerassisted reasoning software will be expanded and tested both
with middle school students and university students. The
computerassisted reasoning software will be used to collect data
about the reasoning strategies of the students. These strategies
will be modeled, published, and used to improve the computerassisted
reasoning software.
Funding: NSF DGE 0338215, $1,836,231, 200409, TRACK 2, GK12:
EdGrid Graduate Fellows Program, A. Hübler coPI.

Selected publications:
Eisenack, K,; Scheffran, J.; Kropp, J. Viability analysis of
management frameworks for fisheries. Environmental Modeling and
Assessment, 2006, 11: 6979. Elhefnawy, N. Societal Complexity and
Diminishing Returns in Security. International Security, Vol. 29, No. 1
(Summer 2004); pp. 152174. 
Ipsen, D.; Rösch, R.; Scheffran, J. Cooperation in global
climate policy: potentialities and limitations. Energy Policy 29/4
(Jan. 2001); pp. 315326. 
Jathe, M.; Scheffran, J. Modelling international security
and stability in a complex world. In: Chaos and Complexity. Berge, P.;
Conte, R.; Dubois, M.; Van Thran Thanh, J., Eds.; Gif sur Yvette:
Editions Frontieres 1995; pp. 331332. 
Jathe, M.; Krabs, W.; Scheffran, J. Control and game
theoretical treatment of a costsecurity model for disarmament.
Mathematical Methods in the Applied Sciences 20; pp. 653666. 
Krabs, W.; Pickl, S.; Scheffran, J. An nperson game under
linear side conditions. In: Optimization, Dynamics and Economic
Analysis. 
Scheffran, J. Strategic Defense,
Disarmament, and Stability
 Modelling Arms Race Phenomena with Security and Costs under Political
and Technical Uncertainties. Doctoral Thesis, Marburg: IAFA No. 9,
1989. 
Scheffran, J. The dynamic interaction between economy and
ecology cooperation, stability and sustainability for a dynamicgame
model of resource conflicts. Mathematics and Computers in Simulation 53
(2000); pp. 371380. 
Scheffran, J. Stability and control of valuecost dynamic
games. Central European Journal of Operations Research 9(7) (Nov.
2001); pp. 197225. 
Scheffran, J. Conflict and Cooperation in Energy and Climate
Change. The Framework of a Dynamic Game of PowerValue Interaction. In:
Yearbook New Political Economy 20. Holler et.al., ed., Mohr Siebeck,
2002; pp. 229254. 
Scheffran, J.; Calculated Security? Mathematical Modelling
of Conflict and Cooperation. In: Mathematics and War. BoossBavnbek,
B.; Hoyrup, J., eds.; Birkhäuser, 2003; pp. 390412. 
Scheffran, J. Interaction in Climate Games: The Case of
Emissions Trading. In: Entscheidungstheorie und praxis in
industrieller Produktion und Umweltforschung. Geldermann, J.; Treitz,
M., eds.; Aachen: Shaker; 2004, pp. 118. 
Scheffran, J. The Formation of Adaptive Coalitions. In
Advances in Dynamic Games. In: A. Haurie, S. M., L.A. Petrosjan, T.E.S.
Raghavan, Ed.; Birkhäuser, 2006, p 163178. 
