The Center for Complex Systems Research (CCSR) studies experimentally and theoretically systems that display adaptive, self-organizing behavior and systems that are usually characterized by a large throughput, such as patterns in bio-chemical system and morphogenesis, the growth of fractal ramified transportation networks including river networks and lightning strikes, turbulent flow, and the growth and decay of social organizations. To describe these complex systems, we develop models and techniques drawn from nonlinear dynamics and chaos, neural nets, cellular automata, artificial life, and genetic algorithms. We derive overarching principles from the models, such as adaptation to the edge of chaos, nonlinear resonances, and the principle of minimum resistance. We use these principles and models to control and predict nonlinear oscillations, chaos, and network growth, in particular, phase transitions and catastrophes, including the transition from peace to war. CCSR faculty operates an Analytical Chemistry lab for oscillating chemical reactions and a Physics lab with a wind tunnel for turbulence experiments, and several experimental setups on pattern formation and dendritic growth.

Research Interests:
Davis's primary research activity is in developing strategies for the implementation of decision-making and control hierarchies in manufacturing systems. Principal research efforts are the development of new processes for hierarchical programming and real-time decision-making. With respect to the latter topic, he is addressing the concerns that arise in both formulating and implementing a decision in an uncertain planning environment. Using newly developed decomposition strategies, the interaction among decisions is being investigated within a newly initiated Flexible Automation Control Laboratory and Manufacturing Systems Laboratory. In collaboration with the National Institute of Standards and Technology, the Department of Defense and several corporate sponsors; these developments are being applied to computer-integrated manufacturing.

Research Interests:

1. Prediction of chaos
Context: Hübler introduced the dynamical reconstruction methods for modeling and prediction of experimental chaotic data sets (Cremers & Hübler, Z. Naturforsch. 42a, 797 (1987)). In particular his work on reconstructing equations of motions from experimental data with unobserved variables is often cited and widely used (Breeden, Dinkelacker & Hübler, Phys. Rev. A 42, 5827 (1990)).
Most significant recent work: C. Strelioff, A. Hübler, Medium Term Prediction of Chaos, Phys. Rev. Lett. 96, 044101 (2006).
In this paper ensemble predictors are used to determine the trajectory of the most likely state of a chaotic system with singular points, such as the logistic map.

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 high-dimensional time continuous systems with low-dimensional 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 time-continuous version. We will study the role of noise in iterated functions and cellular automata, in particular noise-induced phase transitions.
Funding: NSF DMS 0325939 ITR, $3,991,798, 2003-08, Materials Computation Center, A. Hübler co-PI

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 Self-Adjusting Logistic Map, Phys. Rev. Lett. 84, 5991(2000).
This paper explores systems, where chaos is suppressed due to a low-pass 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 self-adjusting 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, non-stationary systems, and high-dimensional systems.
Funding: NSF 0022948, $167,908, 2002-05, Experimental Study of Adaptation to the Edge of Chaos and Critical Scaling in the Self-adjusting Peroxidase-Oxidase 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 multi-dimensional 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 bi-directional instantaneous coupling with a real time implementation of a model system on a computer.
Funding: NSF DMS 0325939 ITR, $3,991,798, 2003-08, Materials Computation Center, A. Hübler co-PI

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, 2003-08, Materials Computation Center, A. Hübler co-PI

5. Resonances in the student-computer 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 web-based tutoring software with this specific goal in mind (Hübler & Assad, CyberProf: An Intelligent Human-Computer Interface for Asynchronous Wide Area Training and Teaching, World Wide Web Journal 4, 231(1996); Raineri, Hübler & Mertens, CyberprofTM: An Intelligent Human-Computer Interface for Interactive Instruction on the World Wide Web, JALN 1, 20 (1997)).
Most significant recent work: Hübler & Martinez. A complex systems perspective to computer-assisted learning. In T-W Chan, A. Collins and J. Lin (Eds.), Global Education on the Net (Springer: New York, NY, 1998), vol. 1, pp 5-12.
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 Field-Specific 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 student-computer interaction will be tested.
Funding: NSF DGE 0338215, $1,836,231, 2004-09, TRACK 2, GK-12: EdGrid Graduate Fellows Program, A. Hübler co-PI

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, 976-985 (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 computer-assisted 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 389-394.
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 computer-assisted reasoning software will be expanded and tested both with middle school students and university students. The computer-assisted 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 computer-assisted reasoning software.
Funding: NSF DGE 0338215, $1,836,231, 2004-09, TRACK 2, GK-12: EdGrid Graduate Fellows Program, A. Hübler co-PI.

Research Interests:
Real-world systems are characterized by their composition and by their dynamical interactions. We characterize, model, and extend nonlinear systems in the realms of chemical biology and materials.
Oscillatory Enzyme Kinetics: Our first area of research is oscillatory oxido-reductive biochemistry as catalyzed by peroxidases. We have developed electrochemical sensors for use with an array of spectrophotometric, magnetic resonance, and chromatographic instruments. We elucidated the chemical details of the NADH/O2 oscillatory reaction as catalyzed by horseradish peroxidase and now study the dynamics of reactions catalyzed by mammalian myeloperoxidase. We pursue methods of modulating enzyme activity to optimize the bioactivity of reaction cofactors.

Enzyme Kinetics in Microliter Volumes: We have begun to study enzyme kinetics in microliter drops. Ultrasound ensures rapid mixing of the drop, and the absence of wall effects promises to simplify both the complexity of the observedchemistry and the analysis of rate data.
Sensor Arrays for Chemical Kinetics: Efficient study of chemical kinetics requires that many species simultaneously be identified and quantified. Development of multicomponent sensors with rapid response, drift compensation, and some ability to sense when novel interferences are occurring requires microfabriction together with electronics integration. Complementing electrochemical and optical sensors are more powerful than either in isolation. A system on which we are concentrating is the oxidative stress on the inner ear correlated with noise-induced hearing loss. We collaborate with biophysicists and audiological biochemists in the study of these temporally and structurally complicated systems.
Flame Diagnostics: Finally, we continue to work on flame diagnostics, closely related to the plasma diagnostics that were the major focus of our group prior to 1994. Our off-campus collaborators use multi-kilowatt combustors to remove metal ions from waste streams.

Selected publications:

Research Interests:
Technology assessment and international security: dual-use of science and technology; space policy, command and control; missile proliferation, missile defense and missile control; proliferation and arms control of weapons of mass destruction.
Environment and sustainable development: integrated assessment in natural resource management (energy, climate, biodiversity, fishery, waste); global risks, environmental conflicts and energy security.
Physics, mathematical modelling, computer simulation: non-linear dynamics and complex systems analysis; modelling of conflict and cooperation; decisionmaking, game theory, agent-based modelling, complex dynamic networks.

Selected publications: