## Biography

Alfred W. Hubler received his *diplom* in 1983 and Ph.D. in 1987, *summa cum laude*, from the Department of Physics, Technical University of Munich, Germany. After a postdoctoral fellowship at the University of Stuttgart, Germany, he came to the University of Illinois as a visiting assistant professor in 1989, and became assistant professor in 1990. Later that year, he also became the associate director of the Center for Complex Systems Research at Illinois, of which he is now the director. Professor Hubler served as a Toshiba Chair Professor at Keio University, Tokyo, in 1993-94.

Since beginning his thesis research, Professor Hubler has worked on nonlinear dynamics and has investigated a broad range of nonlinear phenomena. He is primarily a theorist, but he is also experienced in and capable of guiding both experimental and computational work. He has made solid contributions to the study of the chaotic dynamics in classical systems, both in idealized physical models and in engineering systems. He has been a pioneer in several important recent developments in nonlinear science research, including the control of chaos, the resonant coupling of nonlinear oscillators, and resonant stimulation and novel spectroscopies in nonlinear systems. Professor Hubler was among the very first to recognize that seemingly erratic, random motions associated with deterministic chaos could, in fact, be controlled, and that "chaotic" systems could be more "flexible" than systems undergoing more regular motion.

A skillful and committed teacher, Professor Hubler has also creatively applied the principles of nonlinear resonance to develop an intuitive, interactive web-based software package used to teach a variety of university science courses, at Illinois and around the world. Dubbed " CyberProf," the software analyzes student homework problems in real time and provides meaningful, intelligent, individualized feedback to each student.

## Research Statement

Hubler is an expert in modeling and controlling the dynamics of open dissipative systems, in particular nonlinear and chaotic oscillations and fractal particle agglomeration processes. He has been a pioneer in several important recent developments in complex systems research, including medium-term prediction of chaos with ensemble predictors, modeling adaptation to the edge of chaos, resonance spectroscopy with chaotic forcing functions, and modeling and prediction of the growth of fractal networks with graph theoretical methods. Hubler's goal is the development of a toolbox of methods and paradigms to predict the dynamics and the response of complex systems.

(1) Modeling and prediction of the growth of fractal networks with graph theoretical methods

Context: Hubler carried out the first systematic experimental studies on the growth and the dynamics of fractal particle agglomerates (Dueweke, Dierker & HÃ¼bler, 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. Hubler, 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 suggests 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 sensitively on the initial condition.

Current and future work: Currently there is no model known which 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 this 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 & HÃ¼bler, Phys. Rev. E 59, 3165(1999)).

Funding: DARPA THERA-PI, $1,300,000, 2010-12, Materials Computation Center, A. Hubler, PI

(2) Medium-term prediction of chaos with ensemble predictors

Context: Hubler introduced the dynamical reconstruction methods for modeling and prediction of experimental chaotic data sets (Cremers & HÃ¼bler, 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 & HÃ¼bler, Phys. Rev. A 42, 5827(1990)).

Most significant recent work: C. Strelioff, A. Hubler, 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: Hublerâ€™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 Eulerâ€™s method, the dynamics can be chaotic due to numerical instabilities. Preliminary studies suggest that this type of chaos can be found in the 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. Hubler co-PI

(3) Modeling adaptation to the edge of chaos

Context: Hubler was the very 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. HÃ¼bler, 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. Hubler, 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 â€œadaptation to the edge of chaosâ€, 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 & HÃ¼bler, Chaos 15, 033902 (2005)) and seem to have a wide range of applicability (Melby, Weber & HÃ¼bler, 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 & HÃ¼bler, 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. Hubler PI.

(4) Resonance spectroscopy with chaotic forcing functions

Context: Another groundbreaking discovery was Hublerâ€™s observation that nonlinear dynamical systems react most sensitive to a forcing function that complements its 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: V. Gintautas, G. Foster, and A. W. Hubler, Resonant forcing of select degrees of freedom of multidimensional chaotic map dynamics. J. Stat. Phys. 130 (3), 617-629 (2008); V. Gintautas and A. Hubler, Resonant forcing of nonlinear systems of differential equations. Chaos 18, 033118 (2008).
These papers explore 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 resonances 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. Hubler co-PI