Introduction to Focus Issue: In Memory of John L. Hudson: Self-Organized Structures in Chemical Systems

John L. “Jack” Hudson, a native of Chicago, studied Chemical Engineering at the University of Illinois, in the class of 1959. One of his classmates was Roger Schmitz; they would later become faculty colleagues back at their alma mater and forge a truly creative collaboration. After obtaining an MSE degree at Princeton, Jack came back to Illinois and obtained his Ph.D. degree at Northwestern, working with an eminent advisor (George Bankoff); his thesis involved work on unsteady flow and heat transfer.^1,2

After a postdoctoral year in Grenoble, Jack joined the Faculty in Chemical Engineering at Illinois in 1963, where his research continued to involve flow and transport. Specifically, he worked on transport in rotating flows/discs due to his interest in the cooling of turbine blades. It was during this period that he started working on the fully nonlinear Navier–Stokes equations and the Boussinesq approximation, looking at start-up and transport in rotating discs^3,4 (including experimental work). Additionally, he also became interested in hydrodynamic stability,⁵ multicomponent reacting mixture transport, and chemical reactors^6,7—and so developed his interest in chemically reacting fluids. Jack also displayed an interest in environmental transport problems,⁸ which included a brief stint at the Illinois Environmental Protection Agency (1974–1975).

A turning point in his work came from the combination of a sabbatical in Belgium, where he was introduced to the ideas of the Prigogine school, and the interaction with his old classmate, Roger Schmitz. This introduced Jack to the area of complex nonlinear chemical kinetics and chemical chaos;^9–11 the observation of chaotic dynamics for the BZ reaction in a stirred tank reactor is a classic. This work opened new windows and connections to the world for Jack, which were enhanced by his move to the University of Virginia (UVA), where he stayed for the remainder of his career (ten of those years as Chairman of the Chemical Engineering Department). Jack frequently invited Otto Roessler to Virginia, and this marked the beginning of a truly prolific collaboration on chaotic dynamics. This was precisely the heyday of “nonlinear dynamics and chaos,” and their collaboration sometimes involved distinguished researchers from mathematics or physics, such as James Yorke or Doyne Farmer.^12–15 At the same time, Jack's interest in reacting system dynamics extended to both periodically forced and coupled reactor configurations,^16,17 marking also the beginning of a collaboration with one of the authors (I.G.K.). This is the period where Jack's interest in electrochemical reactions and their oscillations, patterns, and instabilities, began¹⁸—an interest that developed into a passion and remained an important driving force throughout the remainder of his career. A brief period of interest in nonlinear system identification from experimental time and image series (coincident with a “blossoming” of AI (Artificial Intelligence) research at the time) led to original work that created a connection with another one of the authors (K.K.),^19,20 who very soon also enjoyed collaborating with Jack on nonlinear phenomena in electrochemical systems.²¹ This was also the time when a long and truly fruitful connection with the Fritz-Haber-Institut (FHI) in Berlin started,^22,23 as is also emphasized in Gerhard Ertl's Foreword to this issue.

Jack's move away from reactor studies toward electrochemistry may have originated from his interest in mass transport (e.g., in configurations similar to a rotating disk²⁴), the presence of collaboration opportunities in the highly acclaimed Center of Electrochemical Science and Engineering at UVA (e.g., Mark Orazem, Robert Kelly, and John Scully), or the influence of his electrochemist neighbor, Glenn Stoner.

While early studies focused on chaotic electrodissolution,²⁵ later he used a camera to record spatiotemporal patterns in corrosion.²⁶ He became interested in electrode arrays²⁷ as a means to image the surface of the electrode by individually measuring the current distributions over the array with home-built multichannel ammeters. In part, this research was carried out with another co-author (I.Z.K.), which led to a decade long, productive, and stimulating collaboration. After discussions with Alexander Mikhailov and Yoshiki Kuramoto, Jack explored the emergence of synchronization patterns (e.g., through the Kuramoto transition²⁸) and different forms of dynamical clustering.²⁹ He collaborated with Jürgen Kurths on experimental confirmation of chaotic phase synchronization.³⁰ In parallel, the electrode arrays were also used for looking at metastable corrosion patterns.³¹ In collaboration with Harm Rotermund (in Ertl's Group at FHI) and his colleagues at the Fritz-Haber Institute and John Scully at UVA, they showed that, contrary to a previous assumption, the onset of corrosion is a collective phenomenon.³² As an engineer, Jack was interested in control applications. In collaboration with Hiroshi Kori, they developed a synchronization engineering technique that allowed generation of non-trivial synchronization patterns with delayed feedback.³³ From 2005, Jack began talking to his colleagues at UVA about applications of the synchronization techniques to biological systems. With Mark Quigg, they detected epileptic seizures,³⁴ with Jaideep Kapur they looked at controlling spiking patterns of neurons,³⁵ and with J. Randal Moorman they analyzed data from cardiorespiratory synchronization.³⁶ 

Jack's contribution to science and engineering was acknowledged by his receipt of the prestigious Wilhelm Award in Chemical Reaction Engineering of the American Institute of the Chemical Engineers in 1991 and the Alexander von Humboldt Senior Scientist Prize in 1989. In 2008, he was elected to the National Academy of Engineering. Considering his important role in the discovery of chaotic behavior, it was fitting that he served on the Editorial Board of Chaos. To support the Journal, he ensured that only very high quality manuscripts were submitted; for example, his work on dynamical differentiation of chaotic oscillators was the sixth highest cited contribution in the Journal that year, with a field-weighted citation impact of 14 (i.e., the paper was cited 14 times more than similar articles in the field).²⁹ Jack's impact on educating future engineers went beyond science. His former Ph.D. student, Mark Bassett, now Chairman and CEO of Hemlock Semiconductor Corporation said:³⁷ 

“Jack was obviously a great scientist, but what I remember most about Jack were the personal moments. He was my mentor during some very formative years and ignited my passion to work hard and to try to make a difference. I wouldn't be where I am today if it hadn't been for Jack's coaching. In addition, I got married and had two children while I was his student. He was like an honorary grandfather to my family, always there to help when we needed it.”

This issue highlights recent advances in nonlinear dynamics by Jack Hudson's collaborators, students, and friends. A few years back, Jack Hudson published a paper with Yoshiki Kuramoto and co-workers, in which it was shown that cluster formation was a general behavior for oscillators close to a Hopf bifurcation.³⁸ With some past experimental data from Jack's graduate student in the UVA lab, Kori et al.³⁹ report that negative coupling can generate a “noisy” synchronized state, which can be interpreted using an autocatalytic integrate-and-fire model.

The search for complex behavior was a passion of Jack's. While theoretical and experimental techniques for characterization of chaotic systems are now fairly well established, Jack's early theoretical studies with Otto Rössler and co-workers¹³ and later more experiment-focused studies with Christophe Letellier and co-workers⁴⁰ helped greatly in considering chaotic systems as “friends” rather than “foes.” Identical chaotic synchronization, studied extensively by Jack,²⁹ is still a challenging topic: from previous experimental data from Jack's lab, it is now reported that, in contrast to previous assumptions, synchronization is not a recommendable technique for model validation.⁴¹ The manuscript also highlights Jack's early career and interactions with Otto Rössler. Jack had an exciting collaboration with Arkady Pikovsky and co-workers on the characterization of synchronization of electrochemical oscillators.⁴² Jack's group explored intensively the effect of nonlinear coupling on pattern formation, e.g., through itinerant clustering.²⁸ Pikovsky and his colleagues now report that an oscillatory medium with nonlinear nonlocal coupling can generate breathing chimera through the instability of an inhomogeneous state.⁴³ 

Jack popularized the use of electrode arrays for studies of electrochemical reactions.⁴⁴ Because the current of each electrode in the array can be accurately measured with multichannel ammeters, the dynamics of the electrode arrays can be characterized in temporal domain, and the correlations between the current oscillations are indicative of the interactions among the electrodes. During his visits to Germany, Jack collaborated with Katharina Krischer and co-workers on the dynamics of electrocatalytic reactions.²² In the Gerhard Ertl Festschrift of the Journal of Physical Chemistry, Jack's group showed that CO electro-oxidation on two electrodes coupled through a large resistor exhibits a symmetry-breaking mechanism.⁴⁵ Bozdech et al.⁴⁶ now show that, in an array of microelectrodes in the bistable regime of the same electrochemical system, oscillations can occur in a galvanostatic configuration because of the presence of a globally conserved quantity. A co-author of this manuscript, Rolf Schuster, also collaborated with Jack and others on electrochemical nanomachining.⁴⁷ Results on the synchronization of electrode arrays are reported by Kiss et al.,⁴⁸ where it is shown that as coupling changes from global to local, spatially organized partially synchronized states can occur. The work utilizes the nickel electrodissolution system developed in Jack's lab,⁴⁹ which has served as a prototype for exploration of synchronization patterns in chemical systems.

The BZ reaction played an important role in Jack's early career as he showed the existence of chaos and entrainment to external periodic forcing in a continuous stirred-tank reactor (CSTR).^10,50 In the past, investigations of coupled BZ reactions have transformed the field of coupled oscillators with the use of BZ beads,⁵¹ microdroplets,⁵² and CSTRs.⁵³ Ken Showalter et al.⁵⁴ report that complex frequency cycling between excitatory and inhibitory states occurs with photochemically coupled BZ micro-oscillators. With pulse coupled CSTRs, Epstein and Horváth⁵⁵ report that the conventional “phase” description has to be extended with a frequency modulation term, and with positive or negative frequency modulations complex synchronization patterns can be obtained. Jack served as the co-chair of the “Gordon Research Conference on Oscillations and Dynamic Instabilities in Chemical Systems” (GRC) in 1994, a conference series that was started by Irv Epstein in 1982.

Before focusing on dynamics of discrete systems, Jack and co-workers investigated pattern formation in extended media, e.g., rotating waves in iron electrodissolution²⁶ or chemical wave propagation in a tube driven by an oscillatory chemical reaction.⁵⁶ During Jack's time in Germany, he visited Ronald Imbihl and discussed parallels between the nonlinear dynamics of solid-gas catalytic reactions and electrochemical processes, which are presented in Ref. 57. Imbihl et al.⁵⁸ found stripe and hole patterns during catalytic oxidation reactions with NO on Rh(111) with ultrathin vanadium oxide layers. Steinbock et al.⁵⁹ provide examples in which one-dimensional phase singularities, called filaments, may grow by the effect of pinning. The work of Kunihiko Kaneko on globally coupled logistic maps greatly inspired Jack's experiments on globally coupled oscillators.⁶⁰ Kaneko and Kohsokabe⁶¹ present the role of boundary induced pattern formation for the relevance of biological morphogenesis.

In his late career, Jack and his co-workers explored the applications of chemical nonlinear dynamics to biological systems, e.g., epileptic seizures,³⁴ controlling neuronal spikings,³⁵ and cardiorespiratory synchronization.³⁶ When Martin Falcke visited Jack in Charlottesville, VA, they started to investigate spiral breakup of calcium waves in frog oocytes with their co-workers.⁶² Falcke et al.⁶³ explores the correspondence of mathematical structures and experimental systems in cellular biophysics in Ca²⁺ signaling. Jack's group built an electrochemical system, which consisted of 64 individually addressable electrodes; in collaboration with Eckehard Schöll's group, they showed the existence of peculiar clusters, e.g., secondary cluster states with delay dependent phase lags.⁶⁴ Schöll et al.⁶⁵ show the prevalence of chimeras and partially synchronized states in networks with connectivity derived from magnetic resonance imaging.

Anecdotally, before Arthur Winfree took a position at the University of Arizona, he visited Jack in Charlottesville and encouraged him to use phase models to describe the spatiotemporal dynamics of chemical oscillators. About 20 years later, Jack and his co-workers⁶⁶ did accomplish this feat, and a successful collaboration started with the wider collaboration network of Yoshiki Kuramoto and Jack. For example, Jack, Daido, and co-workers⁶⁷ showed how to extract order parameters from global measurements (in a special issue dedicated to Kuramoto on the occasion of his retirement). Daido now reports super slow relaxation in phase oscillators with frustrated interactions.⁶⁸ The application of collective phase models for networks of oscillators provided a great challenge; Nakao et al.⁶⁹ proposed a phase reduction method for a network of coupled dynamical elements and investigate mutual synchronization between networks of excitable or oscillatory elements with random coupling using the method. We note that Jack's interest in collaboration with researchers in Japan started with extensive discussions with Hirokazu Fujisaka about calculating the Lyapunov exponent using a fitted next return map of a chaotic oscillator.¹¹ 

Jack always stressed the importance of finding new principles for descriptions of collective behavior of interacting units. He collaborated with Isao Tokuda and co-workers⁷⁰ on the extraction of phase models from experimental data on coupled chaotic oscillators. Tokuda with his collaborators⁷¹ find that a scale-free property appears in interacting turbulent fires. Jack collaborated with Punit Parmananda and co-workers^72,73 on noise-induced phenomena (e.g., oscillations and synchronization). Kumar and Parmananda⁷⁴ now use the mercury beating heart system to investigate the effect of coupling and common stochastic forcing on the collective behavior. Jack co-organized the GRC conference in 1994 with Ray Kapral. Kapral with his colleagues⁷⁵ numerically study orientation dynamics of pinned chemically propelled nanorotors in fluid and find that hydrodynamic interactions play a lesser role in the collective rotor dynamics.

I.Z.K. acknowledges support from the National Science Foundation (Grant No. CHE-1465013). H.K. acknowledges support from MEXT KAKENHI (Grant No. 15H05876).