Professional Feature - Mario di Bernardo

By Invited Professional Contribution

I completed a 5-year Laurea (MSc) degree in Control Engineering at the University of Naples Federico II in Naples in 1994. While studying for my engineering degree I came across the popular science book 'Chaos' by James Gleick that had just been translated into Italian and got fascinated by the concept of chaos. I knew I wanted to know more, so while I was in my third year, I took the opportunity to travel overnight to Spoleto in Umbria from Naples to hear Paul Davies talking about his book 'The cosmic blueprint’ at a local science festival. Once back in Naples I decided I wanted somehow to combine my passion for control theory with my interest for chaotic systems. At the time I was taking a course in optimal control taught by Franco Garofalo, who was then to become my mentor and advisor, and asked him if I could work on a Master thesis project on the control of chaos. At around the same time, while a final year student, I heard about the Erasmus student exchange program by the EU and on the advice of my Italian mentor I found myself leaving Naples to work on my Master Thesis at the University of Bath in the UK under the mentorship of Prof Gene Ryan, a renown expert in adaptive and nonlinear control.

Once in Bath, I was captivated by the challenge of controlling chaotic systems through adaptive control. Under Gene's expert guidance, I authored my first paper. I vividly recall the thrill of receiving a letter announcing its acceptance by the International Journal of Bifurcations and Chaos, then led by Leon Chua. This marked the beginning of my adventure in science. Eager to delve deeper into control theory and chaos, I posted a message on a bulletin board in 1994—during the early days of the internet. A response came from Alan Champneys, then a young lecturer in applied nonlinear mathematics specializing in bifurcation theory. He invited me to visit the Applied Nonlinear Mathematics group, established a couple of years earlier by John Hogan after his move from Cambridge to Bristol with some of his students. This unforeseen chain of events led me to start my PhD at Bristol in 1995 under the supervision of David Stoten and Alan Champneys,. As Alan’s first PhD student, I found the group in Bristol to be an exceptional place—rich with interdisciplinary discussion, seminars, and fantastic colleagues.

Figure 1. A snapshot of the title and abstract of my first paper back in 1995.

We initially focused on model reference adaptive control of nonlinear chaotic systems. I remember sending that very first paper of mine through the post to Guanrong Chen, who was then teaching at the University of Houston. He responded with a lot of encouragement and valuable advice, and I have been honored to count him as a friend and collaborator ever since. However, I soon became intrigued by a problem suggested by my supervisor and Francesco Vasca who was one of his collaborators back in Italy—studying the nonlinear dynamics of DC/DC power converters. Together with Alan, we dived into the mathematics and bifurcations of piecewise smooth systems. It soon became apparent that what I was observing in DC-DC converters were instances of a distinct class of bifurcations unique to piecewise smooth systems. This realization spurred further research within the Bristol group, as we began to explore these new phenomena, recognizing that there was much more to discover.

In the mid-nineties, the UK was buzzing with scientific initiatives that aligned perfectly with my interests, such as the series of EPSRC Applied Nonlinear Mathematics schools for PhD students. In 1995, this school was held at the University of Surrey, where Hamill and Deane, pioneers in the experimental work on nonlinear dynamics of power converters, taught. Meeting them in person and discussing my work was an incredible opportunity. That same year marked my first presentation at the Snowbird conference. Just a few months into my PhD, I presented a poster on adaptive control of chaos. I vividly remember Jim Yorke stopping by to chat with me—what a moment!

Jim Yorke was also an invited lecturer at the EPSRC Spring School on Applied Nonlinear Mathematics organized by Paul Bresloff at the University of Loughborough in 1997, a pivotal year for me. That year, Alan Champneys was awarded a five-year EPSRC fellowship, and the department advertised a temporary lectureship to replace him. Surprisingly, I was offered the position while still completing my PhD, and I became a lecturer in Bristol. Later in 1997, I returned to Snowbird to present a poster on my work on the nonlinear dynamics of DC-DC converters, discovering that a similar problem was being independently studied by the Maryland group. This conference helped make our work in Bristol more widely known, and I began forming new collaborations with eminent mathematicians like Chris Budd in Bath and Arne Nordmark at KTH, to whom I am still indebted for their insights, technical skills, and intuition.

It was at the IEEE Conference on Decision and Control in San Diego in 1997, where I was presenting my work on adaptive control of chaotic systems, that I became interested in the dynamics of relay control systems. This new interest was sparked by a talk given by Karl Johansson, who was then a PhD student working with Karl J. Åström at Lund University. Later that year, I spent some time as a visiting student at Lund. While experimenting with relay systems, I observed a strange phenomenon where limit cycles interacted with the discontinuity boundary, giving rise to cycles characterized by segments of sliding motion. I discussed this phenomenon with Alan, who was then my supervisor, as well as Chris Budd and Arne Nordmark, and we gradually realized that we were observing a new class of bifurcations, which we named sliding bifurcations. (A few years later, in 1999, while visiting Mark Feigin in Nizhny Novgorod with John Hogan, I discovered that these phenomena had been observed earlier by Filippov and his students in some of their earlier work in Russian.) From that moment on, I decided that bifurcations of piecewise smooth dynamical systems were worth studying, and I became passionate about exploring further.

Figure 2. From left to right: Steve Coombes, Martin Homer, John Hogan and I at Snowbird 1997 (or 1999) .

Things moved quickly from then on, leading to my first successful EPSRC grant application. Soon after, I appointed Piotr Kowalczyk, my first PhD student (now a member of staff at the Wroclaw University of Science and Technology, in Poland), with whom we analyzed and classified different types of sliding bifurcations of limit cycles in piecewise smooth (PWS) systems. In 2001 I met Bernard Brogliato at Snowbird and we had the idea of putting together a large EU grant application on the analysis and control of piecewise smooth systems (SICONOS) that enabled me to hire more students and postdocs. Along with Alan, John, Piotr, Chris and other people in Bristol we formed a group working on the nonlinear dynamics and bifurcations of PWS systems. Work that was expounded in our now (in)famous 2008 book on piecewise smooth systems that was recently, kindly dubbed as the ‘Bristol book’ in a recent Editorial by Igor Belykh, Rachel Kuske, Maurizio Porfiri and David Simpson which introduced a Focus Issue on Non-smooth Dynamics in the international journal 'Chaos'.

Figure 3. Opening meeting of the European project SICONOS (Simulation and Control of Nonsmooth Systems) – from left to right: Arne Nordmark, Maurice Heemels, Michelle Schatzman, Pierre Alart, Bernard Brogliato, Christoph Glocker, Gerard Olivar, Alan Champneys and a much younger version of myself – Grenoble July 2002.

In 1999, I was promoted to a permanent lectureship at the University of Bristol. However, my plans took a sudden turn in 2001 when my father unexpectedly passed away, prompting me to consider moving back to Italy. At that time, Franco, my former mentor in Italy, informed me of a permanent position opening in Naples. I decided to apply and was subsequently appointed as an associate professor of Systems and Control there. This led to a joint appointment between Bristol and Naples, which turned out to be a fantastic opportunity. In Naples, I was part of the Department of Electrical Engineering and ICT, whereas in Bristol, I belonged to the Department of Engineering Mathematics. This dual role allowed me to work on both theoretical problems in applied mathematics and various applications in mechanics, electronics, and control engineering. Additionally, in Naples, I had the chance to attract outstanding research students and initiate a new research direction on the analysis and control of complex networks.

Moving into this area was another great decision I made, and I am thankful to my students and colleagues for the significant work we accomplished together, which has shaped my career ever since. I became a full professor in Bristol in 2006 and later in Naples in 2012. Thanks to my joint appointment, I formed a group across the two universities. Former members of this group are now colleagues at many other universities around the world. In 2007, the President of Italy awarded me the title of "Cavaliere" of the Order of Merit of the Italian Republic, explicitly mentioning the collaboration between Naples and Bristol in recognition of my scientific work. In 2012, I was elected Fellow of the IEEE for my contributions to the analysis and control of nonlinear systems and complex networks.

Figure 4. With current and former group members at a workshop on the island of Ischia in July 2019.

I currently lead a research group focused on nonlinear control and complex networks at the University of Naples Federico II. Our group engages in various interdisciplinary projects and collaborations across diverse areas, from synthetic biology to the coordination and control of large-scale multiagent systems. (For more information, please visit our website at Recently, I was appointed coordinator of a new research program and PhD course on Modeling and Engineering Risk and Complexity at the Scuola Superiore Meridionale, a new school for advanced studies established by the Italian Government in Naples (see for more info).

Figure 5. With Mattia Frasca and Erik Bollt discussing Science at SIAM Conference on Applied Dynamical Systems in Portland, May 2023.

I believe interdisciplinary work is fundamental for advancing scientific knowledge. My background in control engineering, combined with the knowledge I gained as a PhD student in Applied Nonlinear Mathematics, provided me with a tremendous opportunity to develop new ideas and conduct innovative research. As John Hogan often remarked during my time at Bristol, 'Good research stems from establishing a virtuous cycle between theory and applications; new theory suggesting new potential applications and new applications stimulating novel theoretical work.' I have adopted this philosophy as the mantra of my research career.

Also, it was at one of those EPSRC Spring Schools in Applied Nonlinear Mathematics, back when I was a PhD student in Bristol, that I had the honor of attending a lecture by the late Sir James Lighthill, who had once been the Lucasian Professor of Mathematics at Cambridge, among many other distinctions. He shared a piece of advice that I have never forgotten, which goes like this: 'Never spend more than five years working on the same topic; in five years, you either have become a leader in the field and need to move on, or you haven’t and so you must move on.' I believe this was some of the best advice I have ever received.

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