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Celso Grebogi: The chaos tamer

Physicist from Curitiba creates strategies to intervene in complex systems such as the climate and ecological networks

Adriana Meneguzzo Scaglioni

As a child, Celso Grebogi would wake up at 5 a.m. to help his father deliver bread across southern Curitiba before school. He later took a technical course in high school, did a degree in chemical engineering, left the Paraná state capital to pursue a master’s degree in Rio de Janeiro, and then spent 27 years living in the USA. In 1981, having been hired as a professor at the University of Maryland, he began studying the nascent field of nonlinear systems — mathematical structures with several paths and multiple final scenarios, defined based on initial conditions — which includes chaos theory, in which he has achieved international recognition.

In March 1990, Grebogi and two other Maryland professors, Edward Ott and James Yorke, presented the OGY method (named after the initials of their surnames) in the scientific journal Physical Review Letters. It encompasses strategies for controlling — even partially — chaotic systems, which were later verified in experiments and used in telecommunications and climate predictions (see Pesquisa FAPESP issues 65 and 107).

Age 75
University of Aberdeen, Scotland
Area of expertise
Plasma physics, complex systems, chaos theory
Bachelor’s degree in chemical engineering from UFPR (1970), master’s degree in physics from PUC-RJ and the University of Maryland, USA (1975), PhD in physics from the University of Maryland (1978)

Grebogi is director of the Institute for Complex Systems and Mathematical Biology at King’s College, University of Aberdeen, Scotland, and a visiting professor at two universities in Xi’an, China. He has published 431 scientific articles and is currently working on ecological networks, predicting the effect of migrating animal populations and interactions between the human brain and computers. He is married to São Paulo native Adriana Meneguzzo Scaglioni, with whom he has a 22-year-old son called Mateus, and gave this interview via videoconference from his home in Aberdeen at the beginning of January.

In 2001, you moved back to Brazil from the USA, spent four years at the University of São Paulo [USP], and then moved again to Scotland. Did you find it difficult to settle?
I left the US in 2001, a year after my son was born. I didn’t want him to be educated there, but we actually returned to Brazil for another reason. I came from an extremely poor family and everything I achieved was thanks to Brazil. I didn’t pay for university, I studied at PUC-RJ [Pontifical Catholic University of Rio de Janeiro] with a scholarship from the CNPq [Brazilian National Council for Scientific and Technological Development], and I wanted to repay my country. I taught large classes of 900 students at USP’s Polytechnic School and classes of 150 to 200 students at the Physics Institute. But things were complicated at the university at that time. It was very difficult to work. There were always protests, loud-speakers, and people demanding salary increases. They even closed the buildings, so I couldn’t go to my office. It was unpleasant. Then I received an invitation from the vice chancellor of the University of Aberdeen, asking me to go to Scotland. I was writing a response saying “I don’t want to leave Brazil, I came here to stay,” when my wife, Adriana, said “Celso, you’re not happy here, don’t you want to at least take a look at the offer?” So I read the offer and I accepted it. It all happened very fast. I came to Scotland on June 8, 2005, to sort everything out. Then I went back to Brazil, left USP, and we made the move. I started working here in Scotland on July 31. It was only sometime later that we returned to São Paulo to sell our apartment. I had been given an important position: the Sixth Century Chair, named after the age of the university, which was founded in 1495. It is one of the oldest in Scotland — the oldest is St. Andrews, which was founded in 1413, and is about 150 kilometers from Aberdeen.

Were you able to achieve your goals in Aberdeen?
As founder and director of the Institute for Complex Systems and Mathematical Biology, I oversaw — with support from the administration — 14 professors at various stages in their careers and brought in visiting researchers, graduate students, and postdoctoral fellows. The institute employs an average of 40 to 50 people. I stick more to management duties, leaving the staff free to do what they need to do, but we have weekly meetings where I monitor their work and help where I can. Most of my time is spent on research. Because I had already been teaching for 40 years when I arrived here, I asked if my contract could stipulate that I didn’t need to teach or take care of courses. My participation in teaching is now limited to interacting with graduate and postdoctoral students and colleagues. We have done a lot of good work.

We want to understand natural phenomena, make predictions, and improve the public’s standard of living

Is there anything you would highlight?
One very interesting project looked at how protein production is controlled in cells. Proteins are produced based on the information contained in DNA, which is stored in cell nuclei. A molecule called messenger RNA copies the section of DNA that codes for a specific protein and takes this copy to the periphery of the cell, where the information is read by a structure called a ribosome, which assembles the amino acids to form the protein. What carries the amino acids to the ribosome is another molecule called a transporter RNA. Each amino acid is encoded by a three-letter sequence [nitrogenous bases] of messenger RNA. These trios of nitrogenous bases form 64 possible combinations, but proteins are made up of just 20 amino acids. This means that some combinations are synonymous and represent the same amino acid. Since some types of transport RNA are found in very low quantities, the ribosome has to wait. In this study, we looked at the mathematical part of the problem and tried to improve it. We showed that it is possible to replace some trios with others that are synonymous but work faster. We thus increased protein production, something with potential commercial applications. The basis of the work is mathematics — we developed the theory on the congestion of amino acids in the ribosome, and biologists here carried out the experiments. Then we went further. We licensed the technology, because many companies want to accelerate the production of proteins used in disease diagnosis, for example. This work was done about 10 years ago and to this day we still receive royalties that allow us to continue research on the topic.

What is it like working with specialists from different areas?
Interaction between different fields is fundamental to good research. You can’t talk about twenty-first century science without mentioning interdisciplinarity. We apply concepts from mathematics to create theoretical biology, because until recently biology was completely empirical. We want to understand natural phenomena, make predictions based on experiments, and improve the public’s standard of living. There were biologists at the initial interview I had with the vice chancellor, and they asked specific questions and showed a great interest in working with mathematicians and physicists. I had never worked with biologists, but I immediately started going to seminars and conferences at the Institute of Medical Science and we soon came up with ideas for joint research projects. Together with a mathematician friend from Imperial College London and several biologists, we managed to secure £4.6 million in funding from the UK’s Biotechnology and Biological Sciences Research Council [BBSRC] for new studies into the adaptation and pathogenicity mechanisms of Candida albicans and C. glabrata, two species of fungus that live in the human body and are normally defeated by the immune system, but sometimes cause diseases. On this and other projects, we look at the data, then try to develop and apply a mathematical model. If it doesn’t work, we refine it or discard it and try again. When it works, the equations make predictions about the results of biology experiments and help improve models.

What biological phenomena can be converted into mathematical equations or models?
It depends, there are several levels and systems. For example, we can greatly simplify the functioning of the heart by seeing cardiac electrical signals and manipulating them with small perturbations. The brain too, to a certain extent. In surgery on children with epilepsy, it is possible to manipulate the functioning of the brain with implants that correct the errors. But it is not possible to model the interactions between biomolecules in cells or the networks of neurons between the brain’s two hemispheres. We have to pick a level of interaction and see what can be done. I did some work with a wonderful young Chinese researcher called Lin Gao on the effect of alcohol during pregnancy. We evaluated neuron connectivity in 19 adolescents who were prenatally exposed to alcohol and 21 healthy adolescents in a control group. In a 2019 article in the magazine Chaos, we showed that children who experienced prenatal alcohol exposure are at risk of developing fetal alcohol spectrum disorder (FASD), characterized by connectivity failures between the cerebral hemispheres, which can lead to deficits in cognitive function. This student did fieldwork with 15-year-olds in the USA, then stayed here for six months, and then another three. We developed a technique that reproduces communication between brain neurons and saw that any level of alcohol, even a small amount like many people consume casually once a week, can cause tremendous disruptions in the networks between different parts of the brain. We were also able to trace the direction of information — where it comes from and where it goes. With the help of a biologist who joined the group, this technique resulted in a potential medicine for dementia, which we developed here in Aberdeen and is now being tested in Europe, the USA, and Asia.

Celso GrebogiImage created in 1995 based on a chaos control equation developed by Grebogi and colleaguesCelso Grebogi

With these studies, you and your group are clearly showing that it is possible to apply the concepts of complex systems, which was an objective you announced many years ago.
Yes, but in a nonlinear system, it is practically impossible to obtain closed mathematical functions that account for all phenomena, because there are so many parts. Since the 1970s, there have been two basic techniques for discovering the equations that govern the units of nonlinear systems, one in which mathematical models generate the data, and the other the opposite, in which the data generates the models. Together with a former student of mine from one of the leading physics groups in the USA, I published an article in Physical Review X in 2011 describing new uses for a technique called compressive sensing, through which a set of data is used to obtain the complete picture of a phenomenon. Here’s a simple example: in an MRI exam, the person spends an average of 30 minutes motionless inside the device, because the system has to generate images with 1 to 2 million pixels and transmit them to the computer. That’s the part that takes so much time. We showed that by using compressive sensing, you can send just a few pixels and still obtain a perfect reconstruction of the image.

So you created an information filter?
Not a filter, but a random information selection mechanism, which is sufficient to obtain the equations of complex systems in the form of functions and arrive at the desired result. For example, if I want to determine the friendship network of the students in a classroom, I can choose any one of them. One has one friend, another may have five, another seven. In a room of 100 students, I can determine the entire friendship network by interviewing just 60 of them. You don’t need to examine everything. There is an old mathematical dilemma: how can you find a fake coin among 12 coins? They are all the same, but the fake one is heavier or lighter than the others. How many weighings do you have to perform to find the fake one? Since the counterfeit coin is scarce, just 1 in 12, you don’t need to perform 12 weighings, but just three, with groups of three coins in each, to see if the total weight is larger or smaller. We also carried out experiments to validate the compressive sensing. A Chinese friend of mine at Arizona State University, Ying-Cheng Lai, did a test to identify the friendship network between 22 students. With just 12 data points, he was able to establish the friendship network between them. Incredible.

What have you been doing in China?
In the last few years I have spent six months a year in China, working with some of the main universities there to found physics departments. It is very easy to work in interdisciplinary fields there, just like it is in the USA and the UK. All my research in China is open and the people are extremely well educated, with incredible discipline. When I’m there, I never know if it’s Saturday, Sunday, or Monday, because the workload is the same every day. One of the places I sometimes spend time is Xi’an, which has become renowned for the sophisticated use of chaos theory in communication. We encoded a trajectory containing the message to be transmitted using small perturbations in a chaotic system. I’ve been doing a lot of work on devices for brain-computer communication, coding the trajectories coming from the head. We were even on a television show there. We had students sit in wheelchairs and look at a screen in front of them, and they controlled the movements of the wheelchair using only their minds, nothing else. Transferring data from the brain to the computer, the so-called brain-machine interface, is extremely complicated.

With a small perturbation, we can alter the chaotic system so that it behaves in the way we want

You started in chemical engineering and then moved into physics, mathematics, and biology. What drove you to change fields?
I often tell my son, Mateus: “Don’t worry. If you need to change fields, then change fields.” He’s now majoring in chemistry and biology at the University of St. Andrews, about 150 kilometers away, and going into medicine. I studied chemical engineering because that’s where the best professors at the Federal University of Paraná [UFPR] were at the time. The first three years were great, but the fourth and fifth were not so good. I was establishing ties with the Physics Institute, especially a professor called Hugo Frederico Kremer [1929–1969], who was creating graduate courses in Paraná. But he was murdered at the university itself by another professor and I lost any desire to stay there. Kremer had already suggested that I do a master’s degree at PUC-RJ, which had very good professors at the time. My idea was to study relativity, which I had been starting to look at with Kremer. I spent three years giving seminars, attending classes, and trying to really understand relativity. But at PUC-RJ, things were not moving forward. Many professors left and a brilliant colleague of mine started drinking and ended up dying. One day, a professor said to me: “You will never have a chance to leave Brazil, because to do so you need to do a master’s degree at PUC.” And my master’s degree at PUC was not moving forward. But I rebelled — I went to the US consulate in Curitiba and the vice consul awarded me a scholarship to study in Maryland. It was his personal decision, and he also gave me a one-way ticket via Pan Am [an international airline that closed in 1991].

Was your time at PUC wasted?
No, because it meant I didn’t have to take all the courses at Maryland. I studied relativity with Charles Misner, who is now 90 years old. I finished my master’s degree in just one year, and in 1976 he called me and asked me what my plans were. I said I wanted to continue with my doctorate, which I started the previous year, and then return to Brazil. He said: “Good, go back. Relativity is an important concept and I will only supervise foreigners who return to their countries.” But during my PhD, I switched to plasma physics, working with a talented physicist who had recently arrived from Princeton: Chuan Sheng Liu. In 1978, I took a postdoctoral position at the University of California, Berkeley, and began to learn new things. I spent most of my time at the Mathematics Department, attending classes and seminars on dynamical systems and chaos theory. Then I returned to Maryland as a professor, where I stayed from 1981 to 2001.

Was it during this time that you began dedicating yourself to chaos theory?
Exactly. When I started, it was still a new field, with many differences in terminology. Scholars of application were getting involved in nonlinear dynamics, which includes chaos theory, and were discovering new phenomena and applications. In Maryland, we had an advantage that mathematicians did not: we could make conjectures [ideas or formulas based on foundations that have not been empirically verified] and move forward. Freeman Dyson [1923–2020], an English physicist I met at the Royal Society in London, correctly wrote that chaos cannot be controlled, because a small perturbation can have effects on the entire chaotic structure. He was right. But we show that with a small perturbation, we can alter the chaotic system so that it behaves in the way we want.

Is this the OGY method?
Yes. It was a way of showing that chaos could be altered using small perturbations, which had consequences in communications theory and other fields. Shortly after our article in Physical Review Letters presented the approach in 1990, a group led by Antônio Azevedo da Costa and Sérgio Rezende, of the Federal University of Pernambuco, showed experimental evidence of the theory. They subjected a sample of lithium and iron to a magnetic field and aligned the spins [a property of electrons that defines how they interact with magnetic fields]. Using a microwave frequency, they formed spin waves and perturbed them to control the behavior of the waves, based on one of the strategies we had suggested. It was the first experimental verification of chaos control. Then a team from the University of California, Los Angeles, applied the same strategy to control heart and brain rhythms at a children’s hospital in Washington. Chaos is common in nature. In these cases, because they are highly dissipative systems [meaning they lose or transform energy], small perturbations can change the state of the heart or brain, since the chaotic system accesses different periodic states all the time. We show that it is possible to choose one of these periodic states and with a small perturbation, make the chaotic state transform into it asymptotically [in a very approximate way]. But in higher-dimensional systems, like complex networks, it’s more complicated.

My mother was a nurse and her goal in life was to provide an education for me and my three siblings

Are the basic concepts of chaos control that you proposed with your colleagues still used?
Yes, they are still used. One of them, fractal basin boundaries [the separation of two or more orientations in a chaotic system], is used to study global warming, because environmental and climate changes are based on several factors. A landscape can undergo a change of state, for example, due to small alterations. Today, 50% of the world’s ecological space has already undergone irreversible changes due to forest fragmentation, urban development, etc. By 2040, the Earth will be approaching a change of planetary state. These phenomena can be described in two ways. The first is through resilience, a function that leads to bifurcation, creating two states: survival on one side and extinction on the other. The other is in terms of survival, which also involves bifurcation, with one possibility to reverse the situation and the other to move forward. We may already be in a moment of crisis, one of the concepts we formulated in Maryland. We are past the point of no return and we are now waiting for collapse to occur, even if no further environmental deterioration occurs. There is the sustainable situation and then a moment of waiting, as described by an exponential expression: the Lorentz transformation equation [created by Dutch physicist Hendrik Lorentz, 1853–1928, to describe the differences in values of time, distance, and order of events observed by two people moving at different speeds]. To go back to the situation before the bifurcation, we have to work very hard. Wild bees, which function as pollinators of agricultural crops, are disappearing because of disease, destruction of their habitats, and global warming. This is an extremely serious problem — if bees go extinct, humanity will theoretically only have four more years to live. There is a mutual interaction in which the pollinators benefit from the plants and the plants benefit from the pollinators, and there is discrimination, because not every pollinator pollinates every plant, and one plant does not provide food for all pollinators. So if you lose one of the pollinators, you may also lose the plants associated with it and vice versa. We can demonstrate this situation in data, the state of survival in the habitat, the tipping point, extinction, or mutual recovery.

What are your current research priorities?
Ecological networks. I’m looking at interactions between habitats, the effects of migration, and the inflection points of major changes. For example, if certain species are disappearing from one place and migrating to another, this migration may create mutual relationships with plants in the new location, allowing vegetation to recover. I’m also working on the interface between computers and other devices, to detect the sleep stages or tiredness when driving, for example. I’m working with a group in Arizona on something called reservoir computing. It is a network of neurons that can learn or evolve as we feed it with data. Because errors can occur, I have to prove that the network’s behavior is true, so that we can use it to make predictions.

Do you still visit Curitiba often?
All the time. I was back for two quick visits in 2022, because two of my cousins passed away due to the pandemic. My siblings still live there. My father died in 1997 and my mother in 2012. When I was a child, we lived in a wooden house outside Curitiba, close to a dirt road. When I turned 7 — I remember this as if it were yesterday — my father said, “now you can work.” He worked nights at a bakery. My brothers and I would get up at exactly 5:13 a.m., have a drink of coffee with sugar, then leave at 5:30 a.m. to help distribute the bread across the south of Curitiba. At 7:30 a.m. he put us on the bus to go to school. When I turned 14 and finished high school, he said, “now you don’t need to study anymore.” But my mother supported us because her father, Jarek Chruściel, grew up in a big village house in Poland and had a formal education. He was the cousin of Antoni Chruściel, a general in the Polish uprising against the Germans in 1944. The Russians promised to support the Poles, but they were stationed on the other side of the Vistula River, resulting in the massacre of more than 30,000 Poles involved in the uprising and the total destruction of Warsaw. After that, Jarek moved to a Polish colony in Rio Grande do Sul. My mother worked hard, planted crops, took care of pigs and cows, sold milk, made our clothes, she did everything. She was a surgical nurse at Santa Casa de Curitiba before getting married — she admired the doctors and her goal in life was to provide an education for me and my three siblings. It’s because of her that I studied science, a technical course, and got to where I am today despite never having studied biology.