Scientific Russia:The theory of hidden oscillations is the key to stability of technical, economic, and social systems
Source: https://english.spbu.ru/news-events/university-media/theory-hidden-oscillations-key-stability-technical-economic-and-social Parent: https://english.spbu.ru/
23 March 2026
University in Media
Scientific Russia: The theory of hidden oscillations is the key to stability of technical, economic, and social systems
Professor Nikolay Kuznetsov, a scientist from St Petersburg and Corresponding Member of the Russian Academy of Sciences, won the 2024 Russian Federation National Award in Science and Technology for developing the theory of hidden oscillations. In an interview with the Scientific Russia portal, Professor Kuznetsov explained the core principles of the theory, its practical applications in designing and developing new technical systems, and its role in establishing stable economic and social mechanisms.
Photo: Olga Merzliakova / Scientific Russia
Professor Nikolay Kuznetsov is a Doctor of Physics and Mathematics, a recipient of the Russian Federation National Award in Science and Technology, a Corresponding Member of the Russian Academy of Sciences, and the Head of the Department of Applied Cybernetics at St. Petersburg State University.
What is the theory of hidden oscillations? What data does it analyse, and why is it significant for science?
The primary objective of the theory of hidden oscillations is to study the dynamics of diverse objects. In terms of its relevance, it advances us in addressing applied engineering challenges and solving fundamental mathematical problems.
The theory of hidden oscillations represents the next stage in the development of Academician Aleksandr Andronov’s theory of oscillations. It builds on Andronov’s innovative ideas, which integrated Henri Poincaré’s mathematical concepts with radiophysical experiments. Mathematics serves as a universal language, proving both convenient and effective in addressing problems across various fields. The theory of hidden oscillations applies this mathematical framework — for example, in the development of control systems, the analysis and prediction of complex object dynamics (including engineering and energy systems), and the synthesis and design of new technical solutions.
What are the hidden oscillations you study? Why and how are they hidden?
A simple and illustrative example from medicine and biology is the human respiratory system. While in the womb, a baby’s respiratory system is inactive, receiving oxygen through the umbilical cord. In this state, the system is stable and non-oscillatory. After birth, however, the respiratory system transitions from the state of rest to a stable, cyclical mode of operation. This phenomenon — self-excitation of oscillations — was described by Academician Aleksandr Andronov. A similar process often occurs in physics experiments with technical systems: for instance, when a light is switched on, electrical oscillations emerge.
Let us consider another scenario: a person who has nearly drowned and requires resuscitation. Their respiratory system is in a stationary state — stable, but unresponsive to gentle stimulation. To shift the system’s initial conditions away from this stable equilibrium and into the attraction zone of the desired oscillatory mode (which could restart breathing), a strong stimulus — such as an electric shock or artificial respiration — is necessary. This assumes that the system still retains the capacity for such oscillations. Selecting the correct initial conditions is crucial. These oscillations are called hidden: their attraction zone lies outside the observable ‘visibility’ range and is, mathematically, distinct from the natural steady states we typically encounter.
The respiratory system illustrates beneficial hidden oscillations. In technology, however, such oscillations are usually undesirable. An ideal system behaves like a tumbler toy — returning to stability regardless of disturbances. For example, during an airplane landing, a sudden crosswind gust can disrupt the aircraft’s steady trajectory. To ensure a safe landing, the control system must swiftly restore stability and suppress any unwanted oscillations.
Photo: Olga Merzliakova / Scientific Russia
Speaking of aircraft: Is the flutter effect — once addressed by Academician Mstislav Keldysh — also a manifestation of hidden oscillations?
In such systems, both self-excited and hidden oscillations can occur. The problem Mstislav Keldysh solved before World War II, together with a team from the Zhukovsky Central Institute of Aerodynamics, involved suppressing self-excited oscillations in aircraft control systems at high speeds. At the time, engines had grown more powerful, but the materials used for aircraft controls remained inadequate. As a result, at a certain threshold speed, ailerons or wings would begin to oscillate. If the amplitude of these oscillations increased, it led to structural failure.
The self-excitation of oscillations can be observed in physics experiments when speed is gradually increased. However, in experiments conducted in real-world conditions, it is extremely difficult to predict sudden disturbances — such as strong crosswinds — that might trigger oscillations. It is impossible to account for every possible disturbance.
To thoroughly analyse the dynamics, extensive testing is conducted — such as wind tunnel experiments at various speeds and under different loads. This is a complex challenge. To ensure the elimination of unwanted oscillations, calculations and experiments alone are insufficient; developing a theoretical framework is essential. The theory of hidden oscillations aims to develop analytical methods and synthesise them into both analytical and numerical approaches. These methods enable the detection and prediction of such phenomena already at the system design stage.
Mstislav Keldysh and his research team managed to solve this problem to a significant degree. It is important to note that Germany lost over a hundred prototype aircraft before World War II — a devastating setback for its aviation industry. The Soviet Union, however, avoided this issue thanks to Mstislav Keldysh’s work. However, in these systems, alongside observable self-excited oscillations, hidden oscillations can also emerge. Mstislav Keldysh himself accurately noted in his writings that he relied heavily on intuition, as the necessary mathematical theory did not yet exist. A scientist of extraordinary education and broad scientific culture, he always clarified where he shifted from rigorous mathematical reasoning to engineering intuition. Building on one of his studies, we refined the methods for assessing not only self-excited oscillations but also dangerous hidden oscillations.
How challenging is it to predict and identify the processes that lead to hidden oscillations — especially given that they appear to be unique to each technical system?
On the one hand, mathematical language does allow for the formulation of general principles and the development of universal tools. This universality is what makes such methods widely applicable and appealing. On the other hand, every theory has its limitations — including practical ones — and some problems cannot be resolved through theory alone.
However, developing a theoretical framework is essential for cultivating engineering intuition and for describing, understanding, and predicting effects that are not yet observable but may emerge in complex systems. How did Mstislav Keldysh work? He began by analysing a very simple flutter model, using it to assess potential occurrences, understand the phenomenon, and describe its core nature. Then, drawing on his engineering intuition, he calculated the behaviour of the actual system and devised methods to prevent dangerous effects.
Photo: Olga Merzliakova / Scientific Russia
Can artificial intelligence assist in this work today? Does it have sufficient data for analysis?
In my view, artificial intelligence represents the next evolutionary stage in computer science and information technology. The active development of these fields began in the 1980s, when computer science was introduced in secondary schools and the first All-Union Olympiad was held. Higher education in this area — closely tied to mathematics — expanded significantly in the 1990s. The primary advances in digital technology stem from theoretical computer science and algorithms: the mathematical language of computer science. This connection is reflected in specialisations like ‘Applied Mathematics and Informatics’.
Today, computing power has grown exponentially, data volumes have expanded, and algorithms have become more sophisticated. AI has ushered in a new era in computer science and is now widely applied in science and engineering — including as a tool for generating new knowledge. The theory of hidden oscillations integrates classical numerical modelling with AI, enabling the identification of conditions that may trigger hidden oscillations or the prediction of their occurrence based on the system’s permissible parameters.
Mathematical methods are applied not only in technology and engineering but also in the social sciences. Is the theory of hidden oscillations relevant in economics, for example?
Absolutely. A distinct area of economics relates directly to the theory of hidden oscillations: the challenge of forecasting. For instance, we aim for a stable exchange rate for the national currency, the rouble, to enable more reliable economic predictions. However, the exchange rate is dynamically influenced by numerous factors — foreign exchange earnings, taxes, and other indicators. By adjusting these variables, we observe fluctuations in the rouble’s value and must forecast the system’s behaviour within existing mechanisms.
The ideal scenario is when these mechanisms function predictably, like the laws of physics, rather than requiring constant manual intervention — which undermines long-term economic forecasting. This also raises the issue of undesirable oscillations in economic indicators, particularly during shocks such as corporate collapses, market panics, energy price spikes, or natural and social disasters.
Such as the COVID-19 pandemic?
Precisely. The pandemic serves as a classic example of a shock factor: restrictive measures caused a sharp deviation from baseline data, including reduced production, a shrinking workforce, and surging healthcare costs. Such shocks can push the economic system off its stable path and into an undesirable oscillatory state. Moreover, these are often not self-excited oscillations — easy to detect and mitigate — but hidden and far more difficult to predict.
Generally speaking, oscillatory processes are inherent to social systems. For example, the well-known Lotka-Volterra predator-prey model demonstrates that populations of herbivores and carnivores in confined environments — such as the northern tundra — fluctuate in cycles. Such dynamic patterns are common in social and economic sciences, where the theory of hidden oscillations finds broad application.
In medicine, similar principles govern the cardiovascular and respiratory systems. In a healthy individual, brief breath-holding or physical stress may temporarily alter heart rate, but the body quickly returns to its stable, rhythmic mode. The challenge is to ensure that, despite disturbances, the system consistently reverts to this healthy rhythm — avoiding other stable, yet dangerous oscillatory states, such as a sudden spike in heart rate. This opens another significant area for applying the theory of hidden oscillations in practice.
What aspects of the theory of hidden oscillations remain unresolved? What problems are you currently addressing?
When new classes of dynamic objects emerge, researchers across disciplines seek to uncover the shared mechanisms and conditions that give rise to undesirable oscillations. Certain models — whether in economics or technical systems — have unique characteristics. For instance, control systems often rely on a single equilibrium state that the system must achieve, regardless of initial conditions.
Thus, it is possible to identify universal patterns in the emergence of hidden oscillations across different model classes, describe them in order to develop effective methods for their prevention or suppression. This is the primary applied objective.
At the same time, alongside the advancement of applied fields studying hidden oscillations, a broader theoretical framework is being constructed. This framework establishes a unified foundation and a common language for all these methods.
Your works are widely cited internationally. How does Russian mathematical science currently compare to other countries?
First and foremost, it is essential to recognise that science is inherently global. As Anton Chekhov once noted, ‘There is no such thing as national science, just as there are no national multiplication tables.’ Science is an international endeavour of accumulating and advancing knowledge. That said, national characteristics do influence the development of scientific fields.
In mathematics, Russia takes justifiable pride in its mathematical tradition. The development of mathematics in our country is deeply rooted in the era of Peter the Great and the establishment of the Russian Academy of Sciences. This year, we mark the 325th anniversary of Peter the Great’s decree to advance mathematical and navigational education — fields initially intertwined because they addressed practical challenges, including naval navigation, which demanded substantial mathematical expertise.
With the founding of the Russian Academy of Sciences, the first academicians were brought to Russia, among them Leonhard Euler. This celebrated mathematician and mechanical engineer served as an Adjunct Professor in Physiology and, notably, conducted pioneering research on fluid flow through tubes — effectively modelling blood flow in vessels. His work relied on mathematical methods advanced by other scientists. Over three centuries, the Russian mathematical school achieved remarkable success and earned global recognition. In our field — mathematical control theory — fundamental contributions were made by scholars such as Ivan Vyshnegradsky, Aleksandr Lyapunov, Aleksandr Andronov, and many other internationally acclaimed mathematicians.
Mathematical schools emerged not only in St. Petersburg, where the Academy of Sciences was established, but also in other Russian cities. These influential research teams hold a distinguished position in the international scientific community. It is worth emphasising that, alongside human resources and talented young researchers, government support programmes play a crucial role. Substantial resource investments are essential for effective scientific progress and breakthroughs.
Do mathematicians receive enough support today?
I believe that government support and attention can never be excessive — the more, the better. However, existing programmes to foster and develop mathematics have already made a significant impact during critical periods. History repeats itself, and recurring challenges continually test the state’s resolve. Russian state has overcome such trials before: during the October Revolution and Civil War, when many educated individuals left the country; during Perestroika; and after the collapse of the Soviet Union, when science underwent complex transformations. Today, we face new challenges, both political and in the field of international cooperation. Yet, we have experience in navigating these difficulties, and the state remains committed to providing the necessary support.
At the December 2025 general meeting of the Russian Academy of Sciences (RAS), the President of RAS Gennady Krasnikov addressed Russia’s technological leadership programme — an initiative in which the Academy actively participates — and its corresponding funding. During a meeting of the RAS Department of Energy, Mechanical Engineering, Mechanics, and Control Processes, the RAS Vice President Sergei Chernyshev discussed the sixth subprogramme focused on using applied sciences to develop technologies, in particular, for the military-industrial complex.
Government attention to these efforts is indeed evident. We can see it through the Russian Science Foundation, which allocates grants, and through the active development of research programmes. Nevertheless, there is always substantial work to be done in this area. It must adapt to the current, ever-changing situation — both internationally and domestically. Academic programmes are also undergoing major transformations due to the transition from the Bologna system to the national higher education system. To sustain scientific schools, research teams, and organisations, consistent government support remains essential. This will ensure continued effective work and the achievement of results at the level required to meet the challenges facing the country today.
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