In 1900, David Hilbert, one of the most influential mathematicians in history, presented 23 problems that would shape the future of mathematics. Among them, the 16th problem stood out as one of the most challenging, addressing the intriguing question of limit cycles in dynamic systems described by polynomial differential equations. After over a century without a solution, researchers from São Paulo State University (UNESP), Dr. Vinícius da Silva, Dr. João Vieira, and Professor Edson Denis Leonel, have uncovered a solution using an innovative approach based on information geometry. Their findings are published in the journal Entropy.
What is Hilbert’s 16th problem?
The problem can be divided into two parts. The first deals with oval curves in Cartesian planes, while the second, more complex, seeks to determine the maximum number and location of limit cycles in polynomial dynamic systems of degree n.
Limit cycles represent closed, isolated trajectories in systems that repeat indefinitely, like the oscillations of a pendulum or the behavior of electrical circuits. These cycles are crucial for modeling natural and artificial phenomena, from biological rhythms to communication systems.
Despite numerous attempts, a complete solution remained elusive. Traditional methods identified limit cycles but failed to determine their quantity or precise location.
The Brazilian Breakthrough
To overcome the usual difficulties in studying limit cycles, Dr. Vinícius Barros da Silva, Dr. João Peres Vieira, and Professor Edson Denis Leonel introduced the Geometric Bifurcation Theory (GBT), an advanced method combining geometry and dynamics to analyze system changes. With the aid of Riemannian scalar curvature, the researchers found that the maximum number of limit cycles is directly related to the divergence of this curvature to infinity.
According to Dr. da Silva, “Geometric Bifurcation Theory has revealed not only the number of limit cycles but also their locations. Our research demonstrates that these repetitive patterns are linked to the behavior of the system’s scalar curvature. More precisely, when the curvature is positive and reaches extreme values, it indicates the maximum number of limit cycles the given dynamical systems can have.”
This breakthrough was validated across more than 20 dynamic systems, encompassing both simple configurations with few limit cycles and highly complex systems featuring multiple limit cycles. The results were achieved without relying on perturbation methods, highlighting the robustness and versatility of the approach.
“Up to now, our work has garnered over 4,500 views in less than three months and has received numerous recommendations from researchers worldwide, further emphasizing the reliability and impact of the findings. This broad support from the scientific community underscores the significance and consistency of our solution,” added Dr. Vieira and Dr. Leonel.
Implications and Applications
The Brazilian discovery not only solves a century-old mathematical problem but also opens doors to practical applications. Limit cycles are powerful tools for modeling and predicting behaviors across various fields, such as biology, to understand population dynamics, or engineering, to develop more efficient control systems. Moreover, GBT has the potential to revolutionize fields like cybersecurity and quantum cryptography, where limit cycles can be used to create more robust communication and security systems.
The researchers now aim to expand their findings to higher-dimensional dynamic systems, involving more variables and complex interactions, such as those found in quantum mechanics and neural networks.
A landmark in mathematics history
By bridging concepts of geometry and dynamics, the Brazilian solution to Hilbert’s 16th problem is a brilliant example of how mathematics can transform our understanding of the universe and offer practical tools for scientific and technological challenges.
In summary, this groundbreaking work solves Hilbert’s 16th problem while highlighting the potential of geometry to unlock answers in many fields. By taking a fresh perspective on an old question, the team has not only advanced mathematics but also shown how this knowledge can be applied to real-world systems.
Keywords: limit cycles, dynamic systems, David Hilbert, Geometric Bifurcation Theory, applied mathematics.
Journal Reference
da Silva, V.B., Vieira, J.P., & Leonel, E.D. “Exploring Limit Cycles of Differential Equations through Information Geometry Unveils the Solution to Hilbert’s 16th Problem.” Entropy, 2024, 26, 745. DOI: https://doi.org/10.3390/e26090745
About the Authors
Dr. Vinícius Barros holds a Ph.D. in applied physics from São Paulo State University “Júlio de Mesquita Filho” (UNESP), Brazil, earned in 2023. Before this, he completed a Master’s degree at UNESP in 2018 and obtained a bachelor’s in physics from the same institution. In 2018, Dr. Vinícius was a visiting researcher at the Istituto dei Sistemi Complessi (ISC) of the Consiglio Nazionale delle Ricerche in Italy. Dr. Vinícius has also received notable recognition for academic achievements, including first place in the Graduate Program, Doctorate, at UNESP in 2019. Additionally, he was recognized as the top student in the admission class for the physics course at UNESP in 2018 and acknowledged for outstanding academic performance in the physics degree course at UNESP the same year.
His research interests encompass a wide array of topics within physics, including dynamical systems, chaos, Fisher information geometry, differential geometry, Fisher and Rao metric, scalar curvature, and bifurcation theory. Dr. Barros’s work in statistical physics, information geometry, and dynamical systems aims to contribute significantly to advancing knowledge in these fields.
He is continuing his scientific journey by seeking post-doctoral or associate professor positions.
Dr. João Peres Vieira earned a bachelor’s degree in Mathematics from the São Carlos Federal University in 1984, a Master’s in Mathematics from the São Paulo University in 1988, and a Ph.D. in Mathematics from the same institution in 1995. In 2012, he achieved his Habilitation in Mathematics from São Paulo State University “Júlio de Mesquita Filho”, where he currently serves as an Associate Professor.
With extensive expertise in Mathematics, Dr. Vieira specializes in algebraic topology and dynamical systems. His primary research interests focus on fixed points, coincidence theory, and their applications within topological dynamics, offering important insights into the behavior and structure of complex dynamical systems. His contributions reflect a deep commitment to advancing the understanding of both theoretical and applied aspects of these mathematical fields.
Dr. Edson Denis Leonel is a full professor at the Department of Physics, São Paulo State University (UNESP) at Rio Claro Campus. He holds a Bachelor’s degree in Physics from the Federal University of Viçosa (1997), a Master’s (1999), and a Ph.D. in Physics (2003) from the Federal University of Minas Gerais. Dr. Edson Denis Leonel completed his habilitation at the Institute of Geosciences and Exact Sciences (IGCE) of UNESP in 2009 and conducted postdoctoral research at Lancaster University (2003–2005). In 2009, he was a Visiting Professor at the Georgia Institute of Technology (Georgia Tech).
With expertise in Chaos and Dynamical Systems, his research focuses on time series analysis, scaling laws, discrete mappings, chaotic dynamics, Fermi acceleration, classical billiards, and cellular automata. He has been recognized with the V. Afraimovich Award by the International Conference on Nonlinear Science and Complexity in 2023. As a dedicated educator, he contributes to both undergraduate and graduate programs. Additionally, he served as Vice-Dean of the Institute of Geosciences and Exact Sciences (IGCE) from 2017 to 2021 and is currently the Dean (2021–2025).
