Breaking barriers: Brazilian team solves Hilbert’s 16th problem for the first time in 124 years
In 1900, David Hilbert, one of the most influential mathematicians in history, asked 23 questions that would shape the future of mathematics. Among them, problem 16 stands out as one of the most challenging problems, solving the interesting problem of limit cycles in dynamic systems described by polynomial differential equations. After more than a century without a solution, researchers from the State University of São Paulo (UNESP) Dr. Vinícius da Silva, Dr. João Vieira and Professor Edson Denis Leonel discovered a solution using an innovative approach based on information geometry. Their findings were published in the journal Entropy.
What is Hilbert’s 16th question?
The question can be divided into two parts. The first involves elliptic curves in the Cartesian plane, while the second is more complex and aims to determine the maximum number and location of limit cycles in a polynomial dynamic system of degree n.
A limit cycle represents a closed, isolated trajectory in an infinitely repeating system, such as the oscillations of a pendulum or the behavior of an electrical circuit. These cycles are crucial for simulating natural and artificial phenomena, from biological rhythms to communication systems.
Despite numerous attempts, a complete solution remains elusive. Traditional methods identified limit cycles but failed to determine their number or precise locations.
Brazil’s breakthrough
To overcome common difficulties in limit cycle research, Dr. Vinícius Barros da Silva, Dr. João Peres Vieira, and Professor Edson Denis Leonel introduced geometric bifurcation theory (GBT), a method that combines geometry and dynamics to analyze system changes. Advanced methods. Using Riemannian scalar curvature, the researchers found that the maximum number of limit cycles is directly related to the curvature diverging to infinity.
Dr. da Silva said, “Geometric bifurcation theory reveals not only the number of limit cycles, but also their locations. Our study shows that these repeating patterns are related to the behavior of the scalar curvature of the system. More precisely, when the curvature When positive and reaching extreme values, it indicates the maximum number of limit cycles a given dynamical system can have.
This breakthrough was demonstrated in more than 20 dynamic systems, ranging from simple configurations with few limit cycles to highly complex systems with many limit cycles. The results are obtained without relying on perturbation methods, highlighting the robustness and versatility of the method.
“To date, our work has received more than 4,500 views in less than three months and has received numerous suggestions from researchers around the world, further underscoring the robustness and impact of the findings. Science The broad support from the community highlights the importance and consistency of our solutions.
Impact and application
Brazil’s discovery not only solves a century-old mathematical problem, it also opens the door to practical applications. Limit cycles are powerful tools that can be used to model and predict behavior in various fields, such as biology to understand population dynamics, or engineering to develop more efficient control systems. Additionally, GBT has the potential to revolutionize fields such as cybersecurity and quantum cryptography, where limit cycles can be used to create more powerful communication and security systems.
The researchers now aim to extend their findings to higher-dimensional dynamic systems involving more variables and complex interactions, such as those found in quantum mechanics and neural networks.
Milestones in the History of Mathematics
By connecting concepts from geometry and dynamics, Brazil’s solution to Hilbert’s 16th problem is an excellent example of how mathematics is transforming our understanding of the universe and providing practical tools for solving scientific and technological challenges.
Altogether, this groundbreaking work solves Hilbert’s 16th problem while highlighting the potential of geometry to unlock answers in many fields. By taking a new perspective on an old problem, the team not only mastered advanced mathematics but also showed how to apply this knowledge to real-world systems.
Keywords: Limit cycles, dynamic systems, David Hilbert, geometric bifurcation theory, applied mathematics.
Journal reference
da Silva, VB, Vieira, JP, and Leonel, ED “Exploring limit cycles of differential equations through information geometry reveals solutions to Hilbert’s 16th problem.” Entropy, 2024, 26, 745.
About the author
Dr. Vinicius Barros Holds a Ph.D. In 2023, he received his PhD in Applied Physics from the State University of São Paulo “Júlio de Mesquita Filho” (UNESP), Brazil. . In 2018, Dr. Vinícius served as a visiting researcher at the Istituto dei Sistemi Complessi (ISC) in Consiglio Nazionale delle Ricerche, Italy. Dr. Vinícius has also received notable recognition for his academic achievements, including first place in the UNESP Graduate and Doctoral Programs in 2019. Excellent performance in degree programmes.
His research interests cover a wide range of topics in physics, including dynamical systems, chaos, Fisher information geometry, differential geometry, Fisher and Rao metrics, scalar curvature and bifurcation theory. Dr. Barrows’ work in statistical physics, information geometry, and dynamical systems aims to make significant contributions to advancing knowledge in these areas.
He is seeking a postdoctoral or associate professor position to continue his scientific journey.
Dr. Joao Pérez Vieira He received his bachelor’s degree in mathematics from the Federal University of São Carlos in 1984, his master’s degree in mathematics from the University of São Paulo in 1988, and his PhD in mathematics from the University of São Paulo in 1988. In 1995, he received his PhD in Mathematics from the same institution.
Dr. Vieira has extensive expertise in mathematics, specializing in algebraic topology and dynamical systems. His main research interests focus on fixed points, correspondence theory and their applications in topological dynamics, providing important insights into the behavior and structure of complex dynamical systems. His contributions reflect a strong commitment to advancing the understanding of theory and applications in these areas of mathematics.
Dr. Edson Dennis Lionel is a full professor in the Department of Physics of the Rio Claro Campus of the São Paulo State University (UNESP). He holds a bachelor’s degree (1997), a master’s degree (1999) and a PhD in physics from the Federal University of Viçosa. PhD in Physics from the Federal University of Minas Gerais (2003). Dr. Edson Denis Leonel completed his training at UNESP’s Institute of Geosciences and Exact Sciences (IGCE) in 2009 and conducted postdoctoral research at Lancaster University (2003-2005). In 2009, he served as a visiting professor at Georgia Tech.
With expertise in chaos and dynamical systems, his research focuses on time series analysis, scaling laws, discrete maps, chaotic dynamics, Fermi acceleration, classical billiard balls, and cellular automata. He received the V. Afraimovich Award in 2023 from the International Conference on Nonlinear Science and Complexity. In addition, he served as Vice President of the Institute of Geosciences and Exact Sciences (IGCE) from 2017 to 2021 and is currently President (2021-2025).
