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The enigma of Hilbert's 16th problem may be close to a solution, thanks to Brazilian researchers. This breakthrough promises to revolutionize areas such as cybersecurity and quantum cryptography, opening up new possibilities in the study of dynamic systems.
One of the most complex challenges in modern mathematics may finally be close to a solution, thanks to the work of Brazilian researchers.
Scientists São Paulo State University (Unesp) claim to have made significant progress in solving Hilbert's enigmatic 16th problem, a question that has puzzled mathematicians since 1900 and could revolutionize our understanding of dynamical systems, with potential impacts on areas such as quantum cryptography and data security.
This problem, proposed by David Hilbert, is part of a list of 23 fundamental mathematical questions that the German mathematician launched more than a century ago, aiming to guide mathematical research throughout the 20th century.
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While some of these problems were solved relatively quickly, the 16th proved particularly challenging, especially in its second part, which deals with complex limit cycles in dynamical systems.
What is Hilbert's 16th problem?
Hilbert divided the 16th problem into two parts, each relating to a specific field of mathematics.
The first part investigates the number and arrangement of oval curves in certain configurations in the plane.
The second, which is the focus of the recent Brazilian advance, explores how many limit cycles — closed trajectories in dynamical systems — can exist in systems described by polynomial differential equations.
To understand the question, it is useful to remember some basic concepts. In mathematics, a simple equation, such as a linear equation, represents a straight line on the Cartesian plane.
Equations with terms raised to the square, cube, or higher powers generate more complex curves, such as parabolas and ellipses, which form what mathematicians call polynomials.
Limit cycles are essential for modeling repetitive phenomena in diverse natural and artificial systems, such as population dynamics in ecology and temperature control in data centers.
Studying these cycles allows us to predict the behavior of systems with a cyclical pattern. Hilbert's central question was: how many such cycles can there be in a polynomial dynamical system and where do they lie?
The innovation of Brazilian researchers
As Brazilian researchers explain, the challenge has always been to identify and quantify these cycles.
“Until now, methods only confirmed the existence of limit cycles, but were unable to determine their quantity and location,” says Vinícius, one of the authors of the study published in the journal EntropyIt was precisely this limitation that inspired the team to seek new approaches.
The solution found involved the application of the Geometric Bifurcation Theory (GBT), which offers a more accurate way of analyzing the behavior of dynamic systems.
This theory uses geometric metrics and Riemannian scalar curvature to identify the maximum number of limit cycles.
According to Vinícius, the maximum number of limit cycles in a polynomial differential equation can be determined by the number of divergences of the scalar curvature towards infinity.
The team validated this method on more than 20 dynamic systems with different numbers of cycles.
The implications of the discovery
The advancement opens doors to several applications, especially in the field of cybersecurity.
As the team explains, limit cycles are fundamental to secure communication systems and quantum cryptography, which are crucial for data protection in sectors such as finance and banking.
A researchers' discovery It can also be applied in biology to understand population dynamics and chemical reactions, and in engineering to develop more effective control systems.
According to the authors, the next steps include applying the Geometric Bifurcation Theory to higher-dimensional systems and extending it to areas such as quantum mechanics and neural networks.
What are the next steps?
The authors, João Peres Vieira and Edson Denis Leonel, both from Unesp, intend to explore the method in even more complex dynamic systems.
With this, they hope to uncover other applications and expand understanding of limit cycles in different mathematical and scientific contexts.
