2024/11/14 by Buzzi, Claudio A., Novaes, Douglas D.
#34C07 #34C23 #37G15 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2411.09594
For a given natural number n, the second part of Hilbert's 16th Problem asks whether there exists a finite upper bound for the maximum number of limit cycles that planar polynomial vector fields of degree n can have. This maximum number of limit cycle, denoted by H(n), is called the nth Hilbert number. It is well-established that H(n) grows asymptotically as fast as n2 log n. A direct consequence of this growth estimation is that H(n) cannot be bounded from above by any quadratic polynomial function of n. Recently, the authors of the paper [Exploring limit cycles of differential equations through information geometry unveils the solution to Hilbert's 16th problem. Entropy, 26(9), 2024] affirmed to have solved the second part of Hilbert's 16th Problem by claiming that H(n) = 2(n - 1)(4(n - 1) - 2). Since this expression is quadratic in n, it contradicts the established asymptotic behavior and, therefore, cannot hold. In this note, we further explore this issue by discussing some counterexamples.