On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

DOI10.1016/J.PHYSD.2022.133523arXiv2201.02019WikidataQ114141862 ScholiaQ114141862MaRDI QIDQ6387637

Douglas D. Novaes

Publication date: 6 January 2022

Abstract: The second part of the Hilbert's sixteenth problem consists in determining the upper bound $mathcalH(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $ngeq2$, it is still unknown whether $mathcalH(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $mathcalH(n)$. Regarding asymptotic behavior, the best result says that $mathcalH(n)$ grows as fast as $n^{2}log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper "Some open problems in low dimensional dynamical systems" by A. Gasull, Problem 18 proposes another Hilbert's sixteenth type problem, namely improving the lower bounds for $mathcalL(n)$, $ninmathbbN$, which is defined as the maximum number of limit cycles that planar piecewise linear differential systems with two zones separated by a branch of an algebraic curve of degree $n$ can have. So far, $mathcalL(n)geq[n/2],$ $ninmathbbN$, is the best known general lower bound. Again, better lower bounds for small values of $n$ are known in the research literature. Here, by using a recently developed second order Melnikov method for nonsmooth systems with nonlinear discontinuity manifold, it is shown that $mathcalL(n)$ grows as fast as $n^2.$ This will be achieved by providing lower bounds for $mathcalL(n)$, which improves every previous estimates for $ngeq4$.

Mathematics Subject Classification ID

Topological structure of integral curves, singular points, limit cycles of ordinary differential equations (34C05) Bifurcation theory for ordinary differential equations (34C23) Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) for ordinary differential equations (34C07) Discontinuous ordinary differential equations (34A36) Perturbations, asymptotics of solutions to ordinary differential equations (34E10)

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