The 16th Hilbert problem for discontinuous piecewise isochronous centers of degree one or two separated by a straight line

Abstract

In this paper we deal with discontinuous piecewise differential systems formed by two differential systems separated by a straight line when these two differential systems are linear centers (which always are isochronous) or quadratic isochronous centers. It is known that there is a unique family of linear isochronous centers and four families of quadratic isochronous centers. Combining these five types of isochronous centers we obtain fifteen classes of discontinuous piecewise differential systems.We provide upper bounds for the maximum number of limit cycles that these fifteen classes of discontinuous piecewise differential systems can exhibit, so we have solved the 16th Hilbert problem for such differential systems. Moreover in seven of the classes of these discontinuous piecewise differential systems the obtained upper bound on the maximum number of limit cycles is reached.

The number of limit cycles for regularized piecewise polynomial systems is unbounded

“…In this section we suppose that β − < 0. Then Z − has a hyperbolic saddle at (x, y) = ( γ− β− , 1 β− ) with eigenvalues κ − < 0 and κ + > 0 given in (16). From ( 17) it follows that the stable manifold of the hyperbolic saddle is given by x = κ + y + x R and the unstable manifold is given by x = κ − y + x L where x L < 0 and x R > 0 are defined in (19).…”

“…Parallel to this effort, there has been an attempt to bound the number limit cycles in PWS systems in the plane where n = 2. In contrast to the smooth linear setting, limit cycles can exist for piecewise linear systems and J. Llibre and co-workers have obtained upper bounds for a number of cases [26,49,52]. Of course, the interest in bounding the number of limit cycles, comes from Hilbert's 16th problem [48] which seeks to bound the number of limit cycles of polynomial systems: ẋ = P N (x, y), ẏ = Q N (x, y), (1.5) with P N and Q N of fixed degree N .…”

“…At the same time, there has in recent years been an attempt, for example by J. Llibre and co-workers, to determine the maximum number of crossing limit cycles in PWL systems. Certain subcases have been solved [16,34,37] and more generally it has been shown in [6] that the number of crossing limit cycles is bounded by 8. To the best of our knowledge, only 3 crossing limit cycles have been realized (see [22,36]).…”

Integrability and Limit Cycles via First Integrals

“…x = 0.2x − 0.1y − 0.778814,ẏ = 0.4x − 0.2y + 0.00727332, (30) in the region R 2 , its Hamiltonian is:…”

The number of limit cycles for regularized piecewise polynomial systems is unbounded

“…Parallel to this effort, there has been an attempt to bound the number limit cycles in PWS systems in the plane where n = 2. In contrast to the smooth setting, limit cycles can exist for piecewise linear systems and J. Llibre and co-workers have obtained upper bounds for a number of cases [25,47,50]. Of course, the interest in bounding the number of limit cycles, comes from Hilbert's 16th problem [46] which seeks to bound the number of limit cycles of polynomial systems: ẋ = Q N (x, y),…”