Search

We study the crossing periodic orbits and limit cycles of the planar piecewise-continuous differential systems separated by the straight-line x = 0 having in x > 0 the general quadratic isochronous center ẋ = − y + x 2 , ẏ = x (1 + y) after an affine transformation, and in x < 0 an arbitrary quadratic isochronous center except for the quadratic isochronous center ẋ = − y + x 2 − y 2 , ẏ = x (1 + 2 y) which has been studied in [Ghermoul et al., 2021]. For these piecewise-continuous differential systems the upper bound of crossing limit cycles is 2, and there are specific examples having one crossing limit cycle. [ABSTRACT FROM AUTHOR]

First, we study the planar continuous piecewise differential systems separated by the straight line x = 0 formed by a linear isochronous center in x > 0 and an isochronous quadratic center in x < 0. We prove that these piecewise differential systems cannot have crossing periodic orbits, and consequently they do not have crossing limit cycles. Second, we study the crossing periodic orbits and limit cycles of the planar continuous piecewise differential systems separated by the straight line x = 0 having in x > 0 the general quadratic isochronous center ẋ = − y + x 2 − y 2 , ẏ = x (1 + 2 y) after an affine transformation, and in x < 0 an arbitrary quadratic isochronous center. For these kind of continuous piecewise differential systems the maximum number of crossing limit cycles is one, and there are examples having one crossing limit cycles. In short for these families of continuous piecewise differential systems we have solved the extension of the 16th Hilbert problem. [ABSTRACT FROM AUTHOR]