2.
Leonardo P. C. da Cruz; Douglas D. Novaes; Joan Torregrosa
New lower bound for the Hilbert number in piecewise quadratic differential systems Journal Article
Em: Journal of Differential Equations, vol. 266, não 7, pp. 4170-4203, 2019.
@article{CruNovTor2018,
title = {New lower bound for the Hilbert number in piecewise quadratic differential systems},
author = {Leonardo P. C. da Cruz and Douglas D. Novaes and Joan Torregrosa},
url = {http://dx.doi.org/10.1016/j.jde.2018.09.032
https://arxiv.org/abs/1809.03433},
doi = {10.1016/j.jde.2018.09.032},
year = {2019},
date = {2019-03-15},
urldate = {2019-03-15},
journal = {Journal of Differential Equations},
volume = {266},
number = {7},
pages = {4170-4203},
abstract = {We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree $n$ piecewise polynomial differential systems, then $H_p(2)geq 16.$ As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree $n$ piecewise polynomial differential systems, then $H_p(2)geq 16.$ As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers.
1.
Douglas D. Novaes; Joan Torregrosa
On the extended Chebyshev systems with positive accuracy Journal Article
Em: J. Math. Anal. Appl., vol. 488, não 1, pp. 171-186, 2017.
@article{NovTor2017,
title = {On the extended Chebyshev systems with positive accuracy},
author = {Douglas D. Novaes and Joan Torregrosa},
url = {http://dx.doi.org/10.1016/j.jmaa.2016.10.076},
doi = {10.1016/j.jmaa.2016.10.076},
year = {2017},
date = {2017-04-01},
urldate = {2017-04-01},
journal = {J. Math. Anal. Appl.},
volume = {488},
number = {1},
pages = {171-186},
abstract = {A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to counting the number of isolated periodic orbits for a family of nonsmooth systems is done.
},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
A classical necessary condition for an ordered set of n+1 functions F to be an ECT-system in a closed interval is that all the Wronskians do not vanish. With this condition all the elements of Span(F) have at most n zeros taking into account the multiplicity. Here the problem of bounding the number of zeros of Span(F) is considered as well as the effectiveness of the upper bound when some Wronskians vanish. For this case we also study the possible configurations of zeros that can be realized by elements of Span(F). An application to counting the number of isolated periodic orbits for a family of nonsmooth systems is done.
