TY - JOUR

T1 - On the number of limit cycles in piecewise planar quadratic differential systems

AU - Braun, Francisco

AU - Pereira Costa da Cruz, Leonardo

AU - Torregrosa, Joan

N1 - Publisher Copyright:
© 2024 Elsevier Ltd

PY - 2024/10

Y1 - 2024/10

N2 - We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.

AB - We consider piecewise quadratic perturbations of centers of piecewise quadratic systems in two zones determined by a straight line through the origin. By means of expansions of the displacement map, we are able to find isolated zeros of it, without dealing with the unsurprising difficult integrals inherent in the usual averaging approach. We apply this technique to non-smooth perturbations of the four families of isochronous centers of the Loud family, S1, S2, S3, and S4, as well as to non-smooth perturbations of non-smooth centers given by putting different Si's in each zone. To show the coverage of our approach, we apply its first order, which is equivalent to averaging theory of the first order, in perturbations of the already mentioned centers considering all the straight lines through the origin. Then we apply the second order of our approach to perturbations of the above centers for a specific oblique straight line. Here in order to argue we introduce certain blow-ups in the perturbative parameters. As a consequence of our study, we obtain examples of piecewise quadratic systems with at least 12 limit cycles. By analyzing two previous works of the literature claiming much more limit cycles we found some mistakes in the calculations. Therefore, the best lower bound for the number of limit cycles of a piecewise quadratic system is up to now the 12 limit cycles found in the present paper.

KW - Periodic solution

KW - Averaging method

KW - Non-smooth differential system

UR - http://www.scopus.com/inward/record.url?scp=85192106544&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/ce8b37eb-54e2-3048-8a40-d5e06721b7c5/

U2 - 10.1016/j.nonrwa.2024.104124

DO - 10.1016/j.nonrwa.2024.104124

M3 - Article

SN - 1468-1218

VL - 79

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

M1 - 104124

ER -