(Sichuan University. School of Mathematics (China))
(Universitat Autònoma de Barcelona. Departament de Matemàtiques)
| Título variante: |
Limit cycles of discontinuous piecewise quadratic perturbations of a linear center separated by the curve y = xn |
| Fecha: |
2022 |
| Resumen: |
Recently there is increasing interest in studying the limit cycles of the piecewise differential systems due to their many applications. In this paper we prove that the linear system ẋ = y, ẏ = -x, can produce at most seven crossing limit cycles for n ≥ 4 using the averaging theory of first order, where the bounds ≤ 4 for n ≥ 4 even and the bounds ≤ 7 for n ≥ 5 odd are reachable, when it is perturbed by discontinuous piecewise polynomials formed by two pieces separated by the curve y = xn (n ≥ 4), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has 0, 1, 2, 3, 4, 5 or 6 crossing limit cycles for n ≥ 4 even and, furthermore, under a particular condition we prove that the number of crossing limit cycles does not exceed 9 (resp. , 11) for 4 ≤ n ≤ 74 even (resp. , n ≥ 76 even). The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if n ≥ 5 is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems. |
| Resumen: |
The study of the limit cycles of planar differential systems is one of the main problems in the qualitative theory of differential systems. These last years a big interest appeared for studying the limit cycles of the piecewise differential systems due to their many applications. Here we prove that the linear center x˙ = y, y˙ = −x, can produce at most 6 crossing limit cycles for n ≥ 4 even and at most 7 crossing limit cycles for n ≥ 5 odd using the averaging theory of first order, when it is perturbed by discontinuous piecewise differential systems formed by two pieces separated by the curve y = xn (n ≥ 4), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has 0, 1, 2, 3, 4, 5, 6, 7 or 9 crossing limit cycles if 4 ≤ n ≤ 74 is even and 0, 1, 2, 3, 4, 5, 6, 7, 9 or 11 crossing limit cycles if n ≥ 76 is even. The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if n ≥ 5 is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems. |
| Ayudas: |
Agencia Estatal de Investigación PID2019-104658GB-I00
European Commission 777911
|
| Derechos: |
Tots els drets reservats.  |
| Lengua: |
Anglès |
| Documento: |
Article ; recerca ; Versió acceptada per publicar |
| Materia: |
Limit cycles ;
The method of averaging ;
Discontinuous piecewise differential systems |
| Publicado en: |
International journal of bifurcation and chaos in applied sciences and engineering, Vol. 32, Issue 12 (September 2022) , art. 2250187, ISSN 1793-6551 |
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identificación
(Sichuan University. School of Mathematics (China))
