Counting configurations of limit cycles and centers

Armengol Gasull ; Antoni Guillamon ; Víctor Mañosa

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2023), pp. 78-96.

Résumé

We present several results on the determination of the number and distribution of limit cycles or centers for planar systems of differential equations. In most cases, the study of a recurrence is one of the key points of our approach. These results include the counting of the number of configurations of stabilities of nested limit cycles, the study of the number of different configurations of a given number of limit cycles, the proof of some quadratic lower bounds for Hilbert numbers and some questions about the number of centers for planar polynomial vector fields.

Bibliographie

[1] M. J. Álvarez, B. Coll, P. De Maesschalck, R. Prohens, “Asymptotic lower bounds on Hilbert numbers using canard cycles”, J. Differential Equations, 268 (2020), 3370–3391 | DOI | MR

[2] M. J. Álvarez, A. Gasull, R. Prohens, “Topological classification of polynomial complex differential equations with all the critical points of center type”, J. Difference Equ. Appl., 16 (2010), 411–423 | DOI | MR

[3] J. C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe, Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case, Birkhäuser, 2021 | MR | Zbl

[4] Amer. Math. Soc. Transl., 100 (1954), 1–19 | MR | MR

[5] N. L. Biggs, E. K. Lloyd, R. Wilson, Graph Theory. 1736–1936, Oxford University Press, 1974 | MR

[6] T. R. Blows, “Center configurations of Hamiltonian cubic systems”, Rocky Mountain J. Math., 40 (2010), 1111–1122 | DOI | MR | Zbl

[7] A. Cayley, “On the analytical forms called trees, with application to the theory of chemical combinations”, Reports British Assoc. Advance. Sci, 45 (1875), 257–305; The Collected Math. Papers, v. 9, Cambridge University Press, Cambridge, 2009, 427–460 | MR

[8] L. A. Cherkas, “Dulac function for polynomial autonomous systems on a plane”, Differential Equations, 33 (1997), 692–701 | MR | Zbl

[9] C. J. Christopher, N. G. Lloyd, “Polynomial systems: A lower bound for the Hilbert numbers”, Proc. Roy. Soc. London Ser. A, 450 (1995), 219–224 | DOI | MR | Zbl

[10] A. Cima, A. Gasull, F. Mañosas, “Some applications of the Euler-{J}acobi formula to differential equations”, Proc. Amer. Math. Soc., 118 (1993), 151–163 | DOI | MR | Zbl

[11] A. Cima, A. Gasull, F. Mañosas, “On polynomial Hamiltonian planar vector fields”, J. Differential Equations, 106 (1993), 367–383 | DOI | MR | Zbl

[12] A. Cima, J. Llibre, “Configurations of fans and nests of limit cycles for polynomial vector fields in the plane”, J. Differential Equations, 82 (1989), 71–97 | DOI | MR | Zbl

[13] F. Dumortier, J. Llibre and J. C. Artés, Qualitative theory of planar differential systems, UniversiText, Springer–Verlag, New York, 2006 | MR | Zbl

[14] R. M. Fedorov, “Upper bounds for the number of orbital topological types of polynomial vector fields on the plane “modulo limit cycles””, Proc. Steklov Inst. Math., 254 (2006), 238–254 | DOI | MR | Zbl

[15] P. Flajolet, R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009 | MR | Zbl

[16] A. Gasull, “Difference equations everywhere: some motivating examples”, Recent Progress in Difference Equations, Discrete Dynamical Systems and Applications, Proceedings of ICDEA 2017, Springer Proceedings in Mathematics Statistics, 2019, 129–167 | DOI | MR | Zbl

[17] A. Gasull, “Some open problems in low dimensional dynamical systems”, SeMA J., 78 (2021), 233–269 | DOI | MR | Zbl

[18] A. Gasull, H. Giacomini, “A new criterion for controlling the number of limit cycles of some generalized Liénard equations”, J. Differential Equations, 185 (2002), 54–73 | DOI | MR | Zbl

[19] A. Genitrini, “Full asymptotic expansion for Pólya structures”, Proceedings of the 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (Kraków, Poland, 4-8 July 2016), arXiv: 1605.00837 [math.CO] | MR

[20] P. Griffiths, J. Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley Sons], New York, 1978 | MR | Zbl

[21] A. Guillamon, Study of some problems of qualitative theory in the plane with applications to predator-prey systems, PhD thesis, Univ. Autònoma de Barcelona, 1995 | Zbl

[22] M. Han, J. Li, “Lower bounds for the Hilbert number of polynomial systems”, J. Differential Equations, 252 (2012), 3278–3304 | DOI | MR | Zbl

[23] T. Johnson, “A quartic system with twenty-six limit cycles”, Exp. Math., 20 (2011), 323–328 | DOI | MR | Zbl

[24] T. Johnson, W. Tucker, “An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 1451–1458 | DOI | MR | Zbl

[25] A. G. Khovanskiĭ, “The index of a polynomial vector field”, Funktsional. Anal. i Prilozhen., 13 (1979), 49–58 ; 96 | DOI | MR | Zbl

[26] J. Li, “Hilbert's 16th problem and bifurcations of planar vector fields”, Inter. J. Bifur. and Chaos, 13 (2003), 47–106 | DOI | MR | Zbl

[27] J. Li, H. Chan and K. Chung, “Some lower bounds for $H(n)$ in Hilbert's 16th problem”, Qualitative Theory of Dynamical Systems, 3 (2003), 345–360 | MR

[28] H. Liang, J. Torregrosa, “Parallelization of the Lyapunov constants and cyclicity for centers of planar polynomial vector fields”, J. Differential Equations, 259 (2015), 6494–6509 | DOI | MR | Zbl

[29] J. Llibre, G. Rodríguez, “Configurations of limit cycles and planar polynomial vector fields”, J. Differential Equations, 198 (2004), 374–380 | DOI | MR | Zbl

[30] N. G. Lloyd, “A note on the number of limit cycles in certain two-dimensional systems”, J. London Math. Soc. (2), 20 (1979), 277–286 | DOI | MR | Zbl

[31] J. Margalef-Bentabol, D. Peralta-Salas, “Realization problems for limit cycles of planar polynomial vector fields”, J. Differential Equations, 260 (2016), 3844–3859 | DOI | MR | Zbl

[32] I. Niven, H. S. Zuckerman, An introduction to the theory of numbers, John Wiley and Sons, Inc., New York, 1969 | MR

[33] H. Prodinger, R. F. Tichy, “Fibonacci numbers of graphs”, Fibonacci Quart., 20 (1982), 16–21 | MR | Zbl

[34] R. Prohens, J. Torregrosa, “New lower bounds for the Hilbert numbers using reversible centers”, Nonlinearity, 32 (2019), 331–355 | DOI | MR | Zbl

[35] F. Rong, “A note on the topological classification of complex polynomial differential equations with only centre singularities”, J. Difference Equ. Appl., 18 (2012), 1947–1949 | DOI | MR | Zbl

[36] K. S. Sibirskii, “On the number of limit cycles in the neighborhood of a singular point”, Differ. Equ., 1 (1965), 36–47 | MR

[37] N. J. A. Sloane, Bijection between rooted trees and arrangements of circles, Note on A000081, (accessed 6 April 2023) https://oeis.org/A000081/a000081b.txt

[38] N. J. A. Sloane, Entry A000081, The On-Line Encyclopedia of Integer Sequences, , OEIS Foundation Inc. (accessed 6 April 2023) https://oeis.org/A000081

[39] N. J. A. Sloane, Entry A006082, The On-Line Encyclopedia of Integer Sequences, , OEIS Foundation Inc. (accessed 6 April 2023) https://oeis.org/A006082

[40] X. B. Sun, M. Han, “On the number of limit cycles of a $Z_4$-equivariant quintic near-Hamiltonian system”, Acta Math. Sin. (Engl. Ser.), 31 (2015), 1805–1824 | DOI | MR | Zbl

[41] K. Yamato, “An effective method of counting the number of limit cycles”, Nagoya Math. J., 76 (1979), 35–114 | DOI | MR | Zbl