Counting configurations of limit cycles and centers. (English) Zbl 1538.34118
Summary: 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.
MSC:
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
Software:
OEISReferences:
| [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. · Zbl 1460.34039 |
| [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. · Zbl 1201.37022 |
| [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. · Zbl 1493.37001 |
| [4] | N. N. Bautin. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type. Mat. Sb. 30 (1952), 181-196, Amer. Math. Soc. Transl. 100 (1954), 1-19. · Zbl 0059.08201 |
| [5] | N. L. Biggs, E. K. Lloyd, R. Wilson. Graph Theory, 1736-1936. Oxford University Press, 1974. |
| [6] | T. R. Blows. Center configurations of Hamiltonian cubic systems. Rocky Mountain J. Math. 40 (2010), 1111-1122. · Zbl 1210.34042 |
| [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. Also in The Collected Math. Papers. Vol. 9, 427-460. Cambridge University Press, Cambridge, 2009. |
| [8] | L. A. Cherkas. Dulac function for polynomial autonomous systems on a plane. Differential Equations 33 (1997), 692-701. · Zbl 0911.34026 |
| [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. · Zbl 0839.34033 |
| [10] | A. Cima, A. Gasull, F. Mañosas. Some applications of the Euler-Jacobi formula to differ-ential equations. Proc. Amer. Math. Soc. 118 (1993), 151-163. · Zbl 0787.34030 |
| [11] | A. Cima, A. Gasull, F. Mañosas. On polynomial Hamiltonian planar vector fields. J. Dif-ferential Equations 106 (1993), 367-383. · Zbl 0792.34026 |
| [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. · Zbl 0703.34040 |
| [13] | F. Dumortier, J. Llibre and J. C. Artés. Qualitative theory of planar differential systems. UniversiText, Springer-Verlag, New York, 2006. · Zbl 1110.34002 |
| [14] | R. M. Fedorov. Upper bounds for the number of orbital topological types of polynomial vector fields on the plane “modulo limit cycles”. (Russian) Translation in Proc. Steklov Inst. Math. 254 (2006), 238-254. · Zbl 1351.34029 |
| [15] | P. Flajolet, R. Sedgewick. Analytic Combinatorics. Cambridge University Press, Cam-bridge, 2009. · Zbl 1165.05001 |
| [16] | A. Gasull. Difference equations everywhere: some motivating examples. In Recent Progress in Difference Equations, Discrete Dynamical Systems and Applications (Proceedings of ICDEA2017), 129-167. Springer Proceedings in Mathematics & Statistics, 2019. · Zbl 1426.39001 |
| [17] | A. Gasull. Some open problems in low dimensional dynamical systems. SeMA J. 78 (2021), 233-269. · Zbl 1487.37024 |
| [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. · Zbl 1032.34028 |
| [19] | A. Genitrini. Full asymptotic expansion for Pólya structures. Proceedings of the 27th Inter-national Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms, Kraków, Poland, 4-8 July 2016. arXiv:1605.00837 [math.CO] · Zbl 1409.05053 |
| [20] | P. Griffiths, J. Harris. Principles of algebraic geometry. Wiley-Interscience [John Wiley & Sons], Pure and Applied Mathematics, New York, 1978. · Zbl 0408.14001 |
| [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. |
| [22] | M. Han, J. Li. Lower bounds for the Hilbert number of polynomial systems. J. Differential Equations 252 (2012), 3278-3304. · Zbl 1247.34047 |
| [23] | T. Johnson. A quartic system with twenty-six limit cycles. Exp. Math. 20 (2011), 323-328. · Zbl 1267.34059 |
| [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. · Zbl 1193.34061 |
| [25] | A. G. Khovanskiȋ. The index of a polynomial vector field. Funktsional. Anal. i Prilozhen. 13 (1979), 49-58, 96. · Zbl 0422.57009 |
| [26] | J. Li. Hilbert’s 16th problem and bifurcations of planar vector fields. Inter. J. Bifur. and Chaos 13 (2003), 47-106. · Zbl 1063.34026 |
| [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. · Zbl 1050.34039 |
| [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. · Zbl 1334.34070 |
| [29] | J. Llibre, G. Rodríguez. Configurations of limit cycles and planar polynomial vector fields. J. Differential Equations 198 (2004), 374-380. · Zbl 1055.34061 |
| [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. · Zbl 0407.34027 |
| [31] | J. Margalef-Bentabol, D. Peralta-Salas. Realization problems for limit cycles of planar polynomial vector fields. J. Differential Equations 260 (2016), 3844-3859. · Zbl 1338.34072 |
| [32] | I. Niven, H. S. Zuckerman. An introduction to the theory of numbers. John Wiley and Sons, Inc., New York, 1969. · Zbl 0186.36601 |
| [33] | H. Prodinger, R. F. Tichy. Fibonacci numbers of graphs. Fibonacci Quart. 20 (1982), 16-21. · Zbl 0475.05046 |
| [34] | R. Prohens, J. Torregrosa. New lower bounds for the Hilbert numbers using reversible centers. Nonlinearity 32 (2019), 331-355. · Zbl 1414.34024 |
| [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. · Zbl 1333.37044 |
| [36] | K. S. Sibirskii. On the number of limit cycles in the neighborhood of a singular point. Differ. Equ. 1 (1965), 36-47. |
| [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 Z4-equivariant quintic near-Hamiltonian system. Acta Math. Sin. (Engl. Ser.) 31 (2015), 1805-1824. · Zbl 1351.34035 |
| [41] | K. Yamato. An effective method of counting the number of limit cycles. Nagoya Math. J. 76 (1979), 35-114. · Zbl 0422.34036 |
| [42] | A. GASULL, A. GUILLAMON, V.MAÑOSA |
| [43] | Armengol Gasull (1,4) |
| [44] | Departament de Matemàtiques, Universitat Autònoma de Barcelona. Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès, Spain. E-mail: gasull@mat.uab.cat |
| [45] | Departament de Matemàtiques, Universitat Politècnica de Catalunya. EPSEB. Av. Dr. Marañón 44-50, 08028 Barcelona, Spain. E-mail: antoni.guillamon@upc.edu |
| [46] | Víctor Mañosa (3,5) |
| [47] | Departament de Matemàtiques, Universitat Politècnica de Catalunya. |
| [48] | ESEIAAT. Colom 11, 08222 Terrassa, Spain. E-mail: victor.manosa@upc.edu |
| [49] | Centre de Recerca Matemàtica. Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès, Spain. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
