Skip to main content
SpringerLink
Log in

Some open problems in low dimensional dynamical systems

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years in my research. I believe that it is worth to think about them and, hopefully, solve some of the problems or make some substantial progress. Many of them are about planar differential equations but there are also questions about other mathematical aspects: Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials and some recreational problems with a dynamical component.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Acary, V., Bonnefon, O., Brogliato, B.: Nonsmooth Modeling and Simulation for Switched Circuits. Lecture Notes in Electrical Engineering, vol. 69. Springer, Dordrecht (2011)

  2. Álvarez, M.J., Coll, B., De Maesschalck, P., Prohens, R.: Asymptotic lower bounds on Hilbert numbers using canard cycles. J. Differ. Equ. 268(7), 3370–3391 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Álvarez, M.J., Gasull, A., Giacomini, H.: A new uniqueness criterion for the number of periodic orbits of Abel equations. J. Differ. Equ. 234(1), 161–176 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Álvarez, M.J., Gasull, A., Prohens, R.: Global behaviour of the period function of the sum of two quasi-homogeneous vector fields. J. Math. Anal. Appl. 449(2), 1553–1569 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Álvarez, M.J., Gasull, A., Yu, J.: Lower bounds for the number of limit cycles of trigonometric Abel equations. J. Math. Anal. Appl. 342(1), 682–693 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ando, N.: An umbilical point on a non-real-analytic surface. Hiroshima Math. J. 33(1), 1–14 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Andrade, K.D.S., Cespedes, O.A.R., Cruz, D.R., Novaes, D.D.: Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve. Preprint (2020)

  8. Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Translated from the Russian by F. Immirzi; Translation Edited and Abridged by W. Fishwick. Pergamon Press, Oxford (1966)

  9. Argémi, J.: Sur les points singuliers multiples de systèmes dynamiques dans \(R^{2}\). Ann. Mat. Pura Appl. 4(79), 35–69 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  10. Artigue, A.: Periodic orbits in triangular billiards. Miscelánea Mat. 59, 19–40 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Bastos, J.L.R., Buzzi, C.A., Llibre, J., Novaes, D.D.: Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold. J. Differ. Equ. 267(6), 3748–3767 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bernat, J., Llibre, J.: Counterexample to Kalman and Markus–Yamabe conjectures in dimension larger than \(3\). Dyn. Contin. Discret. Impuls. Syst. 2(3), 337–379 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Bihan, F., Rojas, J.M., Sottile, F.: On the sharpness of fewnomial bounds and the number of components of fewnomial hypersurfaces. In: Algorithms in Algebraic Geometry, vol. 146. The IMA Volumes in Mathematics and its Applications, pp. 15–20. Springer, New York (2008)

  14. Blows, T.R.: Center configurations of Hamiltonian cubic systems. Rocky Mt. J. Math. 40(4), 1111–1122 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Boshernitzan, M., Galperin, G., Krüger, T., Troubetzkoy, S.: Periodic billiard orbits are dense in rational polygons. Trans. Am. Math. Soc. 350(9), 3523–3535 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bravo, J.L., Fernández, M., Gasull, A.: Limit cycles for some Abel equations having coefficients without fixed signs. Int. J. Bifurc. Chaos Appl. Sci. Eng. 19(11), 3869–3876 (2009)

  17. Bravo, J.L., Torres, P.J.: Periodic solutions of a singular equation with indefinite weight. Adv. Nonlinear Stud. 10(4), 927–938 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brogliato, B.: Nonsmooth Mechanics: Models, Dynamics and Control. Communications and Control Engineering Series, 3rd edn. Springer, Cham (2016)

    Book  MATH  Google Scholar 

  19. Brousseau, B.A.: Palindromes by addition in base two. Math. Mag. 42, 254–256 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  20. Buzzi, C., Carvalho, Y.R., Gasull, A.: Limit cycles for some families of smooth and non-smooth planar systems. Nonlinear Anal. 207 (2021), 112298

  21. Buzzi, C.A., Gasull, A., Torregrosa, J.: Algebraic limit cycles in piecewise linear differential systems. Int. J. Bifurc. Chaos Appl. Sci. Eng. 28(3), 1850039 (2018) 14 pp

  22. Cairó, L., Llibre, J.: Phase portraits of planar semi-homogeneous vector fields. II. Nonlinear Anal. Ser. A: Theory Methods 39(3), 351–363 (2000)

  23. Casas-Alvero, E.: Higher order polar germs. J. Algebra 240(1), 326–337 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Cen, X.: New lower bound for the number of critical periods for planar polynomial systems. J. Differ. Equ. 271, 480–498 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  25. Chavarriga, J., Grau, M.: Some open problems related to 16b Hilbert problem. Sci. Ser. A Math. Sci. (N.S.) 9, 1–26 (2003)

  26. Chavarriga, J., Sabatini, M.: A survey of isochronous centers. Qual. Theory Dyn. Syst. 1(1), 1–70 (1999)

    Article  MathSciNet  Google Scholar 

  27. Cherkas, L.A., Artés, J.C., Llibre, J.: Quadratic systems with limit cycles of normal size. Bul. Acad. Ştiinţe Repub. Mold. Mat. 1, 31–46 (2003)

  28. Cherkas, L.A., Grin’, A.A.: On the Dulac function for the Kukles system. Differ. Uravn. 46(6), 811–819 (2010)

    MathSciNet  MATH  Google Scholar 

  29. Christopher, C., Li, C.: Limit Cycles of Differential Equations. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser, Basel (2007)

    Google Scholar 

  30. Christopher, C., Llibre, J., Świrszcz, G.: Invariant algebraic curves of large degree for quadratic system. J. Math. Anal. Appl. 303(2), 450–461 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Christopher, C.J., Lloyd, N.G.: Polynomial systems: a lower bound for the Hilbert numbers. Proc. R. Soc. Lond. Ser. A 450(1938), 219–224 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Cima, A., Gasull, A., and Mañosa, V. Dynamics of some rational discrete dynamical systems via invariants. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16(3), 631–645 (2006)

  33. Cima, A., Gasull, A., Mañosa, V.: On Poncelet’s maps. Comput. Math. Appl. 60(5), 1457–1464 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Cima, A., Gasull, A., Mañosa, V.: Stability index of linear random dynamical systems. Electron. J. Qual. Theory Differ. Equ. (2021, to appear)

  35. Cima, A., Gasull, A., Mañosa, V., and Mañosas, F. Different approaches to the global periodicity problem. In: Difference Equations, Discrete Dynamical Systems and Applications, vol. 180. Springer Proceedings in Mathematics and Statistics, pp. 85–106. Springer, Berlin (2016)

  36. Cima, A., Gasull, A., Mañosas, F.: On polynomial Hamiltonian planar vector fields. J. Differ. Equ. 106(2), 367–383 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  37. Cima, A., Gasull, A., Mañosas, F.: Some applications of the Euler–Jacobi formula to differential equations. Proc. Am. Math. Soc. 118(1), 151–163 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  38. Cima, A., Gasull, A., Mañosas, F.: Limit cycles for vector fields with homogeneous components. Appl. Math. (Wars.) 24(3), 281–287 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  39. Cima, A., Gasull, A., Mañosas, F.: The discrete Markus–Yamabe problem. Nonlinear Anal. Ser. A: Theory Methods 35(3), 343–354 (1999)

  40. Cima, A., Gasull, A., Mañosas, F.: A Note on LaSalle’s Problems, Ann. Polon. Math. vol. 76, pp. 33–46 (2001). Polynomial automorphisms and related topics (Kraków, 1999)

  41. Cima, A., Gasull, A., Mañosas, F.: On periodic rational difference equations of order \(k\). J. Differ. Equ. Appl. 10(6), 549–559 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Cima, A., Gasull, A., Mañosas, F.: An explicit bound of the number of vanishing double moments forcing composition. J. Differ. Equ. 255(3), 339–350 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Cima, A., Gasull, A., Mañosas, F.: Around some extensions of Casas-Alvero conjecture for non-polynomial functions. Extr. Math. 35(2), 221–228 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cima, A., Llibre, J.: Configurations of fans and nests of limit cycles for polynomial vector fields in the plane. J. Differ. Equ. 82(1), 71–97 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  45. Cima, A., van den Essen, A., Gasull, A., Hubbers, E., Mañosas, F.: A polynomial counterexample to the Markus–Yamabe conjecture. Adv. Math. 131(2), 453–457 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  46. Cipra, B., Hanson, R.M., Kolan, A.: Periodic trajectories in right-triangle billiards. Phys. Rev. E (3) 52(2), 2066–2071 (1995)

  47. Collins, C.B.: The period function of some polynomial systems of arbitrary degree. Differ. Integral Equ. 9(2), 251–266 (1996)

    MathSciNet  MATH  Google Scholar 

  48. Constantin, A., Villari, G.: Particle trajectories in linear water waves. J. Math. Fluid Mech. 10(1), 1–18 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Conti, R.: Uniformly isochronous centers of polynomial systems in \({\bf R}^2\). In: Differential Equations, Dynamical Systems, and Control Science, vol. 152. Lecture Notes in Pure and Applied Mathematics, pp. 21–31. Dekker, New York (1994)

  50. Coppel, W.A., Gavrilov, L.: The period function of a Hamiltonian quadratic system. Differ. Integral Equ. 6(6), 1357–1365 (1993)

    MathSciNet  MATH  Google Scholar 

  51. de Faria, E., Tresser, C.: On Sloane’s persistence problem. Exp. Math. 23(4), 363–382 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  52. De Maesschalck, P., Dumortier, F.: Classical Liénard equations of degree \(n\ge 6\) can have \([\frac{n-1}{2}]+2\) limit cycles. J. Differ. Equ. 250(4), 2162–2176 (2011)

    Article  MATH  Google Scholar 

  53. De Maesschalck, P., Wynen, J.: Private communication (2020)

  54. Derksen, H., van den Essen, A., Zhao, W.: The Gaussian moments conjecture and the Jacobian conjecture. Isr. J. Math. 219(2), 917–928 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  55. di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P.: Piecewise-Smooth Dynamical Systems: Theory and Applications. Applied Mathematical Sciences, vol. 163. Springer-Verlag London, Ltd., London (2008)

    MATH  Google Scholar 

  56. Dickenstein, A., Rojas, J.M., Rusek, K., Shih, J.: Extremal real algebraic geometry and \({{\cal{A}}} \)-discriminants. Mosc. Math. J. 7(3), 425–452 (2007)

  57. Draisma, J., de Jong, J.P.: On the Casas-Alvero conjecture. Eur. Math. Soc. Newsl. 80, 29–33 (2011)

    MathSciNet  MATH  Google Scholar 

  58. Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin (2006)

    MATH  Google Scholar 

  59. Dumortier, F., Panazzolo, D., Roussarie, R.: More limit cycles than expected in Liénard equations. Proc. Am. Math. Soc. 135(6), 1895–1904 (2007)

    Article  MATH  Google Scholar 

  60. Eisenbud, D., Levine, H.I.: An algebraic formula for the degree of a \({\cal C\it }^{\infty }\) map germ. Ann. Math. (2) 106(1), 19–44 (1977)

  61. Feßler, R.: A proof of the two-dimensional Markus–Yamabe stability conjecture and a generalization. Ann. Pol. Math. 62(1), 45–74 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  62. Filippov, A.F.: Differential Equations with Discontinuous Righthand Sides, vol. 18. Mathematics and Its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht (1988). Translated from the Russian

  63. Françoise, J.-P.: From Abel equations to Jacobian conjecture. Publ. Mat. 58(suppl), 209–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  64. Gantmacher, F.R.: The Theory of Matrices. vols. 1, 2. Translated by K. A. Hirsch. Chelsea Publishing Co., New York (1959)

  65. Gasull, A., Giacomini, H.: A new criterion for controlling the number of limit cycles of some generalized Liénard equations. J. Differ. Equ. 185(1), 54–73 (2002)

    Article  MATH  Google Scholar 

  66. Gasull, A., Giacomini, H.: Upper bounds for the number of limit cycles through linear differential equations. Pac. J. Math. 226(2), 277–296 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  67. Gasull, A., Giacomini, H.: Some applications of the extended Bendixson–Dulac theorem. In: Progress and Challenges in Dynamical Systems, vol. 54. Springer Proceedings in Mathematics and Statistics, pp. 233–252. Springer, Heidelberg (2013)

  68. Gasull, A., Guillamon, A.: Limit cycles for generalized Abel equations. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16(12), 3737–3745 (2006)

  69. Gasull, A., Guillamon, A., Mañosa, V., Mañosas, F.: The period function for Hamiltonian systems with homogeneous nonlinearities. J. Differ. Equ. 139(2), 237–260 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  70. Gasull, A., Liu, C., Yang, J.: On the number of critical periods for planar polynomial systems of arbitrary degree. J. Differ. Equ. 249(3), 684–692 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  71. Gasull, A., Llibre, J.: Limit cycles for a class of Abel equations. SIAM J. Math. Anal. 21(5), 1235–1244 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  72. Gasull, A., Mañosa, V.: Periodic orbits of discrete and continuous dynamical systems via Poincaré–Miranda theorem. Discrete Contin. Dyn. Syst. Ser. B 25(2), 651–670 (2020)

    MathSciNet  MATH  Google Scholar 

  73. Gasull, A., Prohens, R., Torregrosa, J.: Limit cycles for rigid cubic systems. J. Math. Anal. Appl. 303(2), 391–404 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  74. Gasull, A., Torregrosa, J., Zhang, X.: Piecewise linear differential systems with an algebraic line of separation. Electron. J. Differ. Equ. (2020). Paper No. 19, 14

  75. Giacomini, H., Grau, M.: Transversal conics and the existence of limit cycles. J. Math. Anal. Appl. 428(1), 563–586 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  76. Giné, J.: On some open problems in planar differential systems and Hilbert’s 16th problem. Chaos Solitons Fractals 31(5), 1118–1134 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  77. Glutsyuk, A.A.: The asymptotic stability of the linearization of a vector field on the plane with a singular point implies global stability. Funktsional. Anal. i Prilozhen. 29(4), 17–30 (1995)

  78. Graf von Bothmer, H.-C., Labs, O., Schicho, J., van de Woestijne, C.: The Casas-Alvero conjecture for infinitely many degrees. J. Algebra 316(1), 224–230 (2007)

  79. Grau, M., Villadelprat, J.: Bifurcation of critical periods from Pleshkan’s isochrones. J. Lond. Math. Soc. (2) 81(1), 142–160 (2010)

  80. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York (1978)

  81. Gutiérrez, C.: A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 12(6), 627–671 (1995)

  82. Gutiérrez, C., Sánchez-Bringas, F.: Planar vector field versions of Carathéodory’s and Loewner’s conjectures. In: Proceedings of the Symposium on Planar Vector Fields (Lleida, 1996). Publ. Mat., vol. 41, no. 1, pp. 169–179 (1997)

  83. Gutiérrez, C., Sotomayor, J.: Lines of curvature, umbilic points and Carathéodory conjecture. Resenhas 3(3), 291–322 (1998)

    MathSciNet  MATH  Google Scholar 

  84. Haas, B.: A simple counterexample to Kouchnirenko’s conjecture. Beiträge Algebra Geom. 43(1), 1–8 (2002)

    MathSciNet  MATH  Google Scholar 

  85. Halbeisen, L., Hungerbühler, N.: On periodic billiard trajectories in obtuse triangles. SIAM Rev. 42(4), 657–670 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  86. Hille, E.: Ordinary Differential Equations in the Complex Domain. Dover Publications, Inc., Mineola (1997). Reprint of the 1976 original

  87. Huang, J., Torregrosa, J., Villadelprat, J.: On the number of limit cycles in generalized Abel equations. SIAM J. Appl. Dyn. Syst. 19(4), 2343–2370 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  88. Ilyashenko, Y.: Centennial history of Hilbert’s 16th problem. Bull. Am. Math. Soc. (N.S.) 39(3), 301–354 (2002)

  89. Ilyashenko, Y., Yakovenko, S.: Lectures on Analytic Differential Equations. Graduate Studies in Mathematics, vol. 86. American Mathematical Society, Providence (2008)

  90. Kaloshin, V., Sorrentino, A.: On the integrability of Birkhoff billiards. Philos. Trans. Roy. Soc. A 376(2131), 20170419, 16 pp (2018)

  91. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems, vol. 54. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1995). With a supplementary chapter by Katok and Leonardo Mendoza

  92. Khovanskiĭ, A.G.: The index of a polynomial vector field. Funktsional. Anal. i Prilozhen. 13(1), 49–58 (1979)

  93. Khovanskiĭ, A.G.: A class of systems of transcendental equations. Dokl. Akad. Nauk SSSR 255(4), 804–807 (1980)

    MathSciNet  Google Scholar 

  94. Kunze, M.: Non-Smooth Dynamical Systems. Lecture Notes in Mathematics, vol. 1744. Springer, Berlin (2000)

  95. Lagarias, J.C.: The \(3x+1\) problem: an annotated bibliography (1963–1999). In: The Ultimate Challenge: The \(3x+1\) Problem, pp. 267–341. American Mathematical Society, Providence (2010)

  96. LaSalle, J.P.: The Stability of Dynamical Systems. Society for Industrial and Applied Mathematics, Philadelphia (1976). With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics

  97. Lehmer, D.: Sujets d’étude. Sphinx 8, 12–13 (1938)

    Google Scholar 

  98. Li, C., Liu, C., Yang, J.: A cubic system with thirteen limit cycles. J. Differ. Equ. 246(9), 3609–3619 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  99. Li, C., Llibre, J.: Uniqueness of limit cycles for Liénard differential equations of degree four. J. Differ. Equ. 252(4), 3142–3162 (2012)

    Article  MATH  Google Scholar 

  100. Li, J.: Hilbert’s 16th problem and bifurcations of planar polynomial vector fields. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13(1), 47–106 (2003)

  101. Li, T.-Y., Rojas, J.M., Wang, X.: Counting real connected components of trinomial curve intersections and \(m\)-nomial hypersurfaces. Discret. Comput. Geom. 30(3), 379–414 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  102. Lins, A., de Melo, W., Pugh, C.C.: On Liénard’s equation. In: Geometry and Topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976). Lecture Notes in Mathematics, vol. 597, pp. 335–357. Springer, Berlin (1977)

  103. Lins Neto, A.: On the number of solutions of the equation \(dx/dt=\sum ^{n}_{j=0}\, a_{j}(t)x^{j}\), \(0\le t\le 1\), for which \(x(0)=x(1)\). Invent. Math. 59(1), 67–76 (1980)

    MathSciNet  MATH  Google Scholar 

  104. Llibre, J., Martinez-Alfaro, J.: An upper bound of the index of an equilibrium point in the plane. J. Differ. Equ. 253(8), 2460–2473 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  105. Llibre, J., Ponce, E.: Three nested limit cycles in discontinuous piecewise linear differential systems with two zones. Dyn. Contin. Discret. Impuls. Syst. Ser. B Appl. Algorithms 19(3), 325–335 (2012)

  106. Llibre, J., Zhang, X.: A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems. Expo. Math. (2021, to appear)

  107. Llibre, J., Zhang, X.: Limit cycles of the classical Liénard differential systems: a survey on the Lins Neto, de Melo and Pugh’s conjecture. Expo. Math. 35(3), 286–299 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  108. Mardešić, P., Marín, D., Villadelprat, J.: The period function of reversible quadratic centers. J. Differ. Equ. 224(1), 120–171 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  109. Mawhin, J.: Simple proofs of the Hadamard and Poincaré–Miranda theorems using the Brouwer fixed point theorem. Am. Math. Mon. 126(3), 260–263 (2019)

    Article  MATH  Google Scholar 

  110. Miranda, C.: Un’osservazione su un teorema di Brouwer. Boll. Un. Mat. Ital. 2(3), 5–7 (1940)

    MathSciNet  MATH  Google Scholar 

  111. Nishiyama, Y.: Numerical palindromes and the 196 problem. Int. J. Pure Appl. Math. 80, 37–384 (2012)

    MATH  Google Scholar 

  112. Pakovich, F. On rational functions orthogonal to all powers of a given rational function on a curve. Mosc. Math. J. 13(4), 693–731 (2013)

  113. Panov, A.A.: On the number of periodic solutions of polynomial differential equations. Mat. Zametki 64(5), 720–727 (1998)

    MathSciNet  Google Scholar 

  114. Perko, L.M.: Limit cycles of quadratic systems in the plane. Rocky Mt. J. Math. 14(3), 619–645 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  115. Prohens, R., Torregrosa, J.: New lower bounds for the Hilbert numbers using reversible centers. Nonlinearity 32(1), 331–355 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  116. Rothe, F.: The periods of the Volterra–Lotka system. J. Reine Angew. Math. 355, 129–138 (1985)

    MathSciNet  MATH  Google Scholar 

  117. Sansone, G., Conti, R.: Non-Linear Differential Equations, revised edn. Translated from the Italian by Ainsley H. Diamond. International Series of Monographs in Pure and Applied Mathematics, vol. 67. A Pergamon Press Book. The Macmillan Co., New York (1964)

  118. Schwartz, R.E.: Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Exp. Math. 18(2), 137–171 (2009)

  119. Singmaster, D.: Research problems: how often does an integer occur as a binomial coefficient? Am. Math. Mon. 78(4), 385–386 (1971)

    Article  MathSciNet  Google Scholar 

  120. Singmaster, D.: Repeated binomial coefficients and Fibonacci numbers. Fibonacci Q. 13(4), 295–298 (1975)

    MathSciNet  MATH  Google Scholar 

  121. Sloane, N.: The persistence of a number. J. Recreat. Math. 6, 97–98 (1973)

    Google Scholar 

  122. Smale, S.: Mathematical problems for the next century. Math. Intell. 20(2), 7–15 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  123. Tabachnikov, S.: Billiards. Panor. Synth. 1, vi+142 (1995)

  124. Titus, C.J.: A proof of a conjecture of Loewner and of the conjecture of Carathéodory on umbilic points. Acta Math. 131, 43–77 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  125. Trigg, C.W.: Palindromes by addition. Math. Mag. 40(1), 26–28 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  126. Ureña, A.J.: Periodic solutions of singular equations. Topol. Methods Nonlinear Anal. 47(1), 55–72 (2016)

    MathSciNet  MATH  Google Scholar 

  127. Ureña, A.J.: A counterexample for singular equations with indefinite weight. Adv. Nonlinear Stud. 17(3), 497–516 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  128. van den Essen, A.: Polynomial Automorphisms and the Jacobian Conjecture. Progress in Mathematics, vol. 190. Birkhäuser, Basel (2000)

  129. Waldvogel, J.: The period in the Lotka–Volterra system is monotonic. J. Math. Anal. Appl. 114(1), 178–184 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  130. Wang, S., Yang, J.: Period functions and critical periods of piecewise linear system. Electron. J. Differ. Equ. 2020(79), 12 pp (2020)

Download references

Acknowledgements

The author thanks José Luis Bravo and Joan Torregrosa for their feedback on previous versions of this paper.

This work has received funding from the Ministerio de Ciencia e Innovación (PID2019-104658GB-I00 Grant) and the Agència de Gestió d’Ajuts Universitaris i de Recerca (2017 SGR 1617 Grant).

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici Cc, 08193, Cerdanyola del Vallès, Barcelona, Spain

    Armengol Gasull

  2. Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193, Cerdanyola del Vallès, Barcelona, Spain

    Armengol Gasull

Authors
  1. Armengol Gasull
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Armengol Gasull.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gasull, A. Some open problems in low dimensional dynamical systems. SeMA 78, 233–269 (2021). https://doi.org/10.1007/s40324-021-00244-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-021-00244-3

Keywords

Mathematics Subject Classification

Search

Navigation