Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four

Eduardo Sáez; Eduardo Stange; Iván Szántó

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 1, page 105-114
  • ISSN: 0011-4642

Abstract

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A class of degree four differential systems that have an invariant conic x 2 + C y 2 = 1 , C ∈ ℝ , is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.

How to cite

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Sáez, Eduardo, Stange, Eduardo, and Szántó, Iván. "Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four." Czechoslovak Mathematical Journal 57.1 (2007): 105-114. <http://eudml.org/doc/31116>.

@article{Sáez2007,
abstract = {A class of degree four differential systems that have an invariant conic $ x^2+Cy^2=1$, $C\in \{\mathbb \{R\}\}$, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.},
author = {Sáez, Eduardo, Stange, Eduardo, Szántó, Iván},
journal = {Czechoslovak Mathematical Journal},
keywords = {stability; limit cycle; center; bifurcation; stability; limit cycles; center; bifurcations},
language = {eng},
number = {1},
pages = {105-114},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four},
url = {http://eudml.org/doc/31116},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Sáez, Eduardo
AU - Stange, Eduardo
AU - Szántó, Iván
TI - Coexistence of small and large amplitude limit cycles of polynomial differential systems of degree four
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 105
EP - 114
AB - A class of degree four differential systems that have an invariant conic $ x^2+Cy^2=1$, $C\in {\mathbb {R}}$, is examined. We show the coexistence of small amplitude limit cycles, large amplitude limit cycles, and invariant algebraic curves under perturbations of the coefficients of the systems.
LA - eng
KW - stability; limit cycle; center; bifurcation; stability; limit cycles; center; bifurcations
UR - http://eudml.org/doc/31116
ER -

References

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  1. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus, Mat. Sbornik 30(72) (1952), 181–196. (1952) Zbl0059.08201MR0045893
  2. Sur les courbes définies par des équations différentielles, Acta Math. 24 (1900), 1–88. (French) (1900) 
  3. Some quadratics systems with at most one limit cycle, Dynam. report. Expositions Dynam. Systems (N.S.) 2 (1989), 61–88. (1989) MR1000976
  4. The solution of the problem of center for cubic differential systems with four invariant straight lines, An. Stiint. Univ. Al. I. Cuza Iasi, Ser. Nuova, Mat. 44 (Suppl.) (1999), 517–530. (1999) MR1814187
  5. 10.1007/BF02969386, Qual. Theory Dyn. Syst. 2 (2001), 129–143. (2001) MR1844982DOI10.1007/BF02969386
  6. 10.1016/S0362-546X(04)00278-0, Nonlinear Anal., Theory Methods Appl. 59 (2004), 673–693. (2004) MR2096323DOI10.1016/S0362-546X(04)00278-0
  7. Relative position and number of limit cycles of a quadratic differential system, Acta Math. Sin. 22 (1979), 751–758. (Chinese) (1979) MR0559742
  8. The limit cycles of some differential equations, Differ. Uravn. 8 (1972), 924–929. (Russian) (1972) 
  9. 10.1017/S0308210500028195, Proc. R. Soc. Edinb. Sect.  A 112 (1989), 113–134. (1989) Zbl0677.34034MR1007539DOI10.1017/S0308210500028195
  10. Closed orbits and straight line invariants in E 3   systems, Acta Mathematica Sci. 9 (1989), 251–261. (Chinese) (1989) 
  11. Sur les cycles limites, S. M. F. Bull. 51 (1923), 45–188. (French) (1923) MR1504823
  12. 10.1090/S0002-9904-1902-00923-3, American Bull (2) 8 (1902), 437–479. (1902) MR1557926DOI10.1090/S0002-9904-1902-00923-3
  13. 10.1093/imamat/47.2.163, IMA  J.  Appl. Math. 47 (1991), 163–171. (1991) MR1130524DOI10.1093/imamat/47.2.163
  14. Limit cycles in polynomial systems, PhD. thesis, University of Technology, Delft, 1993. (1993) 
  15. Hilbert’s 16th problem for n = 3 H ( 3 ) ≥ 11 , Kexue Tongbao 31 (1984), 718. (1984) 
  16. Etude des oscillations entreteneues, Re. générale de l’électricité 23 (1928), 901–912. (French) (1928) 
  17. 10.11650/twjm/1500575064, Taiwanese J.  Math. 7 (2003), 275–281. (2003) MR1978016DOI10.11650/twjm/1500575064
  18. 10.1016/0893-9659(95)00095-X, Appl. Math. Lett. 9 (1996), 15–18. (1996) MR1389591DOI10.1016/0893-9659(95)00095-X
  19. 10.1016/S0898-1221(02)00161-X, Computers Math. Appl. 44 (2002), 445–455. (2002) MR1912841DOI10.1016/S0898-1221(02)00161-X
  20. Five limit cycles for a simple cubic system, Publications Mathematiques 41 (1997), 199–208. (1997) MR1461651
  21. Some cubic systems with several limit cycles, Nonlinearity (1988), 653–669. (1988) MR0967475
  22. 10.1016/0893-9659(94)90005-1, Appl. Math. Lett. 7 (1994), 23–27. (1994) MR1350389DOI10.1016/0893-9659(94)90005-1
  23. Mémorie sur les courbes définies par leś équations differentialles I–VI, Oeuvre I, Gauthier-Villar, Paris, 1880–1890. (French) (1880–1890) 
  24. On dynamical systems close to Hamiltonian ones, Zh. Eksper. Teoret. Fiz. 4 (1934), 883–885. (Russian) (1934) 
  25. A bibliography of the qualitative theory of quadratic systems of differential equation in the plane, Report TU Delft 92-17, second edition, 1992. (1992) 
  26. 10.1016/S0362-546X(01)00565-X, Nonlinear Anal., Theory Methods Appl. 47 (2001), 4521–4525. (2001) MR1975846DOI10.1016/S0362-546X(01)00565-X
  27. A concrete example of the existence of four limit cycles for planar quadratics systems, Sci. Sin. XXIII (1980), 153–158. (1980) MR0574405
  28. System of equation  ( E 3 ) has five limit cycles, Acta Math. Sin. 18 (1975). (Chinese) (1975) 
  29. The n -degree differential system with 1 2 ( n - 1 ) ( n + 1 ) straight line solutions has no limit cycles, Proc. Conf. Ordinary Differential Equations and Control Theory, Wuhan 1987 (1987), 216–220. (Chinese) (1987) MR1043472
  30. 10.1080/14786442608564127, Philos. Magazine 7 (1926), 978–992. (1926) DOI10.1080/14786442608564127
  31. 10.1016/0022-0396(90)90004-9, J.  Differ. Equations 87 (1990), 305–315. (1990) Zbl0712.34044MR1072903DOI10.1016/0022-0396(90)90004-9
  32. A survey of cubic systems, Ann. Differ. Equations 7 (1991), 323–363. (1991) Zbl0747.34019MR1139341
  33. Cubic Kolmogorov differential system with two limit cycles surrounding the same focus, Ann. Differ. Equations 1 (1985), 201–207. (1985) MR0834242
  34. Twelve limit cycles in a cubic order planar system with Z 2 -symmetry, Communications on pure and applied analysis 3 (2004), 515–526. (2004) MR2098300
  35. 10.1088/0951-7715/8/5/011, Nonlinearity 8 (1995), 843–860. (1995) Zbl0837.34042MR1355046DOI10.1088/0951-7715/8/5/011
  36. A System for Doing Mathematics by Computer, Wolfram Research Mathematica, 1988. (1988) Zbl0671.65002

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