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Volumn 252, Issue 4, 2012, Pages 3278-3304

Lower bounds for the Hilbert number of polynomial systems

Author keywords

Hilbert number; Limit cycle bifurcation; Lower bound; Polynomial system

Indexed keywords


EID: 84455208077     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2011.11.024     Document Type: Article
Times cited : (85)

References (36)
  • 2
    • 0001287885 scopus 로고
    • The relative position and the number of limit cycles of a quadratic differential system
    • Chen L., Wang M. The relative position and the number of limit cycles of a quadratic differential system. Acta Math. Sinica 1979, 22:751-758.
    • (1979) Acta Math. Sinica , vol.22 , pp. 751-758
    • Chen, L.1    Wang, M.2
  • 3
    • 0029142118 scopus 로고
    • Polynomial systems: A lower bound for the Hilbert numbers
    • Christopher C.J., Lloyd N.G. Polynomial systems: A lower bound for the Hilbert numbers. Proc. R. Soc. Lond. Ser. A 1995, 450:219-224.
    • (1995) Proc. R. Soc. Lond. Ser. A , vol.450 , pp. 219-224
    • Christopher, C.J.1    Lloyd, N.G.2
  • 5
    • 0002383108 scopus 로고    scopus 로고
    • Cyclicity of planar homoclinic loops and quadratic integrable systems
    • Han M. Cyclicity of planar homoclinic loops and quadratic integrable systems. Sci. China Ser. A 1997, 40:1247-1258.
    • (1997) Sci. China Ser. A , vol.40 , pp. 1247-1258
    • Han, M.1
  • 7
    • 34548395982 scopus 로고    scopus 로고
    • On the number of limit cycles in near-Hamiltonian polynomial systems
    • Han M., Chen G., Sun C. On the number of limit cycles in near-Hamiltonian polynomial systems. Internat. J. Bifur. Chaos 2007, 17(6):2033-2047.
    • (2007) Internat. J. Bifur. Chaos , vol.17 , Issue.6 , pp. 2033-2047
    • Han, M.1    Chen, G.2    Sun, C.3
  • 8
    • 39449110485 scopus 로고    scopus 로고
    • Bifurcation of limit cycles in a 4th-order near-Hamiltonian polynomial systems
    • Han M., Shang D., Wang Z., Yu P. Bifurcation of limit cycles in a 4th-order near-Hamiltonian polynomial systems. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2007, 17(11):4117-4144.
    • (2007) Internat. J. Bifur. Chaos Appl. Sci. Engrg. , vol.17 , Issue.11 , pp. 4117-4144
    • Han, M.1    Shang, D.2    Wang, Z.3    Yu, P.4
  • 9
    • 18144375228 scopus 로고    scopus 로고
    • A new cubic system having eleven limit cycles
    • Han M., Wu Y., Bi P. A new cubic system having eleven limit cycles. Discrete Contin. Dyn. Syst. 2005, 12(4):675-686.
    • (2005) Discrete Contin. Dyn. Syst. , vol.12 , Issue.4 , pp. 675-686
    • Han, M.1    Wu, Y.2    Bi, P.3
  • 10
    • 14844304659 scopus 로고    scopus 로고
    • On the number and distribution of limit cycles in a cubic system
    • Han M., Zhang T., Zang H. On the number and distribution of limit cycles in a cubic system. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2004, 14(12):4285-4292.
    • (2004) Internat. J. Bifur. Chaos Appl. Sci. Engrg. , vol.14 , Issue.12 , pp. 4285-4292
    • Han, M.1    Zhang, T.2    Zang, H.3
  • 11
    • 0000832248 scopus 로고
    • Mathematical problems
    • (M. Newson, Transl.)
    • Hilbert D. Mathematical problems. Bull. Amer. Math. 1902, 8:437-479. (M. Newson, Transl.).
    • (1902) Bull. Amer. Math. , vol.8 , pp. 437-479
    • Hilbert, D.1
  • 12
    • 0003137914 scopus 로고
    • The origin of limit cycles under perturbation of the equation dw/dz=-Rz/Rw, where R(z,w) is a polynomial
    • Ilyashenko Yu.S. The origin of limit cycles under perturbation of the equation dw/dz=-Rz/Rw, where R(z,w) is a polynomial. Math. USSR-Sb. 1969, 7:353-364.
    • (1969) Math. USSR-Sb. , vol.7 , pp. 353-364
    • Ilyashenko, Y.1
  • 13
    • 0003278590 scopus 로고
    • Finiteness for Limit Cycles
    • Amer. Math. Soc., Providence, RI
    • Ilyashenko Yu.S. Finiteness for Limit Cycles. Transl. Math. Monogr. 1991, vol. 94. Amer. Math. Soc., Providence, RI.
    • (1991) Transl. Math. Monogr. , vol.94
    • Ilyashenko, Y.1
  • 14
    • 0035997405 scopus 로고    scopus 로고
    • Centennial history of Hilbert's 16th problem
    • Ilyashenko Yu.S. Centennial history of Hilbert's 16th problem. Bull. Amer. Math. Soc. (N.S.) 2002, 39(3):301-354.
    • (2002) Bull. Amer. Math. Soc. (N.S.) , vol.39 , Issue.3 , pp. 301-354
    • Ilyashenko, Y.1
  • 15
    • 77954594525 scopus 로고    scopus 로고
    • An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7
    • Johnson T., Tucker W. An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7. Internat. J. Bifur. Chaos 2010, 20(5):1451-1458.
    • (2010) Internat. J. Bifur. Chaos , vol.20 , Issue.5 , pp. 1451-1458
    • Johnson, T.1    Tucker, W.2
  • 16
    • 63149147133 scopus 로고    scopus 로고
    • A cubic system with thirteen limit cycles
    • Li C., Liu C., Yang J. A cubic system with thirteen limit cycles. J. Differential Equations 2009, 246:3609-3619.
    • (2009) J. Differential Equations , vol.246 , pp. 3609-3619
    • Li, C.1    Liu, C.2    Yang, J.3
  • 17
    • 0037246396 scopus 로고    scopus 로고
    • Hilbert's 16th problem and bifurcations of planar polynomial vector fields
    • Li J. Hilbert's 16th problem and bifurcations of planar polynomial vector fields. Internat. J. Bifur. Chaos 2003, 13(1):47-106.
    • (2003) Internat. J. Bifur. Chaos , vol.13 , Issue.1 , pp. 47-106
    • Li, J.1
  • 18
    • 52649094247 scopus 로고    scopus 로고
    • 6-equivariant planar vector field of degree 5
    • 6-equivariant planar vector field of degree 5. Sci. China Ser. A 2002, 45(7):817-826.
    • (2002) Sci. China Ser. A , vol.45 , Issue.7 , pp. 817-826
    • Li, J.1    Chan, H.2    Chung, K.3
  • 19
    • 84896693675 scopus 로고    scopus 로고
    • Some lower bounds for H(n) in Hilbert's 16th problem
    • Li J., Chan H., Chung K. Some lower bounds for H(n) in Hilbert's 16th problem. Qual. Theory Dyn. Syst. 2003, 3:345-360.
    • (2003) Qual. Theory Dyn. Syst. , vol.3 , pp. 345-360
    • Li, J.1    Chan, H.2    Chung, K.3
  • 20
    • 0003132091 scopus 로고
    • Planar cubic Hamiltonian systems and distribution of limit cycles of (E3)
    • Li J., Li C. Planar cubic Hamiltonian systems and distribution of limit cycles of (E3). Acta Math. Sinica 1985, 28(4):509-521.
    • (1985) Acta Math. Sinica , vol.28 , Issue.4 , pp. 509-521
    • Li, J.1    Li, C.2
  • 21
    • 33745295874 scopus 로고    scopus 로고
    • 2-equivariant planar polynomial vector field of degree 7
    • 2-equivariant planar polynomial vector field of degree 7. Internat. J. Bifur. Chaos 2006, 16(4):925-943.
    • (2006) Internat. J. Bifur. Chaos , vol.16 , Issue.4 , pp. 925-943
    • Li, J.1    Zhang, M.2    Li, S.3
  • 22
    • 84905098097 scopus 로고    scopus 로고
    • q-equivariant planar polynomial vector fields
    • q-equivariant planar polynomial vector fields. Qual. Theory Dyn. Syst. 2010, 9:167-219.
    • (2010) Qual. Theory Dyn. Syst. , vol.9 , pp. 167-219
    • Li, J.1    Liu, Y.2
  • 23
    • 0001165830 scopus 로고
    • On the number of limit cycles of a differential equation in a neighbourhood of a singular point
    • (in Russian)
    • Otrokov N.T. On the number of limit cycles of a differential equation in a neighbourhood of a singular point. Mat. Sb. 1954, 34:127-144. (in Russian).
    • (1954) Mat. Sb. , vol.34 , pp. 127-144
    • Otrokov, N.T.1
  • 24
    • 0001304970 scopus 로고
    • Algebraic and geometric aspects of the theory of polynomial vector fields
    • Kluwer Academic, London, D. Schlomiuk (Ed.) Bifurcations and Periodic Orbits of Vector Fields
    • Schlomiuk D. Algebraic and geometric aspects of the theory of polynomial vector fields. NATO ASI Ser. C 1993, vol. 408:429-467. Kluwer Academic, London. D. Schlomiuk (Ed.).
    • (1993) NATO ASI Ser. C , vol.408 , pp. 429-467
    • Schlomiuk, D.1
  • 25
    • 0001012446 scopus 로고
    • A concrete example of the existence of four limit cycles for plane quadratic systems
    • Shi S. A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica 1980, 23:153-158.
    • (1980) Sci. Sinica , vol.23 , pp. 153-158
    • Shi, S.1
  • 26
    • 0002587254 scopus 로고    scopus 로고
    • Mathematical problems for the next century
    • Smale S. Mathematical problems for the next century. Math. Intelligencer 1998, 20:7-15.
    • (1998) Math. Intelligencer , vol.20 , pp. 7-15
    • Smale, S.1
  • 28
    • 20444455852 scopus 로고    scopus 로고
    • Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation
    • Wang S., Yu P. Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation. Chaos Solitons Fractals 2005, 26(5):1317-1335.
    • (2005) Chaos Solitons Fractals , vol.26 , Issue.5 , pp. 1317-1335
    • Wang, S.1    Yu, P.2
  • 29
    • 33745121921 scopus 로고    scopus 로고
    • Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11
    • Wang S., Yu P. Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11. Chaos Solitons Fractals 2006, 30(3):606-621.
    • (2006) Chaos Solitons Fractals , vol.30 , Issue.3 , pp. 606-621
    • Wang, S.1    Yu, P.2
  • 30
    • 33745297890 scopus 로고    scopus 로고
    • 10-equivariant vector fields of degree 9
    • 10-equivariant vector fields of degree 9. Internat. J. Bifur. Chaos 2006, 16(8):2309-2324.
    • (2006) Internat. J. Bifur. Chaos , vol.16 , Issue.8 , pp. 2309-2324
    • Wang, S.1    Yu, P.2    Li, J.3
  • 31
    • 52149110343 scopus 로고    scopus 로고
    • On the number and distributions of limit cycles in a quintic planar vector field
    • Wu Y., Gao Y., Han M. On the number and distributions of limit cycles in a quintic planar vector field. Internat. J. Bifur. Chaos 2008, 18:1939-1955.
    • (2008) Internat. J. Bifur. Chaos , vol.18 , pp. 1939-1955
    • Wu, Y.1    Gao, Y.2    Han, M.3
  • 34
    • 24144462859 scopus 로고    scopus 로고
    • Twelve limit cycles in a cubic case of the 16th Hilbert problem
    • Yu P., Han M. Twelve limit cycles in a cubic case of the 16th Hilbert problem. Internat. J. Bifur. Chaos 2005, 15(7):2191-2205.
    • (2005) Internat. J. Bifur. Chaos , vol.15 , Issue.7 , pp. 2191-2205
    • Yu, P.1    Han, M.2
  • 35
    • 33846866253 scopus 로고    scopus 로고
    • The number and distributions of limit cycles for a class of quintic near-Hamiltonian systems
    • Zang H., Han M., Zhang T., Tade M. The number and distributions of limit cycles for a class of quintic near-Hamiltonian systems. Comput. Math. Appl. 2006, 52:1577-1594.
    • (2006) Comput. Math. Appl. , vol.52 , pp. 1577-1594
    • Zang, H.1    Han, M.2    Zhang, T.3    Tade, M.4
  • 36
    • 2942666280 scopus 로고    scopus 로고
    • Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations
    • Zhang T., Han M., Zang H., Meng X. Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations. Chaos Solitons Fractals 2004, 22:1127-1138.
    • (2004) Chaos Solitons Fractals , vol.22 , pp. 1127-1138
    • Zhang, T.1    Han, M.2    Zang, H.3    Meng, X.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.