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author:

Chen, Hebai (Chen, Hebai.) [1] | Chen, Xingwu (Chen, Xingwu.) [2]

Indexed by:

Scopus SCIE

Abstract:

The degenerate Bogdanov-Takens system (x) over dot = y-(a(1)x+a(2)x(3)), (y) over dot = a(3)x + a(4)x(3) has two normal forms, one of which is investigated in [Disc. Cont. Dyn. Syst. B (22)2017, 1273-1293] and global behavior is analyzed for general parameters. To continue this work, in this paper we study the other normal form and perform all global phase portraits on the Poincare disc. Since the parameters are not restricted to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid. We find necessary and sufficient conditions for existences of limit cycles and homoclinic loops respectively by constructing a distance function among orbits on the vertical isocline curve and further give the number of limit cycles for parameters in different regions. Finally we not only give the global bifurcation diagram, where global existences and monotonicities of the homoclinic bifurcation curve and the double limit cycle bifurcation curve are proved, but also classify all global phase portraits.

Keyword:

Bifurcation degenerate Bogdanov-Takens system figure-eight loop global phase portrait limit cycle

Community:

  • [ 1 ] [Chen, Hebai]Fuzhou Univ, Coll Math & Comp Sci, Fuzhou 350116, Fujian, Peoples R China
  • [ 2 ] [Chen, Xingwu]Sichuan Univ, Dept Math, Chengdu 610064, Sichuan, Peoples R China

Reprint 's Address:

  • [Chen, Xingwu]Sichuan Univ, Dept Math, Chengdu 610064, Sichuan, Peoples R China

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Source :

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B

ISSN: 1531-3492

Year: 2018

Issue: 10

Volume: 23

Page: 4141-4170

1 . 0 0 8

JCR@2018

1 . 3 0 0

JCR@2023

ESI Discipline: MATHEMATICS;

ESI HC Threshold:68

JCR Journal Grade:3

CAS Journal Grade:3

Cited Count:

WoS CC Cited Count: 12

SCOPUS Cited Count: 12

ESI Highly Cited Papers on the List: 0 Unfold All

WanFang Cited Count:

Chinese Cited Count:

30 Days PV: 1

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