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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to be sufficiently small, some classic bifurcation methods for small parameters, such as the Melnikov method, are no longer valid.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.
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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
Affiliated Colleges: