An analytical method of the degenerate Hopf bifurcation is proposed for vibro-impact systems. The phenomenon of the bifurcation and its complicated dynamics are observed. This type of bifurcation originates multi-coexisting solutions dependent of the initial state of the system.

1.
Luo
,
G. W.
, and
Xie
,
J. H.
,
1998
, “
Hopf Bifurcation of a Two-Degree-of-freedom Vibrato Impact System
,”
J. Sound Vib.
,
213
(
3
), pp.
391
408
.
2.
Luo
,
G. W.
, and
Xie
,
J. H.
,
2002
, “
Hopf Bifurcation and Chaos of a Two-Degree-of-Freedom Vibro-Impact System in Two Strong Resonance Cases
,”
Int. J. Non-Linear Mech.
,
37
, pp.
19
34
.
3.
Aidanpaa
,
J. O.
, and
Gupta
,
R. B.
,
1992
, “
Periodic and Chaotic Behavior of a Threshold-Limited Two-Degree-of-Freedom System
,”
J. Sound Vib.
,
165
(
2
), pp.
305
327
.
4.
Luo
,
G. W.
,
Xie
,
J. H.
, and
Guo
,
S. H. L.
,
2001
, “
Periodic Motions and Global Bifurcations of a Two-degree-of-Freedom System With Plastic Vibro-Impact
,”
J. Sound Vib.
,
240
(
5
), pp.
837
858
.
5.
Chenciner
,
A.
,
1985
, “
Bifurcations de Points Fixes Elliptiques I-Courbes Invariantes
,”
IHES Pub. Math.
,
61
, pp.
67
127
.
6.
Carr, J., 1981, Applications of Center Manifold Theory (Applied Mathematical Sciences 35), Springer-Verlag, New York, pp. 33–36.
7.
Iooss, G., 1979, Bifurcation of Maps and Applications (Mathematics Studies 36), North-Holland, Amsterdam.
8.
Wen
,
G. L.
, and
Xu
,
D.
,
2003
, “
Control of Degenerate Hopf Bifurcations in Three-Dimensional Maps
,”
Chaos
,
13
(
2
), pp.
486
494
.
You do not currently have access to this content.