In this paper, the dynamics near a 2:3 resonant Hopf-Hopf bifurcation is studied. The main result is the identification of a distinctive structure connecting 1:2 and 1:3 strong resonances of limit cycles. This structure is found near the Hopf-Hopf point revealing that it may be associated to the resonant case, and may provide useful information about the dynamics generated by this codimension 3 bifurcation.
REFERENCES
1.
K. B.
Hilger
and D. S.
Luciani
, “Forced oscillators: A detailed numerical analysis
,” M.S. thesis, Technical University of Denmark
, 1998
.2.
F. O. O.
Wagener
, “Semi-local analysis of the and resonances in quasi-periodically forced systems
,” in Global Analysis of Dynamical Systems, Festschrift Dedicated to Floris Takens for His 60th Birthday
, edited by H. W.
Broer
, B.
Krauskopf
, and G.
Vegter
(Institute of Physics
, Bristol
, 2001
), pp. 113
–129
.3.
H. W.
Broer
, R.
van Dijk
, and R.
Vitolo
, “Survey of strong normal-internal resonances in quasi-periodically driven oscillators for
,” in Proceedings of the International Conference on SPT 2007
, edited by G.
Gaeta
, R.
Vitolo
, and S.
Walcher
(World Scientific
, Otranto, Italy
, 2007
), pp. 45
–55
.4.
H. W.
Broer
, V.
Naudot
, R.
Roussarie
, K.
Saleh
, and F. O. O.
Wagener
, “Organising centres in the semi-global analysis of dynamical systems
,” Int. J. Appl. Math. Stat.
12
, 7
(2007
).5.
P.
Yu
, “Analysis on double Hopf bifurcation using computer algebra with the aid of multiple scales
,” Nonlinear Dyn.
27
, 19
(2002
).6.
P. A.
Chamara
and B. D.
Coller
, “A study of double flutter
,” J. Fluids Struct.
19
, 863
(2004
).7.
J.
Xie
and W.
Ding
, “Hopf-Hopf bifurcation and invariant torus of a vibro-impact system
,” Int. J. Non-Linear Mech.
40
, 531
(2005
).8.
G.
Revel
, D. M.
Alonso
, and J. L.
Moiola
, “A gallery of oscillations in a resonant electric circuit: Hopf-Hopf and fold-flip interactions
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
18
, 481
(2008
).9.
S.
Ma
, Q.
Lu
, and Z.
Feng
, “Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control
,” J. Math. Anal. Appl.
338
, 993
(2008
).10.
A.
Sterk
, R.
Vitolo
, H.
Broer
, C.
Simó
, and H.
Dijkstra
, “New nonlinear mechanisms of midlatitude atmospheric low-frequency variability
,” Physica D
239
, 702
(2010
).11.
Y. A.
Kuznetsov
, Elements of Applied Bifurcation Theory
, 3rd ed. (Springer-Verlag
, New York
, 2004
).12.
S.
Chow
, C.
Li
, and D.
Wang
, Normal Forms and Bifurcation of Planar Vector Fields
(Cambridge University Press
, Cambridge, England
, 1994
).13.
S. A.
van Gils
, M.
Krupa
, and W. F.
Langford
, “Hopf bifurcation with non-semisimple 1:1 resonance
,” Nonlinearity
3
, 825
(1990
).14.
N. S.
Namachchivaya
, M. M.
Doyle
, W. F.
Langford
, and N. W.
Evans
, “Normal form for generalized Hopf bifurcation with non-semisimple 1:1 resonance
,” Z. Angew. Math. Phys.
45
, 312
(1994
).15.
E.
Knobloch
and M. R. E.
Proctor
, “The double Hopf bifurcation with 2:1 resonance
,” Proc. R. Soc. London
A415
, 61
(1988
).16.
V. G.
LeBlanc
and W. F.
Langford
, “Classification and unfoldings of 1:2 resonant Hopf bifurcation
,” Arch. Ration. Mech. Anal.
136
, 305
(1996
).17.
S. A.
Campbell
, J.
Bélair
, T.
Ohira
, and J.
Milton
, “Complex dynamics and multistability in a damped harmonic oscillator with delayed negative feedback
,” Chaos
5
, 640
(1995
).18.
S. A.
Campbell
and V. G.
LeBlanc
, “Resonant Hopf-Hopf interactions in delay differential equations
,” J. Dyn. Differ. Equ.
10
, 327
(1998
).19.
J.
Xu
and K. W.
Chung
, “Double Hopf bifurcation with strong resonances in delayed systems with nonlinearities
,” Math. Probl. Eng.
2009
, 759363
(2009
).20.
V. G.
LeBlanc
, “On some secondary bifurcations near resonant Hopf-Hopf interactions
,” Dyn. Contin. Discr. Imp. Syst.: Ser. B: App. Algorithms
7
, 405
(2000
).21.
B.
Krauskopf
, “Bifurcation sequences at 1:4 resonance: An inventory
,” Nonlinearity
7
, 1073
(1994
).22.
B.
Krauskopf
, “Strong resonances and Takens’ Utrecht preprint
,” in Global Analysis of Dynamical Systems, Festschrift Dedicated to Floris Takens for His 60th Birthday
, edited by H. W.
Broer
, B.
Krauskopf
, and G.
Vegter
(Institute of Physics
, Bristol
, 2001
), pp. 89
–111
.23.
A.
Luongo
, A.
Paolone
, and A.
Di Egidio
, “Multiple timescales analysis for 1:2 and 1:3 resonant Hopf bifurcations
,” Nonlinear Dyn.
34
, 269
(2003
).24.
B. B.
Peckham
and I. G.
Kevrekidis
, “Lighting Arnold flames: Resonance in doubly forced periodic oscillators
,” Nonlinearity
15
, 405
(2002
).25.
G. R.
Itovich
and J. L.
Moiola
, “Double Hopf bifurcation analysis using frequency domain methods
,” Nonlinear Dyn.
39
, 235
(2005
).26.
W. J. F.
Govaerts
, Numerical Methods for Bifurcations of Dynamical Equilibria
(SIAM
, Philadelphia
, 2000
).27.
A.
Dhooge
, W.
Govaerts
, Y. A.
Kuznetsov
, W.
Mestrom
, A. M.
Riet
, and B.
Sautois
, “MATCONT and CL-MATCONT continuation toolboxes in MATLAB
,” User Manual, Gent University and Utrech University, 2006
.28.
F.
Schilder
, H. M.
Osinga
, and W.
Vogt
, “Continuation of quasi-periodic invariant tori
,” SIAM J. Appl. Dyn. Syst.
4
, 459
(2005
).© 2010 American Institute of Physics.
2010
American Institute of Physics
You do not currently have access to this content.
