Physics and MathematicsNo Access

DOUBLE HOPF BIFURCATION FOR STUART–LANDAU SYSTEM WITH NONLINEAR DELAY FEEDBACK AND DELAY-DEPENDENT PARAMETERS

SUQI MA

School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China

Department of Mathematics, China Agricultural University, Beijing 100083, P. R. China.

Search for more papers by this author

QISHAO LU

School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, P. R. China

Corresponding author.

Search for more papers by this author

, and

S. JOHN HOGAN

Department of Engineering Mathematics, University of Bristol, BS8 1TR, Bristol, United Kingdom

Search for more papers by this author

https://doi.org/10.1142/S0219525907001227Cited by:5 (Source: Crossref)

Abstract

A Stuart–Landau system under delay feedback control with the nonlinear delay-dependent parameter e^-pτ is investigated. A geometrical demonstration method combined with theoretical analysis is developed so as to effectively solve the characteristic equation. Multi-stable regions are separated from unstable regions by allocations of Hopf bifurcation curves in (p,τ) plane. Some weak resonant and non-resonant oscillation phenomena induced by double Hopf bifurcation are discovered. The normal form for double Hopf bifurcation is deduced. The local dynamical behavior near double Hopf bifurcation points are also clarified in detail by using the center manifold method. Some states of two coexisting stable periodic solutions are verified, and some torus-broken procedures are also traced.

Keywords:

References

J. Dusek, P. Le Gal and P. Fraunie, J. Fluid Mech. 264, 59 (1994). Web of Science, Google Scholar
D. Ormières and M. Provansal, Phys. Rev. Lett. 83, 80 (1999). Web of Science, Google Scholar
M. C. Thompson, Th. Leweke and M. Provansal, J. Fluids Struct. 15, 575 (2001), DOI: 10.1006/jfls.2000.0362. Web of Science, Google Scholar
B. Ghidersa and J. Dusek, J. Fluid Mech. 423, 33 (2000), DOI: 10.1017/S0022112000001701. Web of Science, Google Scholar
W. Wischertet al., Phys. Rev. E 49, 203 (1994), DOI: 10.1103/PhysRevE.49.203. Web of Science, Google Scholar
K. Miyakawa and K. Yamada, Physica D 127, 177 (1999), DOI: 10.1016/S0167-2789(98)00310-8. Web of Science, Google Scholar
C. J. Briggset al., J. Animal Ecol. 69, 352 (2000). Web of Science, Google Scholar
Y. Kuang , Delay Differential Equations with Applications in Population Dynamics ( Academic Press , Boston , 1993 ) . Google Scholar
A. Destexhe, Phys. Lett. A 187, 309 (1994), DOI: 10.1016/0375-9601(94)90006-X. Web of Science, Google Scholar
D. V. R. Reddy, A. Sen and G. L. Johnston, Phys. Rev. Lett. 80, 5109 (1998), DOI: 10.1103/PhysRevLett.80.5109. Web of Science, Google Scholar
H. G. Schuster and P. Wagner, Prog. Theor. Phys. 81, 939 (1989), DOI: 10.1143/PTP.81.939. Web of Science, Google Scholar
D. V. R. Reddy, A. Sen and G. L. Johnston, Physica D 144, 335 (2000), DOI: 10.1016/S0167-2789(00)00086-5. Web of Science, Google Scholar
M. K. S. Yeung and S. H. Strogatz, Phys. Rev. Lett. 82, 648 (1999), DOI: 10.1103/PhysRevLett.82.648. Web of Science, Google Scholar
W. G. Aiello and H. I. Freedman, Math Biosci. 101, 139 (1990), DOI: 10.1016/0025-5564(90)90019-U. Web of Science, Google Scholar
E. Beretta and Y. Kuang, SIAM J. Math. Anal. 33, 1144 (2002), DOI: 10.1137/S0036141000376086. Web of Science, Google Scholar
K. L. Cooke, P. van den Driessche and X. Zou, J. Math Biol. 39, 332 (1999), DOI: 10.1007/s002850050194. Web of Science, Google Scholar
J. Xu and K. W. Chung, Physica D 180, 17 (2003), DOI: 10.1016/S0167-2789(03)00049-6. Web of Science, Google Scholar
P. Yu, Y. Yuan and J. Xu, Commun. Non. Sci. Num. Simu. 7, 69 (2002), DOI: 10.1016/S1007-5704(02)00007-2. Google Scholar
S. A. Campell and J. Bélair, Canad. Appl. Math. 137 (1995). Google Scholar
S. A. Campellet al., J. Dyn. Diff. Equa. 7, 213 (1995). Web of Science, Google Scholar
S. Q. Ma, Q. S. Lu and S. L. Mei, Discret. Cont. Dyn. Syst. Ser. B 5, 735 (2005). Web of Science, Google Scholar
C. Elphicket al., Physica D 29, 95 (1987), DOI: 10.1016/0167-2789(87)90049-2. Web of Science, Google Scholar
J. Hale , Introduction to Functional Differential Equations ( Springer , 1993 ) . Google Scholar
P. L. Buono and J. Bélair, J. Diff. Equa. 189, 234 (2003), DOI: 10.1016/S0022-0396(02)00179-1. Web of Science, Google Scholar
T. Faria and L. T. Magalhães, J. Diff. Equa. 122, 201 (1995), DOI: 10.1006/jdeq.1995.1145. Web of Science, Google Scholar
T. Faria and L. T. Magalhães, J. Diff. Equa. 122, 181 (1995), DOI: 10.1006/jdeq.1995.1144. Web of Science, Google Scholar
J. Guckhenheimerm, Appl. Math. Sci. 42, (1983). Google Scholar