An efficient method for studying weak resonant double hopf bifurcation in nonlinear systems with delayed feedbacks

SIAM Journal on Applied Dynamical Systems

Volumn 6, Issue 1, 2007, Pages 29-60

  Xu, Jian ^a   Chung, Kwok Wai ^b   Chan, Chuen Lit ^b  

^a TONGJI UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Delay differential system; Delayed feedback control; Double Hopf bifurcation; Nonlinear dynamics

Indexed keywords

BIFURCATION (MATHEMATICS); CONTROL THEORY; DYNAMICS; FEEDBACK; FEEDBACK CONTROL; NONLINEAR SYSTEMS;

AMPLITUDE DEATHS; ANALYTICAL EXPRESSIONS; ANALYTICAL PREDICTIONS; ANALYTICAL TOOLS; APPROXIMATE EXPRESSIONS; BIFURCATION PARAMETERS; CLOSED FORMS; COMPLEX DYNAMICS; DELAY DIFFERENTIAL SYSTEM; DELAYED FEEDBACK CONTROL; DELAYED FEEDBACKS; DOUBLE HOPF BIFURCATION; DOUBLE HOPF BIFURCATIONS; EFFICIENT METHODS; FEEDBACK GAINS; HOPF POINTS; INITIAL GUESSES; LANDAU SYSTEMS; NONLINEAR DYNAMICS; NUMERICAL SIMULATIONS; ONE HANDS; PERIODIC SOLUTIONS; TIME- DELAYS; VAN DER POLS;

HOPF BIFURCATION;

EID: 34548767317     PISSN: None     EISSN: 15360040     Source Type: Journal    
DOI: 10.1137/040614207     Document Type: Article

References (36)

1. F. GIANNAKOPOULOS AND A. ZAPP, Bifurcations in a planar system of differential delay equations modeling neural activity, Phys. D, 159 (2001), pp. 215-232.
2. X. F. LIAO, K. W. WONG, AND Z. F. WU, Bifurcation analysis on a two-neuron system with distributed delays, Phys. D, 149 (2001), pp. 123-141.
3. S. KRISE AND S. R. CHOUDHURY, Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos Solitons Fractals, 16 (2003), pp. 59-77.
4. N. MACDONALD, Time Lags in Biological Models, Lecture Notes in Biomath. 27, Springer-Verlag, Berlin, 1978.
5. A. H. NAYFEH, C. M. CHIN, AND J. PARTT, Perturbation methods in nonlinear dynamics-applications to machining dynamics, J. Manu. Sci. Eng., 119 (1997), pp. 485-493.
6. A. RAGHOTHAMS AND S. NARAYANAN, Periodic response and chaos in nonlinear systems with parametric excitation and time delay, Nonlinear Dynam., 27 (2002), pp. 341-365.
7. A. MACCARI, The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dynam., 26 (2001), pp. 105-119.
8. T. KALÁR-NAGY, G. STÉFÁN, AND F. C. MOON, Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, Nonlinear Dynam., 26 (2001), pp. 121-142.
9. K. PYRAGAS, Control of chaos via an unstable delayed feedback controller, Phys. Rev. Lett., 86 (2002), pp. 2265-2268.
10. K. PYRAGAS, Transmission of signals via synchronization of chaotic time-delay systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1998), pp. 1839-1842.
11. D. V. GREGORY AND R. RAJARSHI, Chaotic commmnication using time-delayed optical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), pp. 2129-2156.
12. D. V. R. R. EDDY, A. SEN, AND G. L. JOHNSTON, Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks, Phys. D, 144 (2000), pp. 335-357.
13. E. FRIDMAN, Effects of small delays on stability of singularly perturbed systems, Automatica J. IFAC, 38 (2002), pp. 897-902.
14. J. KALAS, AND L. BARÁKOVÁ, Stability and asymptotic behaviour of a two-dimensional differential system with delay, J. Math. Anal. Appl., 269 (2002), pp. 278-300.
15. K. GOPALSAMY AND I. LEUNG, Delay induced periodicity in a neural netlet of excitation and inhibition, Phys. D, 89 (1996), pp. 395-426.
16. B. F. REDMOND, V. G. LEBLANC, AND A. LONGTIN, Bifurcation analysis of a class of first-order nonlinear delay-differential equations with reflectional symmetry, Phys. D, 166 (2002), pp. 131-146.
17. S. A. CAMPBELL AND V. B. LEBLANC, Resonant Hopf-Hopf interaction in delay differential equations, J. Dynam. Differential Equations, 10 (1998), pp. 327-346.
18. V. LEBLANC AND W. LANGFORD, Classification and unfoldings of 1:2 resonant Hopf bifurcation, Arch. Rational Mech. Anal., 136 (1996), pp. 305-357.
19. S. L. DAS AND A. CHATTERJEE, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynam., 30 (2002), pp. 323-335.
20. S. A. CAMPBELL, J. BÉLAIR, T. OHIRA, AND J. MLLTON, Limit cycles, tori, and complex dynamics in a second-order differential equations with delayed negative feedback, J. Dynam. Differential Equations, 7 (1995), pp. 213-236.
21. J. XU AND K. W. CHUNG, Effects of time delayed position feedback on a van der Pol-Duffing oscillator, Phys. D, 180 (2003), pp. 17-39.
22. P. YU, Y. YUAN, AND J. XU, Study of double Hopf bifurcation and chaos for oscillator with time delay feedback, Commun. Nonlinear Sci. Numer. Simul., 7 (2002), pp. 69-91.
23. P. L. BUONO AND J. BÉLAIR, Restrictions and unfolding of double Hopf bifurcation in functional differential equations, J. Differential Equations, 189 (2003), pp. 234-266.
24. T. FARIA AND L. T. MAGALHÃES, Realisation of ordinary differential equations by retarded functional differential equations in neighborhoods of equilibrium points, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), pp. 759-776.
25. P. L. BUONO AND V. G. LEBLANC, Versal unfoldings for linear retarded functional differential equations, J. Differential Equations, 193 (2003), pp. 307-342.
26. H. S. Y. CHAN, K. W. CHUNG, AND Z. XU, A perturbation-incremental method for strongly non-linear oscillators, Internat. J. Non-Linear Mech., 31 (1996), pp. 59-72.
27. H. S. Y. CHAN, K. W. CHUNG, AND D. W. QI, Some bifurcation diagrams for limit cycles of quadratic differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 197-206.
28. J. XU AND Q. S. LU, Hopf bifurcation of time-delay Liénard equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), pp. 939-951.
29. Y. KURAMOTO AND I. NLSHILKAWA, Statistical macrodynamics of large dynamical systems: Case of a phase trasition in oscillator communities, J. Statist. Phys., 49 (1987), pp. 569-582.
30. S. KIM, S. H. PARK, AND C. S. RYU, Multistability in coupled oscillator system with time delay, Phys. Rev. Lett., 79 (1997), pp. 2911-2914.
31. S. H. STROGATZ, Time delay in the Kuramoto model of coupled oscillator, Phys. Rev. Lett., 82 (1999), pp. 648-652.
32. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1993.
33. A. H. NAYFEH, Method of Normal Forms, John Wiley & Sons, New York, 1993.
34. J. K. HALE AND S. M. VERDUYN LUNCL, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
35. O. DIEKMANN, S. M. VERDUYN LUNCL, AND H. O. WALTHER, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
36. Y. A. KUZNETSOV, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998.