Time-delayed feedback control of dynamical small-world networks at Hopf bifurcation

Nonlinear Dynamics

Volumn 58, Issue 1-2, 2009, Pages 319-344

  Xiao, Min ^a,c   Ho, Daniel W C ^b   Cao, Jinde ^a  

^a SOUTHEAST UNIVERSITY   (China)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^c NANJING XIAOZHUANG UNIVERSITY   (China)

Author keywords

Bifurcation control; Hopf bifurcation; Latency time; Low pass filter; Small world network; Time delayed feedback

Indexed keywords

ANALYTICAL INVESTIGATIONS; BIFURCATION CONTROL; BIFURCATION PARAMETER; CHARACTERISTIC EQUATION; CONTROL FORCE; CONTROL LOOP; DELAYED FEEDBACK; DOMAIN OF STABILITY; FEEDBACK PERTURBATION; FEEDBACK SCHEME; FEEDBACK SIGNAL; INTERACTION STRENGTH; LATENCY TIME; NONLINEAR INTERACTIONS; NONZERO LATENCY TIME; PERIODIC ORBITS; SMALL WORLD NETWORKS; SMALL-WORLD NETWORK; TIME-DELAYED FEEDBACK; TIME-DELAYED FEEDBACK METHODS; UNSTABLE STEADY STATE;

DELAY CIRCUITS; HOPF BIFURCATION; SIGNAL FILTERING AND PREDICTION; SIGNAL PROCESSING; STABILIZATION;

FEEDBACK;

EID: 70349204504     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-009-9485-0     Document Type: Article

References (39)

1. Abed, E.H., Wang, H.O., Tesi, A.: Control of bifurcation and chaos. In: Levinen, W.S. (ed.) The Control Handbook, pp. 951-966. CRC Press, Boca Raton (1995)
2. G.R. Chen J.L. Moiola H.O. Wang 2000 Bifurcation control: Theories, methods and applications Int. J. Bifurc. Chaos 10 511 548 1090.37552 1757214
3. E.H. Abed J.H. Fu 1986 Local feedback stabilization and bifurcation control: I. Hopf bifurcation Syst. Control Lett. 7 11 17 0587.93049 10.1016/0167-6911(86)90095-2 833060 (Pubitemid 16502909)
4. Chen, G.R., Lu, J., Yap, K.C.: Feedback control of Hopf bifurcations. In: Proc. of IEEE Int. Symp. on Circ. Sys., Monterey, CA (1998)
5. P. Yu G.R. Chen 2004 Hopf bifurcation control using nonlinear feedback with polynomial functions Int. J. Bifurc. Chaos 14 1683 1704 1129.37335 10.1142/S0218127404010291 2072357 (Pubitemid 40242133)
6. Yu, P.: Bifurcation dynamics in control systems. In: Chen, G.R., Hill, D.J., Yu, X. (eds.) Bifurcation Control: Theory and Applications, pp. 99-126. Springer, Berlin (2003)
7. J. Sieber B. Krauskopf 2004 Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity Nonlinearity 17 85 103 1054.34121 10.1088/0951-7715/17/1/006 2023433 (Pubitemid 38085736)
8. M.E. Bleich J.E.S. Socolar 1996 Stability of periodic orbits controlled by time-delay feedback Phys. Lett. A 210 87 94 10.1016/0375-9601(95)00827-6
9. M.E. Brandt H.T. Shih G.R. Chen 1997 Linear time-delay feedback control of a pathological rhythm in a cardiac conduction model Phys. Rev. E 56 1334 1337 10.1103/PhysRevE.56.R1334
10. Y.X. Song X.H. Yu G.R. Chen J.X. Xu Y.P. Tian 2002 Time delayed repetitive learning control for chaotic systems Int. J. Bifurc. Chaos 12 1057 1065 1051.93528 10.1142/S0218127402004905 1913986 (Pubitemid 34861996)
11. D.J. Watts S.H. Strogatz 1998 Collective dynamics of small-world networks Nature 393 440 442 10.1038/30918 (Pubitemid 28292183)
12. M.E.J. Newman D.J. Watts 1999 Scaling and percolation in the small-world network model Phys. Rev. E 60 7332 7342 10.1103/PhysRevE.60.7332
13. C.F. Moukarzel 1999 Spreading and shortest paths in systems with sparse long-range connections Phys. Rev. E 60 6263 6266 10.1103/PhysRevE.60.R6263 1732259
14. M.E.J. Newman C. Moore D.J. Watts 2000 Mean-field solution of the small-world network model Phys. Rev. Lett. 84 3201 3204 10.1103/PhysRevLett.84. 3201
15. X.F. Wang G.R. Chen 2002 Synchronization in small-world dynamical networks Int. J. Bifurc. Chaos 12 187 192 10.1142/S0218127402004292 (Pubitemid 34862071)
16. Wang, X.F., Chen, G.R.: Complex networks: small-world, scale-free, and beyond, IEEE Circuits Syst. Mag. (2003)
17. X.S. Yang 2001 Chaos in small-world networks Phys. Rev. E 63 046206 10.1103/PhysRevE.63.046206
18. C.G. Li G.R. Chen 2004 Local stability and Hopf bifurcation in small-world delayed networks Chaos, Solitons Fractals 20 353 361 1045.34047 10.1016/S0960-0779(03)00405-3 2025585
19. Chen, G.R. (Ed.): Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton (1999)
20. K. Pyragas 1992 Continuous control of chaos by self-controlling feedback Phys. Lett. A 170 421 428 10.1016/0375-9601(92)90745-8
21. Z.S. Cheng J. Cao 2007 Hopf bifurcation control for delayed complex networks J. Franklin Inst. 344 6 846 857 1126.37033 10.1016/j.jfranklin.2006.10. 007 2344914 (Pubitemid 47176549)
22. Hale, J.: Theory of Functional Differential Equations. Springer, New York (1977)
23. Hassard, B.D., Kazarinoff, N.D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)
24. J.J. Wei S.G. Ruan 1999 Stability and bifurcation in a neural network model with two delays Phys. D 130 255 272 1066.34511 10.1016/S0167-2789(99) 00009-3 1692866
25. J. Blakely L. Illing D. Gauthier 2004 Controlling fast chaos in delay dynamical systems Phys. Rev. Lett. 92 193901 10.1103/PhysRevLett.92.193901
26. D.W. Sukow M.E. Bleich D.J. Gauthier J.E.S. Socolar 1997 Controlling chaos in a fast diode resonator using extended time-delay autosynchronization: Experimental observations and theoretical analysis Chaos 7 560 576 10.1063/1.166256 (Pubitemid 128005606)
27. W. Just D. Reckwerth E. Reibold H. Benner 1999 Influence of control loop latency on time-delayed feedback control Phys. Rev. E 59 2826 2829 10.1103/PhysRevE.59.2826
28. P. Hövel J.E.S. Socolar 2003 Stability domains for time-delay feedback control with latency Phys. Rev. E 68 036206 10.1103/PhysRevE.68.036206
29. J. Schlesner A. Amann N.B. Janson W. Just E. Schöll 2003 Self-stabilization of high-frequency oscillations in semiconductor superlattices by time-delayed autosynchronization Phys. Rev. E 68 066208 10.1103/PhysRevE.68. 066208
30. K. Pyragas V. Pyragas I.Z. Kiss J.L. Hudson 2004 Adaptive control of unknown unstable steady states of dynamical systems Phys. Rev. E 70 026215 10.1103/PhysRevE.70.026215
31. K.C. Yap G.R. Chen T. Ueta 2001 Controlling bifurcations of discrete maps Lat. Am. Appl. Res. 31 157 162
32. E.H. Abed J.H. Fu 1987 Local feedback stabilization and bifurcation control: II. Stationary bifurcation Syst. Control Lett. 8 467 473 0626.93058 10.1016/0167-6911(87)90089-2 890085 (Pubitemid 17606905)
33. G.R. Chen J. Lu B. Nicholas S. Ranganathan 1999 Bifurcation dynamics in discrete-time delayed-feedback control systems Int. J. Bifurc. Chaos 9 287 293 0967.93511 10.1142/S021812749900016X 1689615
34. E.H. Abed H.O. Wang G.R. Chen 1994 Stabilization of period doubling bifurcations and implications for control of chaos Phys. D 70 154 164 0807.58033 10.1016/0167-2789(94)90062-0 1257854
35. H. Yaghoobi E.H. Abed 2003 Local feedback control of the Neimark-Sacker bifurcation Int. J. Bifurc. Chaos 13 879 893 1074.93022 10.1142/ S0218127403006972 1980772 (Pubitemid 36739293)
36. G. Wen D. Xu J. Xie 2003 Control of degenerate Hopf bifurcations in three-dimensional maps Chaos 13 486 494 1080.93662 10.1063/1.1564892 1982868 (Pubitemid 36814881)
37. Z.D. Wang S. Lauria J.A. Fang X.H. Liu 2007 Exponential stability of uncertain stochastic neural networks with mixed time delays Chaos, Solitons Fractals 32 1 62 72 1152.34058 10.1016/j.chaos.2005.10.061 2271102 (Pubitemid 44374960)
38. Z.D. Wang H.S. Shu J.A. Fang X.H. Liu 2006 Robust stability for stochastic Hopfield neural networks with time delays Nonlinear Anal. Real World Appl. 7 5 1119 1128 1122.34065 10.1016/j.nonrwa.2005.10.004 2260903 (Pubitemid 43903684)
39. Z.D. Wang Y.R. Liu X.H. Liu 2005 On global asymptotic stability of neural networks with discrete and distributed delays Phys. Lett. A 345 4-6 299 308 05314210 10.1016/j.physleta.2005.07.025 (Pubitemid 41262382)