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Bifurcation analysis of bursting solutions of two HindmarshRose neurons with joint electrical and synaptic coupling
Timevarying delayed feedback control for an internet congestion control model
1.  School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China, China 
References:
[1] 
T. Alpcan and T. Basar, Global stability analysis of an endtoend congestion control scheme for general topology networks with delay, in "Proceedings of the 42nd IEEE Conference on Decision and Control," (2003), 10921097. 
[2] 
H. Brunner and S. Maset, Time transformations for delay differential equations, Discrete Contin. Dyn. Syst. Ser A, 25 (2009), 751775. doi: 10.3934/dcds.2009.25.751. 
[3] 
Z. Chen and P. Yu, Hopf bifurcation control for an Internet congestion model, International Journal of Bifurcation and Chaos, 15 (2005), 26432651. doi: 10.1142/S0218127405013587. 
[4] 
Y. Choi, Periodic delay effects on cutting dynamics, Journal of Dynamics and Differential Equations, 17 (2005), 353389. doi: 10.1007/s108840053145y. 
[5] 
S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics, 30 (2002), 323335. doi: 10.1023/A:1021220117746. 
[6] 
D. W. Ding, J. Zhu, X. S. Luo and Y. L. Liu, Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm, Nonlinear Analysis: Real World Applications, 10 (2009), 28732883. doi: 10.1016/j.nonrwa.2008.09.007. 
[7] 
S. Floyd and V. Jacobson, Random early detection gateways for congestion avoidance, IEEE/ACM Transctions on Networks, 1 (1993), 397413. 
[8] 
D. E. Gilsinn, Estimating critical Hopf bifurcation parameters for a secondorder delay differential equation with application to machine tool chatter, Nonlinear Dynamics, 30 (2002), 103154. doi: 10.1023/A:1020455821894. 
[9] 
S. T. Guo, G. Feng, X. F. Liao and Q. Liu, Hopf bifurcation control in a congestion control model via dynamic delayed feedback, Chaos, 18 (2008), 043104113. doi: 10.1063/1.2998220. 
[10] 
S. T. Guo, X. F. Liao and C. D. Li, Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay, Nonlinear Analysis: Real World Applications, 9 (2008), 12921309. doi: 10.1016/j.nonrwa.2007.03.006. 
[11] 
S. T. Guo, X. F. Liao, Q. Liu and C. D. Li, Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay, Nonlinear Analysis: Real World Applications, 9 (2008), 17681793. doi: 10.1016/j.nonrwa.2007.05.014. 
[12] 
J. Hale, "Theory of Functional Differential Equations," World Publishing Corporation, Beijing, China, 2003. 
[13] 
V. Jacobson, Congestion avoidance and control, ACM SIGCOMM Computer Communication Review, 18 (1988), 314329. doi: 10.1145/52325.52356. 
[14] 
K. Jiang, X. F. Wang and Y. G. Xi, Bifurcation analysis of an Internet congestion control model, in "Proceedings of ICARCV," (2004), 590594. 
[15] 
F. P. Kelly, Models for a selfmanaged Internet, Philos Trans Roy Soc A, 358 (2000), 23352348. doi: 10.1098/rsta.2000.0651. 
[16] 
F. P. Kelly, A. Maulloo and D. K. H. Tan, Rate control in communication networks: shadow prices, proportional fairness, and stability, J. Oper. Res. Soc., 49 (1998), 237252. 
[17] 
S. Kunniyur and R. Srikant, Endtoend congestion control: utility functions, random lossed and ECN marks, IEEE/ACM Transactions on Networking, 7 (2003), 689702. doi: 10.1109/TNET.2003.818183. 
[18] 
Y. Kuznetsov, "Elements of Applied Bifurcation Theory," 2^{nd} edition, Springer, 1997. 
[19] 
C. G. Li, G. R. Chen, X. F. Liao and J. B. Yu, Hopf bifurcation in an Internet congestion control model, Chaos Solitons & Fractals, 19 (2004), 853862. doi: 10.1016/S09600779(03)002698. 
[20] 
S. Liu, T. Basar and R. Srikant, Controlling the Internet: A survey and some new results, in "Proceedings of the 42nd IEEE Conference on Decision and Control," (2003), 30483057. 
[21] 
F. Liu, Z. H. Guan and H. O. Wang, Controlling bifurcations and chaos in TCPUDPRED, Nonlinear Analysis: Real World Applications, 11 (2010), 14911501. doi: 10.1016/j.nonrwa.2009.03.005. 
[22] 
F. Paganini, A global stability result in network flow control, Systems & Control Letters, 46 (2002), 165172. doi: 10.1016/S01676911(02)001238. 
[23] 
G. Raina, Local bifurcation analysis of some dual congestion control algorithms, IEEE Transactions on Automatic Control, 50 (2005), 11351146. doi: 10.1109/TAC.2005.852566. 
[24] 
G. Raina and O. Heckmann, TCP: Local stability and Hopf bifurcation, Performance Evaluation, 64 (2007), 266275. doi: 10.1016/j.peva.2006.05.005. 
[25] 
Shigeki Tsuji, Tetsushi Ueta, Hiroshi Kawakami and Kazuyuki Aihara, Bifurcation of burst response in an AmariHopfield Neuron pair with a periodic external forces, Electrical Engineering in Japan, 146 (2004), 4353. doi: 10.1002/eej.10217. 
[26] 
R. Srikant, "The Mathematics of Internet Congestion Control," Birkhäuser, Boston, 2004. 
[27] 
X. F. Wang, G. R. Chen and KingTim Ko, A stability theorem for Internet congestion control, Systems & Control Letters, 45 (2002), 8185. doi: 10.1016/S01676911(01)001657. 
[28] 
Z. F. Wang and T. G. Chu, Delay induced Hopf bifurcation in a simplified network congestion control model, Chaos Solitons & Fractals, 28 (2006), 161172. doi: 10.1016/j.chaos.2005.05.047. 
[29] 
M. Xiao and J. D. Cao, Delayed feedbackbased bifurcation control in an Internet congestion model, J. Math. Anal. Appl., 332 (2007), 10101027. doi: 10.1016/j.jmaa.2006.10.062. 
[30] 
J. Xu and K. W. Chung, A perturbationincremental scheme for studying Hopf bifurcation in delayed differential systems, Science in China Series E, 52 (2009), 698708. doi: 10.1007/s1143100900521. 
[31] 
J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks, SIAM J. Applied Dynamical Sysyems, 6 (2007), 2960. doi: 10.1137/040614207. 
[32] 
H. Y. Yang and Y. P. Tian, Hopf bifurcation in REM algorithm with communication delay, Chaos, Solitons & Fractals, 25 (2005), 10931105. doi: 10.1016/j.chaos.2004.11.085. 
[33] 
H. Y. Yang and S. Y. Zhang, Hopf bifurcation of endtoend network congestion control algorithm, 2007 IEEE International Conference on Control and Automation, Guangzhou, China, 2007. 
show all references
References:
[1] 
T. Alpcan and T. Basar, Global stability analysis of an endtoend congestion control scheme for general topology networks with delay, in "Proceedings of the 42nd IEEE Conference on Decision and Control," (2003), 10921097. 
[2] 
H. Brunner and S. Maset, Time transformations for delay differential equations, Discrete Contin. Dyn. Syst. Ser A, 25 (2009), 751775. doi: 10.3934/dcds.2009.25.751. 
[3] 
Z. Chen and P. Yu, Hopf bifurcation control for an Internet congestion model, International Journal of Bifurcation and Chaos, 15 (2005), 26432651. doi: 10.1142/S0218127405013587. 
[4] 
Y. Choi, Periodic delay effects on cutting dynamics, Journal of Dynamics and Differential Equations, 17 (2005), 353389. doi: 10.1007/s108840053145y. 
[5] 
S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dynamics, 30 (2002), 323335. doi: 10.1023/A:1021220117746. 
[6] 
D. W. Ding, J. Zhu, X. S. Luo and Y. L. Liu, Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm, Nonlinear Analysis: Real World Applications, 10 (2009), 28732883. doi: 10.1016/j.nonrwa.2008.09.007. 
[7] 
S. Floyd and V. Jacobson, Random early detection gateways for congestion avoidance, IEEE/ACM Transctions on Networks, 1 (1993), 397413. 
[8] 
D. E. Gilsinn, Estimating critical Hopf bifurcation parameters for a secondorder delay differential equation with application to machine tool chatter, Nonlinear Dynamics, 30 (2002), 103154. doi: 10.1023/A:1020455821894. 
[9] 
S. T. Guo, G. Feng, X. F. Liao and Q. Liu, Hopf bifurcation control in a congestion control model via dynamic delayed feedback, Chaos, 18 (2008), 043104113. doi: 10.1063/1.2998220. 
[10] 
S. T. Guo, X. F. Liao and C. D. Li, Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay, Nonlinear Analysis: Real World Applications, 9 (2008), 12921309. doi: 10.1016/j.nonrwa.2007.03.006. 
[11] 
S. T. Guo, X. F. Liao, Q. Liu and C. D. Li, Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay, Nonlinear Analysis: Real World Applications, 9 (2008), 17681793. doi: 10.1016/j.nonrwa.2007.05.014. 
[12] 
J. Hale, "Theory of Functional Differential Equations," World Publishing Corporation, Beijing, China, 2003. 
[13] 
V. Jacobson, Congestion avoidance and control, ACM SIGCOMM Computer Communication Review, 18 (1988), 314329. doi: 10.1145/52325.52356. 
[14] 
K. Jiang, X. F. Wang and Y. G. Xi, Bifurcation analysis of an Internet congestion control model, in "Proceedings of ICARCV," (2004), 590594. 
[15] 
F. P. Kelly, Models for a selfmanaged Internet, Philos Trans Roy Soc A, 358 (2000), 23352348. doi: 10.1098/rsta.2000.0651. 
[16] 
F. P. Kelly, A. Maulloo and D. K. H. Tan, Rate control in communication networks: shadow prices, proportional fairness, and stability, J. Oper. Res. Soc., 49 (1998), 237252. 
[17] 
S. Kunniyur and R. Srikant, Endtoend congestion control: utility functions, random lossed and ECN marks, IEEE/ACM Transactions on Networking, 7 (2003), 689702. doi: 10.1109/TNET.2003.818183. 
[18] 
Y. Kuznetsov, "Elements of Applied Bifurcation Theory," 2^{nd} edition, Springer, 1997. 
[19] 
C. G. Li, G. R. Chen, X. F. Liao and J. B. Yu, Hopf bifurcation in an Internet congestion control model, Chaos Solitons & Fractals, 19 (2004), 853862. doi: 10.1016/S09600779(03)002698. 
[20] 
S. Liu, T. Basar and R. Srikant, Controlling the Internet: A survey and some new results, in "Proceedings of the 42nd IEEE Conference on Decision and Control," (2003), 30483057. 
[21] 
F. Liu, Z. H. Guan and H. O. Wang, Controlling bifurcations and chaos in TCPUDPRED, Nonlinear Analysis: Real World Applications, 11 (2010), 14911501. doi: 10.1016/j.nonrwa.2009.03.005. 
[22] 
F. Paganini, A global stability result in network flow control, Systems & Control Letters, 46 (2002), 165172. doi: 10.1016/S01676911(02)001238. 
[23] 
G. Raina, Local bifurcation analysis of some dual congestion control algorithms, IEEE Transactions on Automatic Control, 50 (2005), 11351146. doi: 10.1109/TAC.2005.852566. 
[24] 
G. Raina and O. Heckmann, TCP: Local stability and Hopf bifurcation, Performance Evaluation, 64 (2007), 266275. doi: 10.1016/j.peva.2006.05.005. 
[25] 
Shigeki Tsuji, Tetsushi Ueta, Hiroshi Kawakami and Kazuyuki Aihara, Bifurcation of burst response in an AmariHopfield Neuron pair with a periodic external forces, Electrical Engineering in Japan, 146 (2004), 4353. doi: 10.1002/eej.10217. 
[26] 
R. Srikant, "The Mathematics of Internet Congestion Control," Birkhäuser, Boston, 2004. 
[27] 
X. F. Wang, G. R. Chen and KingTim Ko, A stability theorem for Internet congestion control, Systems & Control Letters, 45 (2002), 8185. doi: 10.1016/S01676911(01)001657. 
[28] 
Z. F. Wang and T. G. Chu, Delay induced Hopf bifurcation in a simplified network congestion control model, Chaos Solitons & Fractals, 28 (2006), 161172. doi: 10.1016/j.chaos.2005.05.047. 
[29] 
M. Xiao and J. D. Cao, Delayed feedbackbased bifurcation control in an Internet congestion model, J. Math. Anal. Appl., 332 (2007), 10101027. doi: 10.1016/j.jmaa.2006.10.062. 
[30] 
J. Xu and K. W. Chung, A perturbationincremental scheme for studying Hopf bifurcation in delayed differential systems, Science in China Series E, 52 (2009), 698708. doi: 10.1007/s1143100900521. 
[31] 
J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks, SIAM J. Applied Dynamical Sysyems, 6 (2007), 2960. doi: 10.1137/040614207. 
[32] 
H. Y. Yang and Y. P. Tian, Hopf bifurcation in REM algorithm with communication delay, Chaos, Solitons & Fractals, 25 (2005), 10931105. doi: 10.1016/j.chaos.2004.11.085. 
[33] 
H. Y. Yang and S. Y. Zhang, Hopf bifurcation of endtoend network congestion control algorithm, 2007 IEEE International Conference on Control and Automation, Guangzhou, China, 2007. 
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