# American Institute of Mathematical Sciences

July  2013, 18(5): 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

## Dynamics of a limit cycle oscillator with extended delay feedback

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Received  July 2012 Revised  November 2012 Published  March 2013

Investigating limit cycle oscillator with extended delay feedback is an efficient way to understand the dynamics of a global coupled ensemble or a large system with periodic oscillation. The stability and bifurcation of the arisen neutral equation are obtained. Stability switches and Hopf bifurcations appear when delay passes through a sequence of critical values. Global continuation of Hopf bifurcating periodic solutions and double--Hopf bifurcation are studied. With the help of the unfolding system near double--Hopf bifurcation obtained by using method of normal forms, quasiperiodic oscillations are found. The number of the coexisted periodic solutions is estimated. Finally, some numerical simulations are carried out.
Citation: Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
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##### References:
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Wei, Bifurcation analysis in a limit cycle oscillator with delayed feedback, Chaos, Solitons & Fractals, 23 (2005), 817-831. doi: 10.1016/j.chaos.2004.05.028.  Google Scholar [6] D. V. R. Reddy, A. Sen and G. L. Johnston, Dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks, Physica D, 144 (2000), 335-357. doi: 10.1016/S0167-2789(00)00086-5.  Google Scholar [7] S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay, Phys. Rev. Lett., 79 (1997), 2911. Google Scholar [8] D. V. R. Reddy, A. Sen and G. L. Johnston, Time delay effects on coupled limit cycle oscillators at Hopf bifurcation, Physica D, 129 (1999), 15-34. doi: 10.1016/S0167-2789(99)00004-4.  Google Scholar [9] Y. Li, W. Jiang and H. Wang, Double Hopf bifurcation and quasi-periodic attractors in delay-coupled limit cycle oscillators, J. Math. Anal. Appl., 387 (2012), 1114-1126. doi: 10.1016/j.jmaa.2011.10.023.  Google Scholar [10] K. 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Kuang, On neutral delay logistic gause-type predator-prey systems, Dynamics and Stability of Systems, 6 (1991), 173-189.  Google Scholar [17] J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations, Acta. Math. Sin., 45 (2002), 94-104.  Google Scholar [18] C. Wang and J. Wei, Normal forms for NFDE with parameters and application to the lossless transmission line, Nonlinear Dynam., 52 (2008), 199-206. doi: 10.1007/s11071-007-9271-9.  Google Scholar [19] M. Weedermann, Normal forms for neutral functional differential equations, in "Topics in Functional Differential and Difference Equations" (eds. T. Faria and P. Freitas), Amer. Math. Soc., Providence, (2001), 361-368.  Google Scholar [20] M. Weedermann, Hopf bifurcation calculations for scalar neutral delay differential equations, Nonlinearity, 19 (2006), 2091-2102. doi: 10.1088/0951-7715/19/9/005.  Google Scholar [21] J. Hale and S. 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Ruan, Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays, Physica D, 239 (2010), 2011-2024. doi: 10.1016/j.physd.2010.07.013.  Google Scholar [28] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 1983.  Google Scholar
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