American Institute of Mathematical Sciences

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

Dynamics of a limit cycle oscillator with extended delay feedback

Ben Niu 1, and  Weihua Jiang 2,

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.
Keywords: Extended delay feedback, quasiperiodic solution., double Hopf bifurcation, limit cycle oscillator, neutral type.
Mathematics Subject Classification: 34K18, 34K40, 37G0.
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
References:
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X. Wu and L. Wang, Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay,, Discrete Continuous Dynam. Systems-B, 13 (2010), 503. doi: 10.3934/dcdsb.2010.13.503. Google Scholar

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W. Jiang and J. Wei, Bifurcation analysis in a limit cycle oscillator with delayed feedback,, Chaos, 23 (2005), 817. doi: 10.1016/j.chaos.2004.05.028. Google Scholar

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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. doi: 10.1016/S0167-2789(00)00086-5. Google Scholar

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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. doi: 10.1016/S0167-2789(99)00004-4. Google Scholar

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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. doi: 10.1016/j.jmaa.2011.10.023. Google Scholar

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K. Pyragas, Continuous control of chaos by self-controlling feedback,, Phys. Lett. A, 170 (1992), 421. Google Scholar

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J. Wei and W. Jiang, Stability and bifurcation analysis in Van der Pol's oscillator with delayed feedback,, J. Sound Vibrat., 283 (2005), 801. doi: 10.1016/j.jsv.2004.05.014. Google Scholar

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J. E. S. Socolar, D. W. Sukow and D. J. Gauthier, Stabilizing unstable periodic orbits in fast dynamical systems,, Phys. Rev. E, 50 (1994). Google Scholar

[14]

K. Pyragas, Control of chaos via extended delay feedback,, Phys. Lett. A, 206 (1995), 323. doi: 10.1016/0375-9601(95)00654-L. Google Scholar

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S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer, (1980). Google Scholar

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Y. Kuang, On neutral delay logistic gause-type predator-prey systems,, Dynamics and Stability of Systems, 6 (1991), 173. Google Scholar

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J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations,, Acta. Math. Sin., 45 (2002), 94. Google Scholar

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C. Wang and J. Wei, Normal forms for NFDE with parameters and application to the lossless transmission line,, Nonlinear Dynam., 52 (2008), 199. doi: 10.1007/s11071-007-9271-9. Google Scholar

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[23]

Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate,, Discrete Continuous Dynam. Systems-B, 15 (2011), 93. doi: 10.3934/dcdsb.2011.15.93. Google Scholar

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S. N. Chow and K. Lu, $C^k$ center unstable manifolds,, Proc. Roy. Soc. Edinburgh., 108 (1988), 303. doi: 10.1017/S0308210500014682. Google Scholar

[26]

J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer, (1995). doi: 10.1007/978-1-4612-4050-1. Google Scholar

[27]

Y. Qu, J.Wei and S. Ruan, Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays,, Physica D, 239 (2010), 2011. 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, (1983). Google Scholar

show all references

References:
[1]

A. Sharma, P. R. Sharma and M. D. Shrimali, Amplitude death in nonlinear oscillators with indirect coupling,, Physics Lett. A, 376 (2012), 1562. Google Scholar

[2]

X. Wu and L. Wang, Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay,, Discrete Continuous Dynam. Systems-B, 13 (2010), 503. doi: 10.3934/dcdsb.2010.13.503. Google Scholar

[3]

M. Rosenblum and A. Pikovsky, Delayed feedback control of collective synchrony: An approach to suppression of pathological brain rhythms,, Phys. Rev. E, 70 (2004). doi: 10.1103/PhysRevE.70.041904. Google Scholar

[4]

Y. Kuramoto and I. Nishikawa, Statistical macrodynamics of large dynamical systems: Case of a phase transition in oscillator communities,, J. Statist. Phys., 49 (1987), 569. doi: 10.1007/BF01009349. Google Scholar

[5]

W. Jiang and J. Wei, Bifurcation analysis in a limit cycle oscillator with delayed feedback,, Chaos, 23 (2005), 817. 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. 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). 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. 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. doi: 10.1016/j.jmaa.2011.10.023. Google Scholar

[10]

K. Pyragas, Continuous control of chaos by self-controlling feedback,, Phys. Lett. A, 170 (1992), 421. Google Scholar

[11]

S. Yuan, Y. Song and J. Li, Oscillations in a plasmid turbidostat model with delayed feedback control,, Discrete Continuous Dynam. Systems-B, 15 (2011), 893. doi: 10.3934/dcdsb.2011.15.893. Google Scholar

[12]

J. Wei and W. Jiang, Stability and bifurcation analysis in Van der Pol's oscillator with delayed feedback,, J. Sound Vibrat., 283 (2005), 801. doi: 10.1016/j.jsv.2004.05.014. Google Scholar

[13]

J. E. S. Socolar, D. W. Sukow and D. J. Gauthier, Stabilizing unstable periodic orbits in fast dynamical systems,, Phys. Rev. E, 50 (1994). Google Scholar

[14]

K. Pyragas, Control of chaos via extended delay feedback,, Phys. Lett. A, 206 (1995), 323. doi: 10.1016/0375-9601(95)00654-L. Google Scholar

[15]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer, (1980). Google Scholar

[16]

Y. Kuang, On neutral delay logistic gause-type predator-prey systems,, Dynamics and Stability of Systems, 6 (1991), 173. Google Scholar

[17]

J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations,, Acta. Math. Sin., 45 (2002), 94. 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. doi: 10.1007/s11071-007-9271-9. Google Scholar

[19]

M. Weedermann, Normal forms for neutral functional differential equations,, in, (2001), 361. Google Scholar

[20]

M. Weedermann, Hopf bifurcation calculations for scalar neutral delay differential equations,, Nonlinearity, 19 (2006), 2091. doi: 10.1088/0951-7715/19/9/005. Google Scholar

[21]

J. Hale and S. Lunel, "Introduction to Functional Differential Equations,", Springer, (1993). Google Scholar

[22]

T. Faria and L. Magalhaes, Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation,, J. Differ. Equations, 122 (1995), 181. doi: 10.1006/jdeq.1995.1144. Google Scholar

[23]

Z. Hu, P. Bi, W. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate,, Discrete Continuous Dynam. Systems-B, 15 (2011), 93. doi: 10.3934/dcdsb.2011.15.93. Google Scholar

[24]

J. Carr, "Applications of Centre Manifold Theory,", Springer, (1981). Google Scholar

[25]

S. N. Chow and K. Lu, $C^k$ center unstable manifolds,, Proc. Roy. Soc. Edinburgh., 108 (1988), 303. doi: 10.1017/S0308210500014682. Google Scholar

[26]

J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer, (1995). doi: 10.1007/978-1-4612-4050-1. Google Scholar

[27]

Y. Qu, J.Wei and S. Ruan, Stability and bifurcation analysis in hematopoietic stem cell dynamics with multiple delays,, Physica D, 239 (2010), 2011. 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, (1983). Google Scholar

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