Dynamics of a limit cycle oscillator with extended delay feedback

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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

Abstract
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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:

[1]	A. Sharma, P. R. Sharma and M. D. Shrimali, Amplitude death in nonlinear oscillators with indirect coupling,, Physics Lett. A, 376 (2012), 1562.
[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.
[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.
[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.
[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.
[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.
[7]	S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay,, Phys. Rev. Lett., 79 (1997).
[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.
[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.
[10]	K. Pyragas, Continuous control of chaos by self-controlling feedback,, Phys. Lett. A, 170 (1992), 421.
[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.
[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.
[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).
[14]	K. Pyragas, Control of chaos via extended delay feedback,, Phys. Lett. A, 206 (1995), 323. doi: 10.1016/0375-9601(95)00654-L.
[15]	S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer, (1980).
[16]	Y. Kuang, On neutral delay logistic gause-type predator-prey systems,, Dynamics and Stability of Systems, 6 (1991), 173.
[17]	J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations,, Acta. Math. Sin., 45 (2002), 94.
[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.
[19]	M. Weedermann, Normal forms for neutral functional differential equations,, in, (2001), 361.
[20]	M. Weedermann, Hopf bifurcation calculations for scalar neutral delay differential equations,, Nonlinearity, 19 (2006), 2091. doi: 10.1088/0951-7715/19/9/005.
[21]	J. Hale and S. Lunel, "Introduction to Functional Differential Equations,", Springer, (1993).
[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.
[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.
[24]	J. Carr, "Applications of Centre Manifold Theory,", Springer, (1981).
[25]	S. N. Chow and K. Lu, $C^k$ center unstable manifolds,, Proc. Roy. Soc. Edinburgh., 108 (1988), 303. doi: 10.1017/S0308210500014682.
[26]	J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer, (1995). doi: 10.1007/978-1-4612-4050-1.
[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.
[28]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer, (1983).

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.
[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.
[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.
[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.
[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.
[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.
[7]	S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay,, Phys. Rev. Lett., 79 (1997).
[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.
[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.
[10]	K. Pyragas, Continuous control of chaos by self-controlling feedback,, Phys. Lett. A, 170 (1992), 421.
[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.
[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.
[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).
[14]	K. Pyragas, Control of chaos via extended delay feedback,, Phys. Lett. A, 206 (1995), 323. doi: 10.1016/0375-9601(95)00654-L.
[15]	S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer, (1980).
[16]	Y. Kuang, On neutral delay logistic gause-type predator-prey systems,, Dynamics and Stability of Systems, 6 (1991), 173.
[17]	J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations,, Acta. Math. Sin., 45 (2002), 94.
[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.
[19]	M. Weedermann, Normal forms for neutral functional differential equations,, in, (2001), 361.
[20]	M. Weedermann, Hopf bifurcation calculations for scalar neutral delay differential equations,, Nonlinearity, 19 (2006), 2091. doi: 10.1088/0951-7715/19/9/005.
[21]	J. Hale and S. Lunel, "Introduction to Functional Differential Equations,", Springer, (1993).
[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.
[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.
[24]	J. Carr, "Applications of Centre Manifold Theory,", Springer, (1981).
[25]	S. N. Chow and K. Lu, $C^k$ center unstable manifolds,, Proc. Roy. Soc. Edinburgh., 108 (1988), 303. doi: 10.1017/S0308210500014682.
[26]	J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer, (1995). doi: 10.1007/978-1-4612-4050-1.
[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.
[28]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer, (1983).

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