# 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
##### 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

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
 [1] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [2] Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434 [3] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346 [4] João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138 [5] Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349 [6] Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263 [7] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [8] Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117 [9] Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048 [10] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [11] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [12] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [13] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016 [14] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [15] Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267 [16] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [17] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [18] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [19] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

2019 Impact Factor: 1.27