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Analysis of a scalar nonlocal peridynamic model with a sign changing kernel
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 |
References:
[1] |
A. Sharma, P. R. Sharma and M. D. Shrimali, Amplitude death in nonlinear oscillators with indirect coupling, Physics Lett. A, 376 (2012), 1562-1566. |
[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), 503-516.
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), 041904.
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), 569-605.
doi: 10.1007/BF01009349. |
[5] |
W. Jiang and J. 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. |
[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. |
[7] |
S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay, Phys. Rev. Lett., 79 (1997), 2911. |
[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. |
[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. |
[10] |
K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428. |
[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), 893-914.
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), 801-819.
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), 3245. |
[14] |
K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A, 206 (1995), 323-330.
doi: 10.1016/0375-9601(95)00654-L. |
[15] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer, New York, 1980. |
[16] |
Y. Kuang, On neutral delay logistic gause-type predator-prey systems, Dynamics and Stability of Systems, 6 (1991), 173-189. |
[17] |
J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations, Acta. Math. Sin., 45 (2002), 94-104. |
[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. |
[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. |
[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. |
[21] |
J. Hale and S. Lunel, "Introduction to Functional Differential Equations," Springer, New York, 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), 181-200.
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), 93-112.
doi: 10.3934/dcdsb.2011.15.93. |
[24] |
J. Carr, "Applications of Centre Manifold Theory," Springer, New York, 1981. |
[25] |
S. N. Chow and K. Lu, $C^k$ center unstable manifolds, Proc. Roy. Soc. Edinburgh., 108 (1988), 303-320.
doi: 10.1017/S0308210500014682. |
[26] |
J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer, New York, 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), 2011-2024.
doi: 10.1016/j.physd.2010.07.013. |
[28] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 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), 1562-1566. |
[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), 503-516.
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), 041904.
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), 569-605.
doi: 10.1007/BF01009349. |
[5] |
W. Jiang and J. 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. |
[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. |
[7] |
S. Kim, S. H. Park and C. S. Ryu, Multistability in coupled oscillator systems with time delay, Phys. Rev. Lett., 79 (1997), 2911. |
[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. |
[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. |
[10] |
K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428. |
[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), 893-914.
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), 801-819.
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), 3245. |
[14] |
K. Pyragas, Control of chaos via extended delay feedback, Phys. Lett. A, 206 (1995), 323-330.
doi: 10.1016/0375-9601(95)00654-L. |
[15] |
S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer, New York, 1980. |
[16] |
Y. Kuang, On neutral delay logistic gause-type predator-prey systems, Dynamics and Stability of Systems, 6 (1991), 173-189. |
[17] |
J. Wei and S. Ruan, Stability and global Hopf bifurcation for neutral differential equations, Acta. Math. Sin., 45 (2002), 94-104. |
[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. |
[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. |
[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. |
[21] |
J. Hale and S. Lunel, "Introduction to Functional Differential Equations," Springer, New York, 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), 181-200.
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), 93-112.
doi: 10.3934/dcdsb.2011.15.93. |
[24] |
J. Carr, "Applications of Centre Manifold Theory," Springer, New York, 1981. |
[25] |
S. N. Chow and K. Lu, $C^k$ center unstable manifolds, Proc. Roy. Soc. Edinburgh., 108 (1988), 303-320.
doi: 10.1017/S0308210500014682. |
[26] |
J. Wu, "Theory and Applications of Partial Functional Differential Equations," Springer, New York, 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), 2011-2024.
doi: 10.1016/j.physd.2010.07.013. |
[28] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer, New York, 1983. |
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