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Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise
Periodic forcing on degenerate Hopf bifurcation
Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450001, China |
This paper is devoted to the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods.
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J. M. González–Miranda,
On the effect of circadian oscillations on biochemical cell signaling by NF–B, J. Theor. Biol., 335 (2013), 283-294.
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[11] |
P. Gross,
On harmonic resonance in forced nonlinear oscillators exhibiting a Hopf bifurcation, IMA J. Appl. Math., 50 (1993), 1-12.
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L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.
doi: 10.1007/978–1–4684–0392–3. |
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J. M. Gambaudo,
Perturbation of a Hopf bifurcation by an external time–periodic forcing, J. Differ. Equations, 57 (1985), 172-199.
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W. L. Kath,
Resonance in periodically perturbed Hopf bifurcation, Stud. Appl. Math., 65 (1981), 95-112.
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Y. A. Kuznetsov, S. Muratori and S. Rinaldi,
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J. L. Ren and Q. G. Yuan, Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate, Chaos, 27 (2017), 083124, 15pp.
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J. L. Ren and L. P. Yu,
Codimension–two bifurcation, chaos control in a discrete–time information diffusion model, J. Nonlinear Sci., 26 (2016), 1895-1931.
doi: 10.1007/s00332–016–9323–8. |
[22] |
J. L. Ren and X. P. Li, Bifurcations in a seasonally forced predator–prey model with generalized Holling type Ⅳ functional response, Int. J. Bifurcat. Chaos, 26 (2016), 1650203, 19pp.
doi: 10.1142/S0218127416502035. |
[23] |
S. Rosenblat and D. S. Cohen,
Periodically perturbed bifurcation–1. Simple bifurcation, Stud. Appl. Math., 63 (1980), 1-23.
doi: 10.1002/sapm19806311. |
[24] |
S. Rosenblat and D. S. Cohen,
Periodically perturbed bifurcation. Ⅱ. Hopf bifurcation, Stud. Appl. Math., 64 (1981), 143-175.
doi: 10.1002/sapm1981642143. |
[25] |
A. Rego–Costa, F. Debarre and L. M. Chevin,
Chaos and the (un)predictability of evolution in a changing environment, Evolution, 72 (2018), 375-385.
doi: 10.1111/evo.13407. |
[26] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems(2nd edition), (Springer, New York, NY), 2007. |
[27] |
Y. W. Tao, X. P. Li and J. L. Ren,
A repeated yielding model under periodic perturbation., Nonlinear Dynam., 94 (2018), 2511-2525.
doi: 10.1007/s11071–018–4506–5. |
[28] |
Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems, (World Scientific), 1996.
doi: 10.1142/9789812830548. |
[29] |
D. M. Xiao and H. P. Zhu,
Multiple focus and Hopf bifurcations in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math., 66 (2006), 802-819.
doi: 10.1137/050623449. |
[30] |
Y. Y. Zhang and M. Golubitsky,
Periodically forced Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 10 (2011), 1272-1306.
doi: 10.1137/10078637X. |
show all references
References:
[1] |
F. Barraquand, S. Louca and K. C. Abbott, Moving forward in circles: Challenges and opportunities in modelling population cycles, Ecol. Lett., 20 (2017), 1074-1092. Google Scholar |
[2] |
A. K. Bajaj,
Resonant parametric perturbations of the Hopf bifurcation, J. Math. Anal. Appl., 115 (1986), 214-224.
doi: 10.1016/0022–247X(86)90035–1. |
[3] |
J. H. Bao and Q. G. Yang,
A new method to find homoclinic and heteroclinic orbits, Appl. Math. Comput., 217 (2011), 6526-6540.
doi: 10.1016/j.amc.2011.01.032. |
[4] |
E. Benincà, B. Ballantine and S. P. Ellner, Species fluctuations sustained by a cyclic succession at the edge of chaos, P. Natl. Acad. Sci. USA, 112 (2015), 6389-6394. Google Scholar |
[5] |
S. N. Chow and M. P. John,
Integral averaging and bifurcation, J. Differ. Equations, 26 (1977), 112-159.
doi: 10.1016/0022–0396(77)90101–2. |
[6] |
Z. B. Cheng and F. F. Li, Positive periodic solutions for a kind of second–order neutral differential equations with variable coefficient and delay, Mediterr. J. Math., 15 (2018), Paper No. 134, 19 pp.
doi: 10.1007/s00009–018–1184–y. |
[7] |
Z. B. Cheng and Q. G. Yuan, Damped superlinear duffing equation with strong singularity of repulsive type, J. Fix. Piont Theory A, 22 (2020), Paper No. 37, 18 pp.
doi: 10.1007/s11784–020–0774–z. |
[8] |
E. J. Doedel and B. E. Oldeman, AUTO–07P: continuation and bifurcation software for ordinary differential equations, http://cmvl.cs.concordia.ca/auto., 2012. Google Scholar |
[9] |
W. W. Farr, C. Z. Li, I. S. Labouriau and W. F. Langford,
Degenerate Hopf bifurcation formulas and Hilbert's 16th problem, SIAM J. Math. Anal., 20 (1989), 13-30.
doi: 10.1137/0520002. |
[10] |
J. M. González–Miranda,
On the effect of circadian oscillations on biochemical cell signaling by NF–B, J. Theor. Biol., 335 (2013), 283-294.
doi: 10.1016/j.jtbi.2013.06.027. |
[11] |
P. Gross,
On harmonic resonance in forced nonlinear oscillators exhibiting a Hopf bifurcation, IMA J. Appl. Math., 50 (1993), 1-12.
doi: 10.1093/imamat/50.1.1. |
[12] |
L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1991.
doi: 10.1007/978–1–4684–0392–3. |
[13] |
J. M. Gambaudo,
Perturbation of a Hopf bifurcation by an external time–periodic forcing, J. Differ. Equations, 57 (1985), 172-199.
doi: 10.1016/0022–0396(85)90076–2. |
[14] |
W. L. Kath,
Resonance in periodically perturbed Hopf bifurcation, Stud. Appl. Math., 65 (1981), 95-112.
doi: 10.1002/sapm198165295. |
[15] |
Y. A. Kuznetsov, S. Muratori and S. Rinaldi,
Bifurcations and chaos in a periodic predator–prey model, Int. J. Bifurcat. Chaos, 2 (1992), 117-128.
doi: 10.1142/S0218127492000112. |
[16] |
X. P. Li, J. L. Ren and S. A. Campbell,
How seasonal forcing influences the complexity of a predator–prey system, Discrete Cont. Dyn.–B, 23 (2018), 785-807.
doi: 10.3934/dcdsb.2018043. |
[17] |
M. A. McKarnin, L. D. Schmidt and R. Aris,
Response of nonlinear oscillators to forced oscillations: Three chemical reaction case studies, Chem. Eng. Sci., 43 (1988), 2833-2844.
doi: 10.1016/0009–2509(88)80026–5. |
[18] |
N. S. Namachchivaya and S. T. Ariaratnam,
Periodically Perturbed Hopf Bifurcation, SIAM J. Appl. Math., 47 (1987), 15-39.
doi: 10.1137/0147002. |
[19] |
L. M. Perko,
Higher order averaging and related methods for perturbed periodic and quasi–periodic systems, SIAM J. Appl. Math., 17 (1969), 698-724.
doi: 10.1137/0117065. |
[20] |
J. L. Ren and Q. G. Yuan, Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate, Chaos, 27 (2017), 083124, 15pp.
doi: 10.1063/1.5000152. |
[21] |
J. L. Ren and L. P. Yu,
Codimension–two bifurcation, chaos control in a discrete–time information diffusion model, J. Nonlinear Sci., 26 (2016), 1895-1931.
doi: 10.1007/s00332–016–9323–8. |
[22] |
J. L. Ren and X. P. Li, Bifurcations in a seasonally forced predator–prey model with generalized Holling type Ⅳ functional response, Int. J. Bifurcat. Chaos, 26 (2016), 1650203, 19pp.
doi: 10.1142/S0218127416502035. |
[23] |
S. Rosenblat and D. S. Cohen,
Periodically perturbed bifurcation–1. Simple bifurcation, Stud. Appl. Math., 63 (1980), 1-23.
doi: 10.1002/sapm19806311. |
[24] |
S. Rosenblat and D. S. Cohen,
Periodically perturbed bifurcation. Ⅱ. Hopf bifurcation, Stud. Appl. Math., 64 (1981), 143-175.
doi: 10.1002/sapm1981642143. |
[25] |
A. Rego–Costa, F. Debarre and L. M. Chevin,
Chaos and the (un)predictability of evolution in a changing environment, Evolution, 72 (2018), 375-385.
doi: 10.1111/evo.13407. |
[26] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems(2nd edition), (Springer, New York, NY), 2007. |
[27] |
Y. W. Tao, X. P. Li and J. L. Ren,
A repeated yielding model under periodic perturbation., Nonlinear Dynam., 94 (2018), 2511-2525.
doi: 10.1007/s11071–018–4506–5. |
[28] |
Y. Takeuchi, Global Dynamical Properties of Lotka–Volterra Systems, (World Scientific), 1996.
doi: 10.1142/9789812830548. |
[29] |
D. M. Xiao and H. P. Zhu,
Multiple focus and Hopf bifurcations in a predator–prey system with nonmonotonic functional response, SIAM J. Appl. Math., 66 (2006), 802-819.
doi: 10.1137/050623449. |
[30] |
Y. Y. Zhang and M. Golubitsky,
Periodically forced Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 10 (2011), 1272-1306.
doi: 10.1137/10078637X. |










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