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May  2021, 26(5): 2857-2877. doi: 10.3934/dcdsb.2020208

Periodic forcing on degenerate Hopf bifurcation

Qigang Yuan and  Jingli Ren ,

Henan Academy of Big Data, Zhengzhou University, Zhengzhou 450001, China

* Corresponding author: renjl@zzu.edu.cn

Received  October 2019 Revised  April 2020 Published  May 2021 Early access  July 2020

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.

Keywords: Periodic Forcing, Degenerate Hopf Bifurcation, Averaging method, Poincaré map, Complexity.
Mathematics Subject Classification: Primary: 34C23, 37G15, 37G10.
Citation: Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
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. 

[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.

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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. 

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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.

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E. J. Doedel and B. E. Oldeman, AUTO–07P: continuation and bifurcation software for ordinary differential equations, http://cmvl.cs.concordia.ca/auto., 2012.

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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.

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. 

[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. 

[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.

[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.

Figure 1.  (a) Phase portrait near a degenerate Hopf bifurcation for $ r_{1} = 0.3 $, $ r_{2} = 0.6 $, $ a_{1} = 0.42 $, $ a_{2} = 0.6 $, $ b_{1} = 1.0857 $, $ b_{2} = 0.25 $. (b) Phase portrait near a degenerate Hopf bifurcation for $ r_{1} = 0.3 $, $ r_{2} = 0.6 $, $ a_{1} = 0.447 $, $ a_{2} = 0.6 $, $ b_{1} = 1.13 $, $ b_{2} = 0.25 $
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Figure 2.  Phase portrait of the averaged system, the red points are the equilibria. (a) For $ \mu = 0.06 $, $ \nu = 3 $, $ \alpha = 1 $, $ \beta = 2 $. (b) For $ \mu = 0 $, $ \nu = 3 $, $ \alpha = 1 $, $ \beta = 2 $
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Figure 3.  (b) Bifurcation diagram of the forced system (8) in $ (a_{1},\epsilon) $-plane for $ r_{1} = 0.3,b_{1} = 1.13,r_{2} = 0.6,a_{2} = 0.6,b_{2} = 0.25 $. (c) and (d) Partial enlargements of (b). The solutions of system (8) are as follows, region 1-unstable period-one solution and stable quasiperiodic solution; region 2-unstable period-one solution and unstable period-two solution; region 3-unstable period-one solution, unstable period-two solutions, and period-four solution; region 4-stable and unstable period-one solutions; region 5-stable and unstable period-one solutions
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Figure 4.  (a) Bifurcation diagram of the forced system (8) in $ (a_{1},\epsilon) $-plane for $ r_{1} = 0.3,b_{1} = 0.85,r_{2} = 0.4,a_{2} = 0.6,b_{2} = 0.4 $. (b), (c) Partial enlargements of (a). The solutions of system (8) are as follows, region 1-unstable period-one solutions, unstable period-two solutions, and period-four solution; region 2-stable and unstable period-one solutions; region 3-unstable period-one solutions and stable quasiperiodic solution; region 4-no periodic solution; region 5-unstable period-one solutions and unstable period-two solutions; region 6-stable and unstable period-one solutions
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Figure 5.  (a) Bifurcation diagram of the forced system (26) in $ (b_{1},\epsilon) $-plane for $ r_{1} = 0.3,a_{1} = 0.2,r_{2} = 0.4,a_{2} = 0.4,b_{2} = 0.162 $. (b) Partial enlargements of (a). The solutions of system (26) are as follows, region 1-stable and unstable period-one solutions, unstable quasiperiodic solution; region 2- unstable period-one solutions; region 3-unstable period-one solutions, unstable period-two solutions, and period-four solution; region 4-unstable period-one solutions, stable period-two solutions; region 5-stable and unstable period-one solutions
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Figure 6.  (a) Bifurcation diagram of the forced system (26) in $ (b_{1},\epsilon) $-plane for $ r_{1} = 0.3,b_{1} = 0.6,r_{2} = 0.4,a_{2} = 0.6,b_{2} = 0.4 $. (b) bifurcation curves of a period-one saddle. (c) and (d) Partial enlargements of (a). The solutions of system (26) are as follows, region 1-stable and unstable period-one solutions; region 2- unstable period-one solutions and stable quasiperiodic solution; region 3-unstable period-one solutions, unstable period-two solutions, and period-four solution; region 4-stable period-one solutions; region 5-unstable period-one solutions, unstable period-two solutions; region 6-stable and unstable period-one solutions
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Figure 7.  (a) Bifurcation diagram of the forced system (27) in $ (a_{2},\epsilon) $-plane for $ r_{1} = 0.3, a_{1} = 0.2, b_{1} = 0.5,r_{2} = 0.4, b_{2} = 0.162 $. (b) Bifurcation diagram of the forced system (27) in $ (a_{2},\epsilon) $-plane for $ r_{1} = 0.3, a_{1} = 0.6, b_{1} = 0.83, r_{2} = 0.4, b_{2} = 0.4 $
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Figure 8.  Bifurcation diagram of the forced system (28) in $ (b_{2},\epsilon) $-plane for the case $ r_{1} = 0.3,a_{1} = 0.2,b_{1} = 0.83,r_{2} = 0.4,a_{2} = 0.4 $
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Figure 9.  (a) Bifurcation diagram of the forced system (28) in $ (b_{2},\epsilon) $-plane for $ r_{1} = 0.3,a_{1} = 0.6,b_{1} = 0.85,r_{2} = 0.4,a_{2} = 0.6 $. (b), (c) and (d) Partial enlargements of (a). The solutions of system (28) are as follows, region 1-unstable period-one solution, unstable period-two solutions, and period-four solution or chaos in some subregion; region 2-stable and unstable period-one solution; region 3-unstable period-one solution and stable quasiperiodic solution; region 4-unstable period-one solutions and unstable period-two solutions; region 5-stable and unstable period-one solutions; region 6-unstable period-one solution, unstable period-two solutions, and period-four solution
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Figure 10.  Phase portraits and Poincaré section of the periodically forced system. (a) A stable period-two orbit in system (27) for $ r_{1} = 0.3 $, $ r_{2} = 0.4 $, $ a_{1} = 0.6 $, $ a_{2} = 0.4 $, $ b_{1} = 0.83 $, $ b_{2} = 0.4 $, $ \epsilon = 0.337 $. (b) A stable period-four orbit in system (27) for $ r_{1} = 0.3 $, $ r_{2} = 0.4 $, $ a_{1} = 0.6 $, $ a_{2} = 0.4 $, $ b_{1} = 0.83 $, $ b_{2} = 0.4 $, $ \epsilon = 0.277 $. (c) Torus in system (8) for $ r_{1} = 0.3 $, $ r_{2} = 0.6 $, $ a_{1} = 0.45183 $, $ a_{2} = 0.6 $, $ b_{1} = 1.13 $, $ b_{2} = 0.25 $, $ \epsilon = 0.277 $. (d) Poincaré section of the torus
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