Singular Hopf bifurcation in singularly perturbed system with delay

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

In this paper, we study the singular Hopf bifurcation in a class of singularly perturbed systems with delay. The key condition for the generation of a singular Hopf bifurcation involves the existence of eigenvalues of the singular pure imaginary part. The necessary and sufficient conditions for the generation of a singular Hopf bifurcation are discussed in a class of general nonlinear ( n, 1)- fast-slow systems with delay on slow variables (n ≥ 1).

Keywords

singularly perturbed system / delay / singular Hopf bifurcation

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] S. M. Baer, T. Erneux, Singular Hopf bifurcation to relaxation oscillations, SIAM J. Appl. Math., 46 (1986), 721-739. https://doi.org/10.1137/0146047

[2] B. Braaksma, Singular Hopf bifurcation in systems with fast and slow variables, J. Nonlinear Sci., 8 (1986), 457-490. https://doi.org/10.1007/s003329900058

[3] L. J. Yang, X. W. Zeng, Stability of singular Hopf bifurcations, J. Differ. Equ., 206 (2004), 30-54. https://doi.org/10.1016/j.jde.2004.08.002

[4] C. Wang, X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow-fast predator- prey model of generalized holling type III, J. Differ. Equ., 267 (2019), 3397-3441. https://doi.org/10.1016/j.jde.2019.04.008

[5] M. Engel, C. Kuehn, M. Petrera, Y. Suris, Discretized fast-slow systems with canards in two dimensions, J. Nonlinear Sci., 32 (2022), 19. https://doi.org/10.1007/s00332-021-09778-2

[6] K. H. M. Nyman, P. Ashwin, P. D. Ditlevsen, Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems, Nonlinearity, 33 (2020), 2853-2904. https://doi.org/10.1088/1361-6544/ab7292

[7] R. Curtu, Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network, Phys. D-Nonlinear Phenom., 239 (2010), 504-514. https://doi.org/10.1016/j.physd.2009.12.010

[8] K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, et al., The forced van der Pol equation II: Canards in the reduced system, SIAM J. Appl. Dyn. Syst., 2 (2003), 570-608. https://doi.org/10.1137/S1111111102419130

[9] M. Desroches, A. Guillamon, E. Ponce, R. Prohens, S. Rodrigues, A. E. Teruel, Canards, folded nodes, and mixed-mode oscillations in piecewise-linear slow-fast systems, SIAM Rev., 58 (2016), 653-691. https://doi.org/10.1137/15M1014528

[10] I. A. Garcia, J. Llibre, S. Maza, On the periodic orbit bifurcating from a zero Hopf bifurcation in systems with two slow and one fast variables, Appl. Math. Comput., 232 (2014), 84-90. https://doi.org/10.1016/j.amc.2013.12.184

[11] M. Wechselberger, Existence and bifurcation of canards in R3 in the case of a folded node, SIAM J. Appl. Dyn. Syst., 4 (2005), 101-139. https://doi.org/10.1137/030601995

[12] P. De Maesschalck, E. Kutafina, N. Popovi, Sector-delayed-hopf-type mixed-mode oscillations in a prototypical three-time-scale model, Appl. Math. Comput., 273 (2016), 337-352. https://doi.org/10.1016/j.amc.2015.09.083

[13] H. Jardon-Kojakhmetov, C. Kuehn, M. Steinert, The hyperbolic umbilic singularity in fast-slow systems, Nonlinearity, 37 (2024), 95036. https://doi.org/10.1088/1361-6544/ad6bde

[14] J. Guckenheimer, Singular Hopf bifurcation in systems with two slow variables, SIAM J. Appl. Dyn. Syst., 7 (2008), 1355-1377. https://doi.org/10.1137/080718528

[15] K. U. Kristiansen, The dud canard: Existence of strong canard cycles in R3, J. Differ. Equ., 375 (2023), 706-749. https://doi.org/10.1016/j.jde.2023.09.008

[16] C. Wang, X. Zhang, Relaxation oscillations in a slow-fast modified leslie-gower model, Appl. Math. Lett., 87 (2019), 147-153. https://doi.org/10.1016/j.aml.2018.07.029

[17] S. M. Li, C. Wang, K. L. Wu, Relaxation oscillations of a slow-fast predator-prey model with a piecewise smooth functional response, Appl. Math. Lett., 113 (2021), 106852. https://doi.org/10.1016/j.aml.2020.106852

[18] S. M. Li, X. L. Wang, X. L. Li, K. L. Wu, Relaxation oscillations for leslie-type predator-prey model with holling type I response functional function, Appl. Math. Lett., 120 (2021), 107328. https://doi.org/10.1016/j.aml.2021.107328

[19] M. Mimura, M. Nagayama, K. Sakamoto, Singular Hopf-bifurcation in a chemical reaction- diffusion system, Appl. Math. Lett., 11 (1998), 127-131. https://doi.org/10.1016/S0893-9659(98)00069-X

[20] J. Mallet-Paret, R. D. Nussbaum, A differential delay equation arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292. https://doi.org/10.1137/0520019

[21] M. C. Soriano, J. Garcia-Ojalvo, C. R. Mirasso, I. Fischer, Complex photonics: Dynamics and applications of delay-coupled semiconductor lasers, Rev. Mod. Phys., 85 (2013), 421-470. https://doi.org/10.1103/RevModPhys.85.421

[22] S. Yanchuk, S. Ruschel, J. Sieber, M. Wolfrum, Temporal dissipative solitons in time-delay feedback systems, Phys. Rev. Lett., 123 (2019), 053901. https://doi.org/10.1103/PhysRevLett.123.053901

[23] M. Wanic, C. Jasiukiewicz, Z. Toklikishvili, V. Jandieri, M. Trybus, E. Jartych, et al., Entanglement properties of photon-magnon crystal from nonlinear perspective, Physica D: Nonlinear Phenom., 476 (2025), 134699. https://doi.org/10.1016/j.physd.2025.134699

[24] O. Diekmann, M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations , Birkhauser Basel, (2007), 187-200. https://doi.org/10.1007/978-3-7643-7794-612

[25] G. Agaba, Y. Kyrychko, K. Blyuss, Dynamics of vaccination in a time-delayed epidemic model with awareness, Math. Biosci., 294 (2017), 92-99. https://doi.org/10.1016/j.mbs.2017.05.006

[26] L. S. Young, S. Ruschel, S. Yanchuk, T. Pereira, Consequences of delays and imperfect implementation of isolation in epidemic control, Sci. Rep., 9 (2019), 3505. https://doi.org/10.1038/s41598-018-38142-7

[27] T. Insperger, G. Stepan, J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dyn., 47 (2007), 275-283. https://doi.org/10.1007/s11071-006-9044-1

[28] A. Otto, G. Radons, The influence of tangential and torsional vibrations on the stability lobes in metal cutting, Nonlinear Dyn., 82 (2015), 1989-2000. https://doi.org/10.1007/s11071-015-2293-9

[29] M. Wanic, Z. Toklikishvili, S. K. Mishra, M. Trybus, L. Chotorlishvili, Magnetoelectric fractals, Magnetoelectric parametric resonance and Hopf bifurcation, Physica D: Nonlinear Phenom., 467 (2024), 134257. https://doi.org/10.1016/j.physd.2024.134257

[30] E. M. Izhikevich, Polychronization: Computation with spikes, Neural Comput., 18 (2006), 245-282. https://doi.org/10.1162/neco.2006.18.2.245

[31] T. W. Ko, G. B. Ermentrout, Effects of axonal time delay on synchronization and wave formation in sparsely coupled neuronal oscillators, Phys. Rev. E, 76 (2007), 56206. https://doi.org/10.1103/PhysRevE.76.056206

[32] W. Xu, H. Shu, Z. Tang, H. Wang, Complex dynamics in a general diffusive predator- prey model with predator maturation delay, J. Dyn. Differ. Equ., 36 (2024), 1879-1904. https://doi.org/10.1007/s10884-022-10176-9

[33] X. Zhang, Z. Liu, Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay, Appl. Math. Model., 91 (2021), 530-548. https://doi.org/10.1016/j.apm.2020.08.054

[34] Y. G. Zheng, Z. H. Wang, Stability analysis of nonlinear dynamic systems with slowly and periodically varying delay, Commun. Nonlinear Sci., 17 (2012), 3999-4009. https://doi.org/10.1016/j.cnsns.2012.02.026

[35] H. Wang, Y. Qian, Dynamics of a delayed predator-prey system in highland pasture, Qual. Theor. Dyn. Syst., 24 (2025), 44. https://doi.org/10.1007/s12346-024-01207-5

[36] X. Ai, Y. Zhang, Bifurcation analysis of a singular predator-prey plankton system with biotoxin delay and constant harvesting, Nonlinear Dyn., 113 (2025), 895-929. https://doi.org/10.1007/s11071-024-10231-8