1:2 and 1:4 resonances in a two-dimensional discrete Hindmarsh–Rose model

Li, Bo; He, Zhen

doi:10.1007/s11071-014-1696-3

Cited by 30 publications

(15 citation statements)

References 37 publications

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“…In [21,27], we proved that map (3) possesses flip bifurcation, Neimark-Sacker bifurcation, and 1 : 1, 1 : 2, and 1 : 4 resonance. The aim of this paper is to prove that this discrete model possesses the 1 : 3 resonance.…”

Section: Journal Of Applied Mathematicsmentioning

confidence: 91%

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

2014

Journal of Applied Mathematics

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1 : 3 resonance of a two-dimensional discrete Hindmarsh-Rose model is discussed by normal form method and bifurcation theory. Numerical simulations are presented to illustrate the theoretical analysis, which predict the occurrence of a closed invariant circle, period-three saddle cycle, and homoclinic structure. Furthermore, it also displays the complex dynamical behaviors, especially the transitions between three main dynamical behaviors, namely, quiescence, spiking, and bursting.

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Section: Journal Of Applied Mathematicsmentioning

confidence: 91%

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

2014

Journal of Applied Mathematics

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“…Moreover, (vi1) if μ = μ 22 and ρ 1 = ρ 10 := 3ρ 2 , a 1:3 resonance occurs at E 2 ; (vi2) if μ = μ 22 and ρ 1 = ρ 11 := 2ρ 2 , a 1:4 resonance occurs at E 2 .…”

Section: Lemma 21mentioning

confidence: 99%

“…In what follows, we investigate the Neimark-Sacker bifurcation of E 2 if the parameter μ varies in the neighbourhood of μ 22 . We consider map (2) as follows:…”

Section: Neimark-sacker Bifurcation For Ementioning

confidence: 99%

“…Theorem 3.1 If μ = μ 22 and ϑ = 0, then map (2) undergoes a Neimark-Sacker bifurcation at E 2 (x 2 , y 2 ) when μ varies in the neighbourhood of μ 22 . Moreover, if ϑ < 0 (resp., ϑ > 0), then an attracting (resp., repelling) invariant closed circle bifurcates from the fixed point for μ > μ 22 (resp., μ < μ 22 ).…”

Section: Neimark-sacker Bifurcation For Ementioning

confidence: 99%

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Neimark–Sacker bifurcation and the generate cases of Kopel oligopoly model with different adjustment speed

Chen

2020

Adv Differ Equ

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In this paper, bifurcations and chaotic behaviours of Kopel oligopoly model with different adjustment speed are discussed. The results imply that the Kopel oligopoly model undergoes flip bifurcation, Neimark-Sacker bifurcation, 1:3 and 1:4 resonances, which could induce complex dynamics, especially global behaviours between different orbits. The conditions for the occurrence of three different kinds of bifurcation are derived. Furthermore, the numerical simulations provide us the case study of theoretical analysis and the corresponding dynamical behaviours, especially the occurrence of global orbits.

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“…This is a codimension two bifurcation point called strong resonance 1:2, and the different types of resonances have been well studied in previous works. 20,21 Thus, passing through the point A there is a bifurcation curve related to period-two orbits. Curve f (1) is the flip bifurcation curve.…”

Section: A Bifurcationsmentioning

confidence: 99%

Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate

Ren

Yuan

2017

Chaos

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A three dimensional microbial continuous culture model with a restrained microbial growth rate is studied in this paper. Two types of dilution rates are considered to investigate the dynamic behaviors of the model. For the unforced system, fold bifurcation and Hopf bifurcation are detected, and numerical simulations reveal that the system undergoes degenerate Hopf bifurcation. When the system is periodically forced, bifurcation diagrams for periodic solutions of period-one and period-two are given by researching the Poincar e map, corresponding to different bifurcation cases in the unforced system. Stable and unstable quasiperiodic solutions are obtained by NeimarkSacker bifurcation with different parameter values. Periodic solutions of various periods can occur or disappear and even change their stability, when the Poincar e map of the forced system undergoes Neimark-Sacker bifurcation, flip bifurcation, and fold bifurcation. Chaotic attractors generated by a cascade of period doublings and some phase portraits are given at last. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.5000152]The microbiological fermentation technique is widely applied in many fields for its economic importance. It is also investigated due to the complex behaviours observed during the process. We study bifurcations of the microbial continuous culture model with two types of dilution rates: the steady dilution rate and periodically forced dilution rate. For the steady dilution rate, we prove that the model undergoes fold bifurcation and Hopf bifurcation. When the dilution rate is periodically forced, we find that the bifurcations of the equilibria of the unforced system can be extended to the forced system as bifurcations of periodic solutions. Furthermore, periodic perturbation can give rise to complex dynamics, such as quasiperiodic solutions, periodic solutions of various periods, and chaos. In addition, the various periodic solutions can well explain the oscillation phenomena observed in laboratory experiments.

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1:2 and 1:4 resonances in a two-dimensional discrete Hindmarsh–Rose model

Cited by 30 publications

References 37 publications

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

1 : 3 Resonance and Chaos in a Discrete Hindmarsh-Rose Model

Neimark–Sacker bifurcation and the generate cases of Kopel oligopoly model with different adjustment speed

Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate

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