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

Li, Bo; He, Zhen

doi:10.1155/2014/896478

Cited by 25 publications

(3 citation statements)

References 30 publications

(44 reference statements)

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“…In 2015, Felicio et al [21] studied the discrete analog of the continuous Hindmarsh-Rose neuron model and had shown the presence of Arnold tongues and the devil's staircase. Researchers in [22][23][24] have explored the dynamical behaviors and bifurcation patterns of the discrete two-dimensional Hindmarsh-Rose neuron model.…”

Section: Introductionmentioning

confidence: 99%

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Muni

2023

Phys. Scr.

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Similar to period-doubling bifurcation of fixed points, periodic orbits, it has been found since 1980's that a corresponding doubling bifurcation can also be found in the case of quasiperiodic orbits. Doubling bifurcations of quasiperiodic orbits has an important consequence on the dynamics of the system under consideration. Recently, it has been shown that subsequent doublings of quasiperiodic closed invariant curves leads to the formation of Shilnikov attractors. In this contribution, we illustrate for the first time in a discrete neuron system, the phenomenon of doubling of closed invariant curves. We also show the presence of mode-locked orbits and the geometry of one-dimensional unstable manifolds associated to them resulting in the formation of a resonant closed invariant curve. Moreover, we illustrate the phenomenon of crisis, and multistability in the system.

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Section: Introductionmentioning

confidence: 99%

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Muni

2023

Phys. Scr.

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“…e codimension-2 bifurcations of a delayed discrete-time Hopfield neural network, discrete epidemic model with nonlinear incidence rate, and discrete SIS epidemic model with standard incidence are analyzed in [22][23][24]. A discrete Hindmarsh-Rose model undergoes codimension-2 bifurcations with 1 : 2 and 1 : 4 strong resonances [25], and 1 : 3 strong resonance [26]. 1 : 1 strong resonance bifurcation of a discrete predator-prey model with nonmonotonic functional response has also been studied in [27].…”

Section: Introductionmentioning

confidence: 99%

Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes

2020

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This paper concerns a discrete wild and sterile mosquito model with a proportional release rate of sterile mosquitoes. It is shown that the discrete model undergoes codimension-2 bifurcations with 1 : 2, 1 : 3, and 1 : 4 strong resonances by applying the bifurcation theory. Some numerical simulations, including codimension-2 bifurcation diagrams, maximum Lyapunov exponents diagrams, and phase portraits, are also presented to illustrate the validity of theoretical results and display the complex dynamical behaviors. Moreover, two control strategies are applied to the model.

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“…Yu and Cao [9] presented the existence of one-parameter bifurcations of three-dimensional discrete Hindmarsh-Rose, and numerical simulations were given to illustrate the bifurcation analysis. One can find more information on bifurcations of the Hindmarsh-Rosetype model in [10][11][12][13][14][15][16][17][18] and on related one(two)-parameter bifurcation theory in [19][20][21][22][23][24][25][26][27]. Herein, we discuss the following revised model [28] in 2007:…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of a two-dimensional discrete Hindmarsh–Rose type model

2019

Adv Differ Equ

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In this paper, bifurcation analysis of a discrete Hindmarsh-Rose model is carried out in the plane. This paper shows that the model undergoes a flip bifurcation, a Neimark-Sacker bifurcation, and 1 : 2 resonance which includes a pitchfork bifurcation, a Neimark-Sacker bifurcation, and a heteroclinic bifurcation. The sufficient conditions of existence of the fixed points and their stability are first derived. The flip bifurcation and Neimark-Sacker bifurcation are analyzed by using the inner product method and normal form theory. The conditions for the occurrence of 1 : 2 resonance are also presented. Furthermore, the sufficient conditions of pitchfork, Neimark-Sacker, and heteroclinic bifurcations are derived and expressed by implicit functions. The numerical analysis shows us consistence with the theoretical results and exhibits interesting dynamics, especially symmetric and invariant closed orbits. The dynamics observed in this paper can be used to mimic the dynamical behaviors of one single neuron and design a humanoid locomotion model for applications in bio-engineering and so on.

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

Cited by 25 publications

References 30 publications

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Mode-locked orbits, doubling of invariant curves in discrete Hindmarsh-Rose neuron model

Codimension-2 Bifurcation Analysis and Control of a Discrete Mosquito Model with a Proportional Release Rate of Sterile Mosquitoes

Bifurcation analysis of a two-dimensional discrete Hindmarsh–Rose type model

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