Hopf bifurcation analysis for Parkinson oscillation with heterogeneous delays: A theoretical derivation and simulation analysis

Wang, Zhizhi; Hu, Bing; Zhu, Luyao; Lin, Jiahui; Xu, Minbo; Wang, Dingjiang

doi:10.1016/j.cnsns.2022.106614

Cited by 13 publications

(8 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Hopf bifurcation critical condition of the system (3) was provided in [3]. However, in models (1) to (3), the delays are discrete, and distributed delays are rarely introduced into neuron models with biological backgrounds.…”

Section: Introductionmentioning

confidence: 99%

“…According to the simulation result in [3], the parameters are τ S = 12.80 ms, τ G = 20 ms, τ E = 10-20 ms, and τ I = 10-20 ms. Therefore, the results in [3] are only for special parameters. Note that model (6) has four discrete delays.…”

Section: Introductionmentioning

confidence: 99%

“…For convenience, we considered K i (i = 1, 2, 3, 4) as the weak gamma functions, setting 3,4), and let…”

Section: Introductionmentioning

confidence: 99%

“…In other words, all of the solutions of system (7) are boundedness. According to the parameter values in [3]: M S = 300 spk/s, B S = 17 spk/s, M G = 400 spk/s, B G = 75 spk/s, M E = 71.77 spk/s, B E = 3.62 spk/s, M I = 276 spk/s, B I = 7.18 spk/s, we know that F Y (x) are monotone increasing functions for Y = S, G, E, and I. Therefore, system (6) has a unique equilibrium point (S * , G * , E * , I * ) T .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Oscillatory Behavior of the Solutions for a Parkinson’s Disease Model with Discrete and Distributed Delays

Feng

2024

Axioms

View full text Add to dashboard Cite

In this paper, the oscillatory behavior of the solutions for a Parkinson’s disease model with discrete and distributed delays is discussed. The distributed delay terms can be changed to new functions such that the original model is equivalent to a system in which it only has discrete delays. Using Taylor’s expansion, the system can be linearized at the equilibrium to obtain both the linearized part and the nonlinearized part. One can see that the nonlinearized part is a disturbed term of the system. Therefore, the instability of the linearized system implies the instability of the whole system. If a system is unstable for a small delay, then the instability of this system will be maintained as the delay increased. By analyzing the linearized system at the smallest delay, some sufficient conditions to guarantee the existence of oscillatory solutions for a delayed Parkinson’s disease system can be obtained. It is found that under suitable conditions on the parameters, time delay affects the stability of the system. The present method does not need to consider a bifurcating equation. Some numerical simulations are provided to illustrate the theoretical result.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For convenience, we considered K i (i = 1, 2, 3, 4) as the weak gamma functions, setting 3,4), and let…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Oscillatory Behavior of the Solutions for a Parkinson’s Disease Model with Discrete and Distributed Delays

Feng

2024

Axioms

View full text Add to dashboard Cite

show abstract

“…According to the simulation result in [16], the parameters τS = 12.80ms, τG = 20ms, τE = 10-20ms, and τI = 10-20ms. Therefore, the results in [14] and [16] are only for special parameters.…”

Section: Introductionmentioning

confidence: 99%

Periodic Oscillation of the Solutions for a Parkinson's Disease Model

Feng

2023

ARJOM

View full text Add to dashboard Cite

In this paper, the oscillation of the solutions for a Parkinson's disease model with multiple delays is discussed. By linearizing the system at the equilibrium point and analyzing the instability of the linearized system, some sufficient conditions to guarantee the existence of periodic oscillation of the solutions for a delayed Parkinson's disease system are obtained. It is found that under suitable conditions on the parameters, time delay affects the stability of the system. The present method does not need to consider a bifurcating equation. Some numerical simulations are provided to illustrate our theoretical prediction.

show abstract

Numerical Bifurcation of a Delayed Diffusive Hematopoiesis Model with Dirichlet Boundary Conditions

Liu,

Wang

2024

Ukr Math J

View full text Add to dashboard Cite

Hopf bifurcation analysis for Parkinson oscillation with heterogeneous delays: A theoretical derivation and simulation analysis

Cited by 13 publications

References 49 publications

Oscillatory Behavior of the Solutions for a Parkinson’s Disease Model with Discrete and Distributed Delays

Oscillatory Behavior of the Solutions for a Parkinson’s Disease Model with Discrete and Distributed Delays

Periodic Oscillation of the Solutions for a Parkinson's Disease Model

Numerical Bifurcation of a Delayed Diffusive Hematopoiesis Model with Dirichlet Boundary Conditions

Contact Info

Product

Resources

About