bifurcation parameter
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2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Maximilian Engel ◽  
Christian Kuehn ◽  
Matteo Petrera ◽  
Yuri Suris

AbstractWe study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem.


Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3324
Author(s):  
Xinxin Qie ◽  
Quanbao Ji

This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca2+ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc.


2021 ◽  
Vol 11 (1) ◽  
pp. 684-701
Author(s):  
Siyu Chen ◽  
Carlos Alberto Santos ◽  
Minbo Yang ◽  
Jiazheng Zhou

Abstract In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.


2021 ◽  
Vol 31 (14) ◽  
Author(s):  
Binfeng Xie ◽  
Zhengce Zhang ◽  
Na Zhang

In this work, a prey–predator system with Holling type II response function including a Michaelis–Menten type capture and fear effect is put forward to be studied. Firstly, the existence and stability of equilibria of the system are discussed. Then, by considering the harvesting coefficient as bifurcation parameter, the occurrence of Hopf bifurcation at the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation are proved. Furthermore, through the analysis of fear effect and capture item, we find that: (i) the fear effect can either stabilize the system by excluding periodic solutions or destroy the stability of the system and produce periodic oscillation behavior; (ii) increasing the level of fear can reduce the final number of predators, but not lead to extinction; (iii) the harvesting coefficient also has significant influence on the persistence of the predator. Finally, numerical simulations are presented to illustrate the results.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Binfeng Xie

AbstractIn this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.


2021 ◽  
Author(s):  
Salvatore Rionero

AbstractThe longtime behaviour of the FitzHugh–Rinzel (FHR) neurons and the transition to instability of the FHR steady states, are investigated. Criteria guaranteeing solutions boundedness, absorbing sets, in the energy phase space, existence and steady states instability via oscillatory bifurcations, are obtained. Denoting by $$ \lambda ^{3} + \sum\nolimits_{{k = 1}}^{3} {A_{k} } (R)\lambda ^{{3 - k}} = 0 $$ λ 3 + ∑ k = 1 3 A k ( R ) λ 3 - k = 0 , with R bifurcation parameter, the spectrum equation of a steady state $$m_0$$ m 0 , linearly asymptotically stable at certain value of R, the frequency f of an oscillatory destabilizing bifurcation (neuron bursting frequency), is shown to be $$ f=\displaystyle \frac{\sqrt{A_2(R_\mathrm{H})}}{2\pi } $$ f = A 2 ( R H ) 2 π with $$R_\mathrm{H}$$ R H location of R at which the bifurcation occurs. The instability coefficient power (ICP) (Rionero in Rend Fis Acc Lincei 31:985–997, 2020; Fluids 6(2):57, 2021) for the onset of oscillatory bifurcations, is introduced, proved and applied, in a new version.


2021 ◽  
Vol 31 (12) ◽  
pp. 2150187
Author(s):  
Marius-F. Danca

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [ Danca & Kuznetsov, 2018 ], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.


Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Rachid Dhifaou ◽  
Houda Brahmi

Intensive and repetitive simulations are required to study static and dynamic behaviours of systems. Particular phenomena such as bifurcation and chaos require long simulation times and analysis. To check the existence of bifurcations and chaos in a dynamic system, a fine-tuning procedure of a bifurcation parameter is to be carried out. This increases considerably the computing time, and a great amount of patience is needed to obtain adequate results. Because of the high switching frequency of a boost inverter, the integration process of the dynamic model used to describe it uses an integration step that is in general less than one microsecond. This makes the integration process time consuming even for a short simulation. Thus, a fast, but accurate, method is suitable to analyse the dynamic behaviour of the converter. This work contains two topics. First, we develop a like-discrete integration process that permits precise results in a very fast manner. For one switching period, we compute only two or a maximum of three breaking points depending on whether we treat a continuous conduction mode (CCM) or a discontinuous conduction mode (DCM) of the inductor current. Furthermore, with each segment of the dynamic trajectory, an exact analytic formula is associated. The second goal is to use this result to develop a discrete iterative map formulated as in standard discrete time series models. The Jacobian matrix of the found iterative map is defined and used to compute Lyapunov exponents to prove existence of chaos. Performance of the developed study is positively evaluated by using classical simulations and fine-tuning a bifurcation parameter to detect chaos. This parameter is the desired reference of the inductor current peak. Results show that the proposed scheme is very fast and accurate. The study can be easily extended to other switching topologies of DC-DC inverters.


2021 ◽  
pp. 1-17
Author(s):  
T. D. Frank ◽  
P. Stowik

Data from three functional magnetic resonance imaging (fMRI) studies that involved in total about 100 participants and showed that the strength of several visual illusions such as the Ebbinghaus, Ponzo, and Muller-Lyer illusions depends on neuroanatomical subject measures such as visual cortex surface area and parahippocampal cortex gray matter volume were evaluated using a dynamical systems perspective to determine brain bifurcation parameters. Bifurcation parameters that involved power laws and captured relational dependencies were fitted separately to the three fMRI studies. The bifurcation parameter hypothesis that states that such parameters show unique quantities and are no longer correlated to structural systems properties was tested. The power law exponents and mean bifurcation parameter values were determined. For all three studies and three illusion types, the bifurcation parameter hypothesis was supported. Accordingly, the constructed parameters characterized the reactions of the participants under the Ebbinghaus, Ponzo, and Muller-Lyer illusions in terms of unique threshold values that no longer depended on neuroanatomical subject measures. Power law exponents in the range from 1 to 7 were found. The fMRI data describing gray matter volume of certain active regions in the parahippocampal cortex showed some interesting relationship between the mean bifurcation parameter values.


Author(s):  
Chengdai Huang ◽  
Jinde Cao

This paper expounds the bifurcations of two-delayed fractional-order neural networks (FONNs) with multiple neurons. Leakage delay or communication delay is viewed as a bifurcation parameter, stability zones and bifurcation conditions with respect to them are commendably established, respectively. It declares that both leakage delay and communication delay immensely influence the stability and bifurcation of the developed FONNs. The explored FONNs illustrate superior stability performance if selecting a lesser leakage delay or communication delay, and Hopf bifurcation generates once they overstep their critical values. The verification of the feasibility of the developed analytic results is implemented via numerical experiments.


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