scholarly journals Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Shengwei Yao ◽  
Huonian Tu
Keyword(s):  
Equilibrium Point ◽  
Resting State ◽  
Neural System ◽  
Multiple Delays ◽  
Stability Criteria ◽  
Periodic Activity ◽  
Stability Switches ◽  
Delay Dependence ◽  
The Stability ◽  
Theoretical Results

A FitzHugh-Nagumo (FHN) neural system with multiple delays has been proposed. The number of equilibrium point is analyzed. It implies that the neural system exhibits a unique equilibrium and three ones for the different values of coupling weight by employing the saddle-node bifurcation of nontrivial equilibrium point and transcritical bifurcation of trivial one. Further, the stability of equilibrium point is studied by analyzing the corresponding characteristic equation. Some stability criteria involving the multiple delays and coupling weight are obtained. The results show that the neural system exhibits the delay-independence and delay-dependence stability. Increasing delay induces the stability switching between resting state and periodic activity in some parameter regions of coupling weight. Finally, numerical simulations are taken to support the theoretical results.

Multitype Activity Coexistence in an Inertial Two-Neuron System with Multiple Delays

2015 ◽  
Vol 25 (13) ◽  
pp. 1530040 ◽  
Author(s):  
Zigen Song ◽  
Jian Xu ◽  
Bin Zhen
Keyword(s):  
Multiple Equilibria ◽  
Activity Patterns ◽  
Pitchfork Bifurcation ◽  
Neural System ◽  
Multiple Delays ◽  
Periodic Activity ◽  
Neuron System ◽  
Fold Bifurcation ◽  
Trivial Equilibrium ◽  
The Stability

In this paper, an inertial two-neuron system with multiple delays is analyzed to exhibit the effect of time delays on system dynamics. The parameter region with multiple equilibria is obtained employing the pitchfork bifurcation of trivial equilibrium. The stability analysis illustrates that two nontrivial equilibria are both stable for any delays. It implies that the neural system exhibits a stability coexistence of two resting states. Further, due to the existence of multiple delays, the neural system has a periodic activity around the trivial equilibrium via Hopf bifurcation. Finally, numerical simulations are employed to illustrate many richness coexistence for multitype activity patterns. Employing the period-adding route and fold bifurcation of periodic orbit, the neural system may have multistability coexistence of two resting states, two ASP-3s (anti-symmetric periodic activity with period three), one SSP-1 (self-symmetric periodic activity with period one), and one quasi-periodic spiking. Additionally, with increasing delay, quasi-periodic spiking evolves into chaos behavior.

scholarly journals Asymptotic Stability of Caputo Type Fractional Neutral Dynamical Systems with Multiple Discrete Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Hai Zhang ◽  
Daiyong Wu ◽  
Jinde Cao
Keyword(s):  
Asymptotic Stability ◽  
Matrix Theory ◽  
Sufficient Conditions ◽  
Algebraic Approach ◽  
Stability Criteria ◽  
Differential Systems ◽  
Discrete Delays ◽  
Transcendental Equations ◽  
The Stability ◽  
Theoretical Results

We discuss the delay-independent asymptotic stability of Caputo type fractional-order neutral differential systems with multiple discrete delays. Based on the algebraic approach and matrix theory, the sufficient conditions are derived to ensure the asymptotic stability for all time-delay parameters. By applying the stability criteria, one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and convenient. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed theoretical results.

scholarly journals Mathematical Modelling of Deforestation Due to Population Density and Industrialization

Jurnal Varian ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 9-16
Author(s):  
Didiharyono D. ◽  
Irwan Kasse
Keyword(s):  
Numerical Simulation ◽  
Population Density ◽  
Mathematical Modelling ◽  
Equilibrium Point ◽  
Equilibrium Points ◽  
Stability Criteria ◽  
Hurwitz Stability ◽  
Stable Conditions ◽  
Linear Differential ◽  
The Stability

The focus of the study in this paper is to model deforestation due to population density and industrialization. To begin with, it is formulated into a mathematical modelling which is a system of non-linear differential equations. Then, analyze the stability of the system based on the Routh-Hurwitz stability criteria. Furthermore, a numerical simulation is performed to determine the shift of a system. The results of the analysis to shown that there are seven non-negative equilibrium points, which in general consist equilibrium point of disturbance-free and equilibrium points of disturbances. Equilibrium point TE7(x, y, z) analyzed to shown asymptotically stable conditions based on the Routh-Hurwitz stability criteria. The numerical simulation results show that if the stability conditions of a system have been met, the system movement always occurs around the equilibrium point.

ATTRACTABILITY AND LOCATION OF EQUILIBRIUM POINT OF CELLULAR NEURAL NETWORKS WITH TIME-VARYING DELAYS

2004 ◽  
Vol 14 (05) ◽  
pp. 337-345 ◽  
Author(s):  
ZHIGANG ZENG ◽  
DE-SHUANG HUANG ◽  
ZENGFU WANG
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Equilibrium Point ◽  
Global Exponential Stability ◽  
Cellular Neural Networks ◽  
Stability Conditions ◽  
Time Varying ◽  
The Stability ◽  
Theoretical Results

This paper presents new theoretical results on global exponential stability of cellular neural networks with time-varying delays. The stability conditions depend on external inputs, connection weights and delays of cellular neural networks. Using these results, global exponential stability of cellular neural networks can be derived, and the estimate for location of equilibrium point can also be obtained. Finally, the simulating results demonstrate the validity and feasibility of our proposed approach.

scholarly journals Hopf Bifurcation and Dynamic Analysis of an Improved Financial System with Two Delays

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. Jin ◽  
C. Z. Wang
Keyword(s):  
Nonlinear Dynamics ◽  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Financial System ◽  
Two Delays ◽  
Normal Form Method ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

The complex chaotic dynamics and multistability of financial system are some important problems in micro- and macroeconomic fields. In this paper, we study the influence of two-delay feedback on the nonlinear dynamics behavior of financial system, considering the linear stability of equilibrium point under the condition of single delay and two delays. The system undergoes Hopf bifurcation near the equilibrium point. The stability and bifurcation directions of Hopf bifurcation are studied by using the normal form method and central manifold theory. The theoretical results are verified by numerical simulation. Furthermore, one feature of the proposed financial chaotic system is that its multistability depends extremely on the memristor initial condition and the system parameters. It is shown that the nonlinear dynamics of financial chaotic system can be significantly changed by changing the values of time delays.

scholarly journals Fractional SEIRP model for COVID-19 dynamics incorporatingsocial distancing and environment

2021 ◽  
Author(s):  
Saheed Ojo Akindeinde ◽  
E. Okyere ◽  
A. O. Adewumi ◽  
R. S. Lebelo ◽  
O. O. Fabelurin ◽  
...  
Keyword(s):  
Compartmental Model ◽  
Reproduction Number ◽  
Banach Fixed Point Theorem ◽  
The Novel ◽  
Disease Dynamics ◽  
Stability Criteria ◽  
Proposed Model ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Theoretical Results

Abstract We propose a Caputo-based fractional compartmental model for the dynamics of the novel COVID-19 epidemic taking into consideration social distancing and the influence of the environment. Using basic concepts such as continuity and Banach fixed-point theorem, the existence and uniqueness of the solution to the proposed model were shown. Furthermore, we analyze the stability of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. The concept of next-generation matrices was used to compute the basic reproduction number $R_0,$ a number that determines the spread or otherwise of the disease into the general population. Numerical simulation of the disease dynamics was carried out using the fractional Adam-Bashforth-Moulton method to validate the obtained theoretical results.

Bifurcations and Synchronization of Delay-Coupled Neuronal Networks with Autaptic Connections

2018 ◽  
Vol 28 (06) ◽  
pp. 1850071 ◽  
Author(s):  
Xiaochen Mao
Keyword(s):  
System Performance ◽  
Time Delays ◽  
Electrical Synapse ◽  
Stability Criteria ◽  
Neural Activities ◽  
Coupled Networks ◽  
Different Types ◽  
The Stability ◽  
Multistability Coexistence ◽  
Theoretical Results

This paper focuses on the dynamic behaviors of delay-coupled networks consisting of an arbitrary number of nonidentical neurons with unidirectional connections and an electrical synapse from one neuron onto itself. The stability criteria and different types of bifurcations are discussed by decomposing and analyzing the associated characteristic equation. Then, the study turns to the validation of theoretical results through numerical simulations and various interesting neural activities are observed, such as nontrivial equilibria, different patterns of oscillations, multistability coexistence, multiple switches between the quiescent and different periodic states, and transitions between the coexisting equilibria and the coexistence of equilibria and periodic oscillations. It is shown that time delays play important roles in the system performance and can be used to control neural activities.

scholarly journals Stability and Hopf Bifurcation Analysis for a Gause-Type Predator-Prey System with Multiple Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Juan Liu ◽  
Changwei Sun ◽  
Yimin Li
Keyword(s):  
Hopf Bifurcation ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
Two Delays ◽  
The Stability ◽  
Theoretical Results

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.

scholarly journals Dynamics Induced by Delay in a Nutrient-Phytoplankton Model with Multiple Delays

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Chuanjun Dai ◽  
Hengguo Yu ◽  
Qing Guo ◽  
He Liu ◽  
Qi Wang ◽  
...  
Keyword(s):  
Numerical Simulations ◽  
Periodic Solutions ◽  
Positive Equilibrium ◽  
Multiple Delays ◽  
Asymptotically Stable ◽  
Phytoplankton Dynamics ◽  
Stability Switches ◽  
Globally Asymptotically Stable ◽  
Analytical Analysis ◽  
The Stability

A nutrient-phytoplankton model with multiple delays is studied analytically and numerically. The aim of this paper is to study how the delay factors influence dynamics of interaction between nutrient and phytoplankton. The analytical analysis indicates that the positive equilibrium is always globally asymptotically stable when the delay does not exist. On the contrary, the positive equilibrium loses its stability via Hopf instability induced by delay and then the corresponding periodic solutions emerge. Especially, the stability switches for positive equilibrium occur as the delay is increased. Furthermore, the numerical simulations show that periodic-2 and periodic-3 solutions can appear due to the existence of delays. Numerical results are consistent with the analytical results. Our results demonstrate that the delay has a great impact on the nutrient-phytoplankton dynamics.

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