THE EFFECT OF TIME DELAY ON SPATIOTEMPORAL DYNAMICS IN ONE-DIMENSIONAL DISCRETE EXCITABLE MEDIA

2001 ◽  
Vol 15 (02) ◽  
pp. 167-176
Author(s):  
TAE-HOON CHUNG ◽  
SEUNGWHAN KIM
Keyword(s):  
Time Delay ◽  
Spatiotemporal Dynamics ◽  
Excitable Media ◽  
Winding Number ◽  
One Dimensional ◽  
Circle Maps ◽  
Pattern Complexity ◽  
Existence And Stability ◽  
Spatiotemporal Intermittency ◽  
The Stability

We investigate the effect of time delay on spatiotemporal dynamics in one-dimensional discrete excitable media with local delayed-interactions using coupled sine circle-maps. With the help of the stability analysis and numerical calculation of the pattern complexity entropy, we construct the phase diagram in the parameter space time delay and the nonlinear coupling. We find that the time delay affects the existence and stability of various regular states including homogeneously phase-locked and checkerboard states. In particular, the time delay induces the breakup of the homogeneously phase-locked state into spatiotemporal intermittency and the occurrence of multi-stability that depends on the winding number.

Stability results for an elastic–viscoelastic wave equation with localized Kelvin–Voigt damping and with an internal or boundary time delay

2020 ◽  
pp. 1-57
Author(s):  
Mouhammad Ghader ◽  
Rayan Nasser ◽  
Ali Wehbe
Keyword(s):  
Wave Equation ◽  
Time Delay ◽  
Frequency Domain ◽  
Viscoelastic Damping ◽  
One Dimensional ◽  
Viscoelastic Wave ◽  
Smooth Coefficients ◽  
The Stability ◽  
Stability Results ◽  
Dimensional Wave

We investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin–Voigt type and with boundary or localized internal delay feedback. The main novelty in this paper is that the Kelvin–Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin–Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t − 4 . However, an exponential decay of the energy of the system is established provided that the Kelvin–Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary. While, when the Kelvin–Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t − 4 . Otherwise, if the above assumptions are not true, we establish instability results.

Stability and Bifurcation in an SI Epidemic Model with Additive Allee Effect and Time Delay

2021 ◽  
Vol 31 (04) ◽  
pp. 2150060
Author(s):  
Yangyang Lv ◽  
Lijuan Chen ◽  
Fengde Chen ◽  
Zhong Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Allee Effect ◽  
Stability Of Equilibria ◽  
Existence And Stability ◽  
Vital Effects ◽  
Strong Allee Effect ◽  
The Stability

In this paper, we consider an SI epidemic model incorporating additive Allee effect and time delay. The primary purpose of this paper is to study the dynamics of the above system. Firstly, for the model without time delay, we demonstrate the existence and stability of equilibria for three different cases, i.e. with weak Allee effect, with strong Allee effect, and in the critical case. We also investigate the existence and uniqueness of Hopf bifurcation and limit cycle. Secondly, for the model with time delay, the stability of equilibria and the existence of Hopf bifurcation are discussed. All the above show that both additive Allee effect and time delay have vital effects on the prevalence of the disease.

scholarly journals Bound states of atomic Josephson vortices

2017 ◽  
Vol 95 (4) ◽  
pp. 336-339 ◽  
Author(s):  
Muhammad Irfan Qadir ◽  
Usama Tahir
Keyword(s):  
Bound States ◽  
Coupling Parameter ◽  
Bound State ◽  
Dark Soliton ◽  
Critical Value ◽  
One Dimensional ◽  
Josephson Vortices ◽  
Existence And Stability ◽  
The Stability ◽  
Bose Einstein

We study the existence and stability of the bound state Josephson vortices solution in two parallel quasi one-dimensional coupled Bose–Einstein condensates. The system can be elucidated by linearly coupled Gross–Pitaevskii equations. The purpose of this study is to investigate the effects of altering the strength of coupling between the two condensates over the stability of the bound-state Josephson vortices. It is found that the stability of bound-state Josephson vortices depends on the value of coupling strength. However, at a critical value of coupling parameter, the Josephson vortices solution transforms into a coupled dark soliton.

scholarly journals ON THE GIERER–MEINHARDT SYSTEM WITH SATURATION

2004 ◽  
Vol 06 (02) ◽  
pp. 259-277 ◽  
Author(s):  
JUNCHENG WEI ◽  
MATTHIAS WINTER
Keyword(s):  
Ground State ◽  
Continuation Method ◽  
Sufficient Conditions ◽  
State Equation ◽  
Smooth Bounded Domain ◽  
One Dimensional ◽  
Stability Behavior ◽  
Existence And Stability ◽  
The Stability ◽  
The One

We consider the following shadow Gierer–Meinhardt system with saturation: [Formula: see text] where ∊>0 is a small parameter, τ≥0, k>0 and Ω⊂Rn is smooth bounded domain. The case k=0 has been studied by many authors in recent years. Here we give some sufficient conditions on k for the existence and stability of stable spiky solutions. In the one-dimensional case we have a complete answer to the stability behavior. Central to our study are a parameterized ground-state equation and the associated nonlocal eigenvalue problem (NLEP) which is solved by functional analysis arguments and the continuation method.

On the asymptotic stability of monotonic electrostatic collisionless shock waves

1969 ◽  
Vol 3 (3) ◽  
pp. 411-416 ◽  
Author(s):  
D. Biskamp
Keyword(s):  
Shock Waves ◽  
Acoustic Waves ◽  
Distribution Functions ◽  
Asymptotic State ◽  
Ion Acoustic Waves ◽  
One Dimensional ◽  
Oscillatory Structure ◽  
Existence And Stability ◽  
The Stability ◽  
Shock Solution

Recently the question of existence and stability of laminar one-dimensional coffisionless shock waves has received some new attention (Montgomery & Joyce 1969). These electrostatic shock waves are usually thought of as being generated by ion acoustic waves (pulses) in a plasma with Te ≫ Ti Apart from the well- known class of shock solution with oscillatory structure (Moiseev & Sagdeev 1963) there exist, at least within the framework of Vlasov theory, monotonic transitions from one asymptotic state to another (Montgomery & Joyce 1969). For these shock-wave like solutions it is easy to see that the electrons cannot be in thermal equilibrium (have a Maxwellian distribution) in the downstream state, if they are so in the upstream state. Hence it appears to be natural to investigate first the stability of the homogeneous downstream state (which tells us if the transition is possible at all) by investigating the properties of the class of permitted distribution functions for the electrons, before attacking the much more involved problem of the possible instabilities arising from the inhomogeneitythe transition region of the shock.

scholarly journals On the stability of the telegraph equation with time delay

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1251-1259
Author(s):  
Allaberen Ashyralyev ◽  
Deniz Agirseven ◽  
Koray Turk
Keyword(s):  
Boundary Condition ◽  
Time Delay ◽  
Test Problem ◽  
Dirichlet Boundary ◽  
Telegraph Equation ◽  
Stability Estimates ◽  
Initial Value ◽  
One Dimensional ◽  
Telegraph Equations ◽  
The Stability

In this study, the initial value problem for telegraph equations with time delay in a Hilbert space is considered. The main theorem on stability estimates for the solution of this problem is established. As a test problem, one-dimensional delay telegraph equation with the Dirichlet boundary condition is considered.

Spatiotemporal Dynamics of a Network of Coupled Time-Delay Digital Tanlock Loops

2016 ◽  
Vol 26 (05) ◽  
pp. 1650076 ◽  
Author(s):  
Bishwajit Paul ◽  
Tanmoy Banerjee ◽  
B. C. Sarkar
Keyword(s):  
Time Delay ◽  
Communication Systems ◽  
Nearest Neighbor ◽  
Circulant Matrix ◽  
Spatiotemporal Chaos ◽  
Spatiotemporal Dynamics ◽  
Random Pattern ◽  
Matrix Formalism ◽  
Extensive Numerical Simulation ◽  
The Stability

The time-delay digital tanlock loop (TDTLs) is an important class of phase-locked loop that is widely used in electronic communication systems. Although nonlinear dynamics of an isolated TDTL has been studied in the past but the collective behavior of TDTLs in a network is an important topic of research and deserves special attention as in practical communication systems separate entities are rarely isolated. In this paper, we carry out the detailed analysis and numerical simulations to explore the spatiotemporal dynamics of a network of a one-dimensional ring of coupled TDTLs with nearest neighbor coupling. The equation representing the network is derived and we carry out analytical calculations using the circulant matrix formalism to obtain the stability criteria. An extensive numerical simulation reveals that with the variation of gain parameter and coupling strength the network shows a variety of spatiotemporal dynamics such as frozen random pattern, pattern selection, spatiotemporal intermittency and fully developed spatiotemporal chaos. We map the distinct dynamical regions of the system in two-parameter space. Finally, we quantify the spatiotemporal dynamics by using quantitative measures like Lyapunov exponent and the average quadratic deviation of the full network.

scholarly journals A Study of Within-Host Dynamics of Dengue Infection incorporating both Humoral and Cellular Response with a Time Delay for Production of Antibodies

2021 ◽  
Vol 9 (1) ◽  
pp. 66-80
Author(s):  
Deva Siva Sai Murari Kanumoori ◽  
D Bhanu Prakash ◽  
D. K. K. Vamsi ◽  
Carani B Sanjeevi
Keyword(s):  
Immune Response ◽  
Sensitivity Analysis ◽  
Time Delay ◽  
Equilibrium Points ◽  
Dengue Infection ◽  
Complex Behaviour ◽  
Proposed Model ◽  
Existence And Stability ◽  
The Stability ◽  
Endemic State

Abstract a. Background: Dengue is an acute illness caused by a virus. The complex behaviour of the virus in human body can be captured using mathematical models. These models helps us to enhance our understanding on the dynamics of the virus. b. Objectives: We propose to study the dynamics of within-host epidemic model of dengue infection which incorporates both innate immune response and adaptive immune response (Cellular and Humoral). The proposed model also incorporates the time delay for production of antibodies from B cells. We propose to understand the dynamics of the this model using the dynamical systems approach by performing the stability and sensitivity analysis. c. Methods used: The basic reproduction number (R0) has been computed using the next generation matrix method. The standard stability analysis and sensitivity analysis were performed on the proposed model. d. Results: The critical level of the antibody recruitment rate(q) was found to be responsible for the existence and stability of various steady states. The stability of endemic state was found to be dependent on time delay(τ). The sensitivity analysis identified the production rate of antibodies (q) to be highly sensitive parameter. e. Conclusions: The existence and stability conditions for the equilibrium points have been obtained. The threshold value of time delay (τ0) has been computed which is critical for change in stability of the endemic state. Sensitivity analysis was performed to identify the crucial and sensitive parameters of the model.

Entrainment and Annihilation of Reentrant Excitation in a Periodically Stimulated Ring of Excitable Media

1997 ◽  
Vol 36 (04/05) ◽  
pp. 290-293
Author(s):  
L. Glass ◽  
T. Nomura
Keyword(s):  
Cardiac Arrhythmias ◽  
Initial Phase ◽  
Human Heart ◽  
Periodic Wave ◽  
Excitable Media ◽  
One Dimensional ◽  
Clinical Observations ◽  
Periodic Stimulation ◽  
Reentrant Excitation ◽  
Stimulation Of

Abstract:Excitable media, such as nerve, heart and the Belousov-Zhabo- tinsky reaction, exhibit a large excursion from equilibrium in response to a small but finite perturbation. Assuming a one-dimensional ring geometry of sufficient length, excitable media support a periodic wave of circulation. As in the periodic stimulation of oscillations in ordinary differential equations, the effects of periodic stimuli of the periodically circulating wave can be described by a one-dimensional Poincaré map. Depending on the period and intensity of the stimulus as well as its initial phase, either entrainment or termination of the original circulating wave is observed. These phenomena are directly related to clinical observations concerning periodic stimulation of a class of cardiac arrhythmias caused by reentrant wave propagation in the human heart.

On the stability of a class of nonlinear time delay systems

1999 ◽  
Author(s):  
Dan Ivancscu ◽  
Silviu-Iulian Niculcscu ◽  
Jcan-Michcl Dion ◽  
Luc Dugard
Keyword(s):  
Time Delay ◽  
Delay Systems ◽  
Time Delay Systems ◽  
The Stability
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