OSCILLATIONS IN AN EXCITABLE SYSTEM WITH TIME-DELAYS

2003 ◽  
Vol 13 (11) ◽  
pp. 3483-3488 ◽  
Author(s):  
N. BURIĆ ◽  
N. VASOVIĆ
Keyword(s):  
Fixed Point ◽  
Time Delays ◽  
Stable Limit Cycle ◽  
Multiple Time ◽  
Time Lags ◽  
Stable Limit ◽  
Excitable System ◽  
Subcritical Hopf Bifurcation ◽  
Intermediate Time ◽  
Nagumo Equations

Transition from excitability to asymptotic periodicity in an excitable system, modeled by the FitzHugh–Nagumo equations, with multiple time-delays, is analyzed. It is demonstrated that, for intermediate time-lags, the system has two coexisting attractors, a hyperbolic stable fixed point and a stable limit cycle. The fixed point is destabilized via subcritical Hopf bifurcation for much larger values of the time-lags.

The orientations of prolate ellipsoids in linear shear flows

2011 ◽  
Vol 690 ◽  
pp. 51-93 ◽  
Author(s):  
Ehud Gavze ◽  
Mark Pinsky ◽  
Alexander Khain
Keyword(s):  
Fixed Point ◽  
Probability Distribution ◽  
Characteristic Time ◽  
Stable Limit Cycle ◽  
Normal Growth ◽  
Characteristic Time Scale ◽  
Stable Limit ◽  
Stable Fixed Point ◽  
Linear Shear ◽  
Pure Imaginary Eigenvalues

AbstractThe dynamics of the orientations of prolate ellipsoids in general linear shear flow is considered. The motivation behind this work is to gain a better understanding of the motion and the orientation probability distribution of ice particles in clouds in order to improve the modelling of their collision. The evolution of the orientations is governed by the Jeffery equation. It is shown that the possible attractors of this equation are fixed points, limit cycles and an infinite set of periodic solutions, named Jeffery orbits, in the case of simple shear. Linear stability analysis shows that the existence and the stability of the attractors are determined by the eigenvalues of the linear part of the equation. If the eigenvalues possess a non-vanishing real part, then there always exists either a stable fixed point or a stable limit cycle. Pure imaginary eigenvalues lead to Jeffery orbits. The convergence to a stable fixed point or to a stable limit cycle may either be monotonic or may be retarded due to the occurrence of non-normal growth. If non-normal growth occurs the convergence rate may be much slower compared with the characteristic time scale of the shear. Expressions for the characteristic time scale of convergence to the stable solutions are derived. In the case of non-normal growth, expressions are derived for the delay in the convergence. The orientation probability distribution function (p.d.f.) is computed via the solution of the Fokker–Planck equation. The p.d.f. is either periodic, in the case of simple shear (pure imaginary eigenvalues), or it converges to singular points or strips in the orientation space (fixed points and limit cycles) on which it grows to infinity. Time-independent p.d.f.s exist only for imaginary eigenvalues. Unlike the case where Brownian diffusion is present, the steady solutions are not unique and depend on the initial conditions.

scholarly journals Amplitude death and restoration in networks of oscillators with random-walk diffusion

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Pau Clusella ◽  
M. Carmen Miguel ◽  
Romualdo Pastor-Satorras
Keyword(s):  
Fixed Point ◽  
Random Walk ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Amplitude Death ◽  
Numerical Stability Analysis ◽  
Unstable Fixed Point ◽  
Reactive Particles ◽  
Collective Oscillations ◽  
Complex Phenomena

AbstractSystems composed of reactive particles diffusing in a network display emergent dynamics. While Fick’s diffusion can lead to Turing patterns, other diffusion schemes might display more complex phenomena. Here we study the death and restoration of collective oscillations in networks of oscillators coupled by random-walk diffusion, which modifies both the original unstable fixed point and the stable limit-cycle, making them topology-dependent. By means of numerical simulations we show that, in some cases, the diffusion-induced heterogeneity stabilizes the initially unstable fixed point via a Hopf bifurcation. Further increasing the coupling strength can moreover restore the oscillations. A numerical stability analysis indicates that this phenomenology corresponds to a case of amplitude death, where the inhomogeneous stabilized solution arises from the interplay of random walk diffusion and heterogeneous topology. Our results are relevant in the fields of epidemic spreading or ecological dispersion, where random walk diffusion is more prevalent.

scholarly journals Analysis of a Model of Leishmaniasis with Multiple Time Lags in All Populations

2019 ◽  
Vol 24 (2) ◽  
pp. 63
Author(s):  
Ephraim Agyingi ◽  
Tamas Wiandt
Keyword(s):  
Time Delays ◽  
System Analysis ◽  
Reservoir Host ◽  
Endemic Equilibrium ◽  
Multiple Time ◽  
Time Lags ◽  
Animal Reservoir ◽  
Sis Model ◽  
Exposed Population ◽  
Disease Free

There are several types of deterministic compartmental models for disease epidemiology such as SIR, SIS, SEIS, SEIR, etc. The exposed population group in, for example SEIS or SEIR, usually represents individuals in the incubation class. Time delays (of which there are several types) when incorporated into a SIR or SIS model, also fulfil the role of the incubation period without necessarily adding another compartment to the model. This paper incorporates time delays into a SIS model that describes the transmission dynamics of cutaneous leishmaniasis. The time lags account for the incubation periods within the sandflies vector, the human hosts and the different animal groups that serve as reservoir hosts. A threshold value, R 0 , of the model is computed and used to study the disease-free equilibrium and endemic equilibrium of the system. Analysis demonstrating local and global stability of the disease-free equilibrium when R 0 < 1 is provided for all n + 1 population groups involved is provided. The existence of an endemic equilibrium is only guaranteed when R 0 > 1 and numerical analysis of the endemic equilibrium for a human host, a vector host and a single animal reservoir host that is globally stable is also provided.

Oscillatory Dynamics of P53 Network With Time Delays

2018 ◽  
Vol 21 (6) ◽  
pp. 411-419 ◽  
Author(s):  
Conghua Wang ◽  
Fang Yan ◽  
Yuan Zhang ◽  
Haihong Liu ◽  
Linghai Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Cell Fate ◽  
Time Delays ◽  
Sustained Oscillation ◽  
Multiple Time ◽  
Cell Fate Decisions ◽  
Oscillatory Dynamics ◽  
Main Components ◽  
Parameter Values ◽  
P53 Network

Aims and Objective: A large number of experimental evidences report that the oscillatory dynamics of p53 would regulate the cell fate decisions. Moreover, multiple time delays are ubiquitous in gene expression which have been demonstrated to lead to important consequences on dynamics of genetic networks. Although delay-driven sustained oscillation in p53-based networks is commonplace, the precise roles of such delays during the processes are not completely known. Method: Herein, an integrated model with five basic components and two time delays for the network is developed. Using such time delays as the bifurcation parameter, the existence of Hopf bifurcation is given by analyzing the relevant characteristic equations. Moreover, the effects of such time delays are studied and the expression levels of the main components of the system are compared when taking different parameters and time delays. Result and Conclusion: The above theoretical results indicated that the transcriptional and translational delays can induce oscillation by undergoing a super-critical Hopf bifurcation. More interestingly, the length of these delays can control the amplitude and period of the oscillation. Furthermore, a certain range of model parameter values is essential for oscillation. Finally, we illustrated the main results in detail through numerical simulations.

Dynamics of a delayed competitive system affected by toxic substances with imprecise biological parameters

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5271-5293
Author(s):  
A.K. Pal ◽  
P. Dolai ◽  
G.P. Samanta
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Toxic Substances ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Competitive System ◽  
Interval Numbers ◽  
Cycle Oscillation

In this paper we have studied the dynamical behaviours of a delayed two-species competitive system affected by toxicant with imprecise biological parameters. We have proposed a method to handle these imprecise parameters by using parametric form of interval numbers. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate our analytical findings.

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