Sustained Periodic Oscillations Research Articles

Vaccination games and imitation dynamics with memory

In this paper, we model dynamics of pediatric vaccination as an imitation game, in which the rate of switching of vaccination strategies is proportional to perceived payoff gain that consists of the difference between perceived risk of infection and perceived risk of vaccine side effects. To account for the fact that vaccine side effects may affect people's perceptions of vaccine safety for some period of time, we use a delay distribution to represent how memory of past side effects influences current perception of risk. We find disease-free, pure vaccinator, and endemic equilibria and obtain conditions for their stability in terms of system parameters and characteristics of a delay distribution. Numerical bifurcation analysis illustrates how stability of the endemic steady state varies with the imitation rate and the mean time delay, and this shows that it is not just the mean duration of memory of past side effects, but also the actual distribution that determines whether disease will be maintained in the population at some steady level, or if sustained periodic oscillations around this steady state will be observed. Numerical simulations illustrate a comparison of the dynamics for different mean delays and different distributions, and they show that even when periodic solutions are observed, there are differences in their amplitude and period for different distributions. We also investigate the effect of constant public health information campaigns on vaccination dynamics. The analysis suggests that the introduction of such campaigns acts as a stabilizing factor for endemic equilibrium, allowing it to remain stable for larger values of mean time delays.

Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response

In this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady state is represented by up to three distinct steady states, whose existence is determined by the values of appropriate Lambert W functions. We investigate how stability of extinction and coexistence steady states is affected by the rate of predation, predator fecundity, and the parameter characterizing the strength of functional response. The results suggest that the extinction steady state can be stable for sufficiently high predation rate and for sufficiently small predator fecundity. Moreover, in certain parameter regimes, a stable extinction steady state can coexist with a stable prey-only equilibrium or with a stable coexistence equilibrium, and it is rather the initial conditions that determine whether prey and predator populations will be maintained at some steady level, or both of them will become extinct. Another possibility is for coexistence steady state to be unstable, in which case sustained periodic oscillations around it are observed. Numerical simulations are performed to illustrate the behavior for all dynamical regimes, and in each case a corresponding phase plane of the transformed system is presented to show a correspondence with stable and unstable extinction steady state.

Controlling draw resonance during extrusion film casting of nanoclay filled linear low-density polyethylene: An experimental study and numerical linear stability analysis

Commercially important extrusion film casting (EFC) processes for manufacturing plastic films or sheets are hampered by several instabilities that severely limits their productivity. In this research we focussed on one important instability: the draw resonance that occurs during the EFC process mainly under extensional flow conditions. Draw resonance is the sustained periodic oscillations in the film dimensions, notably film width and thickness, when the process operates beyond a critical draw ratio (CDR). In this research our goal was to reduce this draw resonance instability by incorporating well dispersed nanoclay fillers in a base polymeric resin (such as a linear low density polyethylene – LLDPE) to determine how these nanocomposite (NC) formulations can prevent or reduce the draw resonance defect. EFC experiments were conducted on the base resin and on the NC formulations under non-isothermal conditions to determine the onset of the draw resonance experimentally. Conventional linear stability analysis was performed to determine the onset of the draw resonance defect numerically. Numerical predictions for the onset of draw resonance were in qualitative agreement with our experimental data. Our results showed that incorporating appropriate nanoclay concentrations in a base polymeric resin indeed enhanced the EFC process stability for those polymer formulations and thus can have important economic implications for processors.

Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay

A predator-prey model with predator-age structure and predator-prey reaction time delay τ1 is investigated. We consider the predator fertility function as a piecewise function related to predator development time τ2. Considering the complexity of the characteristic equation, we theoretically prove that a non-trivial periodic oscillation phenomena through Hopf bifurcation appears when parameter τ1=τ2:=τ passes through some critical values, by employing the integrated semigroup theory and Hopf bifurcation theory for semilinear equations with non-dense domain. We perform model calibration, data fitting and parameter estimation by utilizing real data, and analyze the rationality and deficiencies of our model. The corresponding theoretical results are certificated by numerical simulations. Additionally, when τ1 and τ2 take different values, we numerically analyze their influence on the model and conclude that the dynamic behavior of the system around the positive equilibrium changes between local asymptotic stability and sustained periodic oscillations.

Parametric oscillations in a dissipative bosonic Josephson junction

We study the dynamics of a nonlinear dissipative bosonic Josephson junction with a time-dependent sinusoidal perturbation in the interaction term. We demonstrate parametric resonance, where the system undergoes sustained periodic oscillations even in the presence of dissipation. This happens when the frequency of the perturbation is close to twice the frequency of the unperturbed Josephson oscillations and the strength of perturbation exceeds a critical threshold. We have formulated the threshold conditions for parametric oscillations. To explore the nature of the oscillations, we carry out a multiple time scale analysis of the stability boundaries in terms of the V-shaped Arnold tongue in the parameter space. Full numerical simulations have been performed for the zero-, running-, and π-phase modes of the nonlinear Josephson effect. Our results demonstrate that in the π-phase mode the system is capable of making a transition from regular parametric to chaotic parametric oscillations as one crosses the stability boundary. Also, the phase difference undergoes phase slip before executing sustained parametric oscillations.

Quantifying the Role of Stochasticity in the Development of Autoimmune Disease.

In this paper, we propose and analyse a mathematical model for the onset and development of autoimmune disease, with particular attention to stochastic effects in the dynamics. Stability analysis yields parameter regions associated with normal cell homeostasis, or sustained periodic oscillations. Variance of these oscillations and the effects of stochastic amplification are also explored. Theoretical results are complemented by experiments, in which experimental autoimmune uveoretinitis (EAU) was induced in B10.RIII and C57BL/6 mice. For both cases, we discuss peculiarities of disease development, the levels of variation in T cell populations in a population of genetically identical organisms, as well as a comparison with model outputs.

Multistability driven by inhibitory kinetics in a discrete-time size-structured chemostat model.

In this paper, we consider a discrete size-structured nonlinear chemostat model with inhibitory kinetics and multiple competing species. We show that, same as in models with monotone increasing nutrient uptake response functionals studied in the literature, one essential growth-limiting nutrient can support at most one species. More interestingly, we show that, differing from monotone increasing kinetics, inhibitory kinetics can induce several types of multistability and the competitive outcome is initial condition dependent. The species with the smallest lower break-even value is not always the winning species, and the winning species can exhibit sustained periodic oscillations and even chaotic irregular oscillations.

Study of Two Time Delays in IVGTT Glucose–Insulin Dynamical System

Time delays occurred in insulin secretion and its action play a vital role in maintaining the glucose level in diabetics. Li et al. proposed a model by considering a time delay occurred in insulin secretion from pancreas stimulated by raised glucose concentration. After insulin secretion, a delay is occur in insulin action to lower the raised glucose concentration. To meet with the physiology of human body, the model given by Li et al. has been extended by incorporating a second time delay occurred in insulin action. Stability analysis has been done for the extended model and it possesses positive solutions. From the simulation results, we conclude that sustained periodic oscillations are observed for both of the time delays. Also, the simulations shows that after introducing the delay in insulin action, the delay length of insulin secretion proposed by Li et al. has been shortened, which can be proved important in maintaining the glucose level after delivery of insulin.

Bounded global Hopf branches for stage-structured differential equations with unimodal feedback

We consider a class of stage-structured differential equations with unimodal feedback. By using the time delay as a bifurcation parameter, we show that the number of local Hopf bifurcation values is finite. Furthermore, we analytically prove that these local Hopf bifurcation values are neatly paired, and each pair is jointed by a bounded global Hopf branch. We use the well-known Mackey–Glass equation with a stage structure as an illustrative example to demonstrate that bounded global Hopf branches can induce interesting and rich dynamics. As the delay increases over a finite interval, the stage-structured Mackey–Glass equation exhibits certain symmetric dynamic patterns: the solutions evolve from a stable equilibrium to sustained stable periodic oscillations, to chaotic-like aperiodic oscillations and back to sustained stable periodic oscillations, to a stable equilibrium.

Characterization of nonlinear behavior of an airfoil under stall-induced pitching oscillations

This work presents an investigation on the embedded dynamics of experimental aeroelastic signals of an airfoil under the influence of stall-induced oscillations. Helicopter blades or wind turbines are severely imposed to vibrate in stall conditions, which motivates this research. Despite a significant effort to model the unsteady aerodynamics associated with the stall phenomenon, nonlinear aeroelastic behavior prediction and analysis in such flow regime remain formidable challenges. This modeling requires proper knowledge of the physical events during stall regime, what can be better attained from experimental data. In this work a pitching airfoil is tested in a wind tunnel model. Due to the stall influence the airfoil presents sustained periodic and limit cycle oscillations at high angles of attack. The influence of wind-off preset incidence angles are also included in these analyses. The aeroelastic signals are evaluated using techniques from time series theory and spectral analysis. The method of delays approach is used to reconstruct the state space, revealing indications to possible bifurcations and complex dynamics. Changes in amplitudes of stall-induced oscillation due to airspeed and preset angles increases were also investigated. Frequency domain analysis through power spectra evolutions and higher-order spectra was used to identify and confirm nonlinear couplings and other complex features.

Entrainment of noise-induced and limit cycle oscillators under weak noise

Theoretical models that describe oscillations in biological systems are often either a limit cycle oscillator, where the deterministic nonlinear dynamics gives sustained periodic oscillations, or a noise-induced oscillator, where a fixed point is linearly stable with complex eigenvalues, and addition of noise gives oscillations around the fixed point with fluctuating amplitude. We investigate how each class of models behaves under the external periodic forcing, taking the well-studied van der Pol equation as an example. We find that when the forcing is additive, the noise-induced oscillator can show only one-to-one entrainment to the external frequency, in contrast to the limit cycle oscillator which is known to entrain to any ratio. When the external forcing is multiplicative, on the other hand, the noise-induced oscillator can show entrainment to a few ratios other than one-to-one, while the limit cycle oscillator shows entrain to any ratio. The noise blurs the entrainment in general, but clear entrainment regions for limit cycles can be identified as long as the noise is not too strong.

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.

Nonlinear dynamics and chaotic motion in the isothermal film casting process

In the industrial film casting process with the two-stage cascade (velocity and tension controlled) loop, various aspects of nonlinear dynamics and chaotic motion have been investigated solving simple 1-D viscoelastic model. Most of the extensional deformation processes exhibit sustained-periodic oscillations or stable limit cycles of state variables (called draw resonance instability) beyond the critical onset, even if they are conventionally controlled systems. Besides such well-known instability, some intriguing nonlinear dynamic phenomena such as period-doubling and eventual chaotic motion of film thickness and width could be observed when a disturbance is imposed in the second tension-controlled system. In other words, the bifurcation mode is dramatically changed by a sinusoidal disturbance under the combined velocity-tension control condition in contrast to only constant take-up velocity condition inducing a supercritical Hopf bifurcation of draw resonance.

Wave phenomena and edge of chaos in a diffusive predator-prey system under Allee effect

We investigate a spatial predator-prey system; predator under Allee effect. The predator-prey community dynamics is assumed to be of Holling-Tanner type. When community’s local dynamics displays sustained periodic oscillations, the system’s spatio-temporal dynamics supports periodic traveling waves. Dispersal of species is assumed to be a random process in one-dimension only. Two dimensional (2D) parameter scan studies reveal that the system dynamics is self-organized at Edges of Chaos (EOC). This is a vital result as existence of EOC has not been demonstrated in a reaction-diffusion system yet. As population dynamics of all animals are governed by two basic processes: a) growth and b) dispersal, existence of EOC puts a recent theory of ecological chaos (Upadhyay, 2009; Rai and Upadhyay, 2006) on robust footing. Role of Allee effect in enhancing chaos presents an opportunity for its application in control of biological invasions and conservation of small populations. Extinction-persistence transitions were also explored.

Analysis of liquid circulation in a rectangular tank with a gas source at a corner

Abstract In this work, we investigate experimentally and numerically the hydrodynamics induced by a bubble plume introduced at a corner of a rectangular tank. Such gas–liquid flows are inherently unsteady. Particle Image Velocimetry (PIV) was used to experimentally determine transient velocity fields in the system. For this gas–liquid flow system, both the fluctuating and mean liquid velocities were determined experimentally by PIV. This technique enables us to determine velocity fields in a 2D plane. The behavior of the system was simulated in FLUENT 6.2 using a two fluid Euler–Euler model with a constant bubble size of 3 mm. Water is treated as the continuous phase and the gas bubbles are treated as the dispersed phase. The motion of the bubbles renders the flow turbulence and this effect is captured by the mixture k–ɛ turbulence model. Two and three-dimensional simulations were carried out to predict the flow behavior. The predictions of the time averaged flow field, turbulent intensity etc. are compared with experimental observations. We also calculate the magnitude of the turbulent viscosity from our model. For the case of corner injection of bubbles, we conclude that the velocity at a point does not show sustained periodic oscillations in time.

Media/Psychological Impact on Multiple Outbreaks of Emerging Infectious Diseases

We use a compartmental model to illustrate a possible mechanism for multiple outbreaks or even sustained periodic oscillations of emerging infectious diseases due to the psychological impact of the reported numbers of infectious and hospitalized individuals. This impact leads to the change of avoidance and contact patterns at both individual and community levels, and incorporating this impact using a simple nonlinear incidence function into the model shows qualitative differences of the transmission dynamics.

Thermal Lensing and Optical Bistability in Ferroelectrics

In a recently developed theory of thermal lensing by ferroelectrics under laser illumination, the sample temperature as a function of radius and time was expressed as an integral. It is now shown that this integral can be approximated by known functions in a form which is accurate to within 1% for practical experimental situations. The theory is extended to explain asymmetric thermal focusing by BNN (barium sodium niobate, Ba2NaNb5O15). The presence of optical bistability in PMN (lead magnesium niobate, Pb(Mg1/3Nb2/3)O3), and PLZT (lanthanum doped lead zirconium titanate, Pb(Zr1-xTix)O3:La) and its absence in BNN and Ce:SBN (cerium doped strontium barium niobate, Ce3+:SrxBa1-xNb2O6) is explained by the presence or absence of Fabry-Pérot resonance. Possible sustained periodic relaxation oscillations are discussed.

Periodic coexistence in the chemostat with three species competing for three essential resources

A chemostat model of three species of microorganisms competing for three essential, growth-limiting nutrients is considered. J. Husiman and F.J. Weissing [Nature 402 (1999) 407] show numerically that this model can generate periodic oscillations. The present contribution is concerned with rigorous analysis regarding the existence of periodic oscillations in this model. Our analysis is based on the following observation made by Huisman and Weissing: there is a cyclic replacement of species, if each species becomes limited by the resource for which it is the intermediate competitor. Using a permanence theory, an index theory, and a Poincaré–Bendixson theory for three-dimensional competitive systems, we analytically succeed to give sufficient conditions for the existence of periodic orbits in the limit sets in this model. The results in this paper suggest that with a wide range of parameter values, sustained periodic oscillations of species abundances for the model are possible, without involving external disturbances. Our results also suggest that competition is not necessarily destructive, i.e., in the case of existence of sustained periodic oscillations, if one of three competitors is absent, one of the other two rivals cannot survive.

Bistability of quasi periodicity and period doubling in a delay-induced system

Sustained periodic relaxation oscillations (soft mode) and periodic spiking oscillations (hard mode) leading to chaotic oscillations with different bifurcation situations are found to coexist in lasers with incoherent delayed feedback that can be modeled by particle motion in a Toda-like potential with retarded interaction. Bistability of the soft-mode attractor (quasi periodicity) and the hard-mode attractor (period doubling) and their combined attraction are predicted theoretically.

Discrete Versus Continuous Controlled System

Because decision making processes involve costly mechanisms, they seldom are implemented continuously but instead take place at discrete time intervals. The question is then: what is the impact of this process discreteness on the systems dynamics? This problem is set here in the context of fisheries. Through a very simple though realistic model representing a shrimp fishery, we show that a discrete decision making process may lead to dynamics that differ completely from those obtained with a continuous process. For this purpose, we consider the interaction between a shrimp stock and the exploiting fleet. We then focus on the decision process that governs the allocation of the fleet fishing effort between the two stages of this stock: the young adults living near the coast and the mature adults located offshore. We first analyze the behaviour of the system when the discreteness in the decision making is not accounted for. In that case, the system turns out to be globally stable. We then identify the behaviour of the system when the decision process is discretised. In that case the solutions of the system yield sustained periodic oscillations. Our conclusion is that discreteness which is known to occur in decision making processes of most anthropic systems should be taken into account, especially in studies aiming precisely at investigating the dynamics of such systems.