Stability Of Equilibrium Solutions Research Articles

Stability and Hopf Bifurcation in a HIV-1 System with Multitime Delays

In this paper, we propose a mathematical model for HIV-1 infection with three time delays. The model examines a viral-therapy for controlling infections by using an engineered virus to selectively eliminate infected cells. In our model, three time delays represent the latent period of pathogen virus, pathogen virus production period and recombinant (genetically modified) virus production period, respectively. Detailed theoretical analysis have demonstrated that the values of three delays can affect the stability of equilibrium solutions, can also lead to Hopf bifurcation and oscillated solutions of the system. Moreover, we give the conditions for the existence of stable positive equilibrium solution and Hopf bifurcation. Further, the properties of Hopf bifurcation are discussed. These theoretical results indicate that the delays play an important role in determining the dynamic behavior quantitatively. Therefore, it is a fact that delays are very important, which should not be missed in controlling HIV-1 infections.

Dynamics and stability of a three-dimensional model of cell signal transduction with delay

In this paper, we consider a three-dimensional model of cell signal transduction with delay. The deactivation of signalling proteins occurs throughout the cytosol and activation is localized to specific sites in the cell. The enzyme kinetic functions employ a constant delay to model the time lapse during reactions and also the recovery times associated with conformational changes. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of delayed differential algebraic equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo Hopf bifurcations for certain delay values. In these cases sustained oscillations are observed. The Poincaré–Lindstedt3 method is used to improve upon the asymptotic approximations. The simulations of the full three-dimensional system correspond well with simulations of the reduced delayed differential algebraic equations.

On the Budyko-Sellers energy balance climate model with ice line coupling

Over 40 years ago, M. Budyko and W. Sellers independently introduced low-order climate models that continue to play an important role in the mathematical modeling of climate. Each model has one spatial variable, and each was introduced to investigate the role ice-albedo feedback plays in influencing surface temperature. This paper serves in part as a tutorial on the Budyko-Sellers model, with particular focus placed on the coupling of this model with an ice sheet that is allowed to respond to changes in temperature, as introduced in recent work by E. Widiasih. We review known results regarding the dynamics of this coupled model, with both continuous (``Sellers-type) and discontinuous (``Budyko-type) equations. We also introduce two new Budyko-type models that are highly effective in modeling the extreme glacial events of the Neoproterozoic Era. We prove in each case the existence of a stable equilibrium solution for which the ice sheet edge rests in tropical latitudes. Mathematical tools used in the analysis include geometric singular perturbation theory and Filippov's theory of differential inclusions.

Towing simulation in wind through a nonlinear 4-DOF model: Bifurcation analysis and occurrence of fishtailing

Towing dynamics is investigated by means of a simplified mathematical model that approximates the tug as a point particle with prescribed motion, while the tow is modelled as a 4-DOF (surge/sway/yaw/roll) nonlinear dynamical system. Manoeuvring forces and moments are calculated via a model capable of dealing with high drift angles; aerodynamic effects are also taken into account and the connection between the tug and the tow is simulated by means of a single elastic towline. A numerical bifurcation analysis is carried out to determine the equilibria of the system when a relevant parameter (such as wind direction/speed) is modified. Time-domain simulations are performed wherever necessary. An example application is proposed, in which the behaviour of two vessels under towing is examined. It is shown that multiple stable/unstable towing equilibria may exist and that the response of the system is deeply affected by the wind direction and speed; in the absence of stable equilibrium solutions, a potentially dangerous fishtailing phenomenon takes place, sometimes leading to evident roll oscillations and large snap-loads in the towing line. Two different towing speeds are examined, in order to analyse how such an important operational parameter can affect the equilibria of the system and their stability.

A theoretical analysis of pitch stability during gliding in flying snakes

Flying snakes use their entire body as a continuously morphing ‘wing’ to produce lift and shallow their glide trajectory. Their dominant behavior during gliding is aerial undulation, in which lateral waves are sent posteriorly down the body. This highly dynamic behavior, which is unique among animal gliders, should have substantial effects on the flight dynamics and stability of the snakes, resulting from the continuous redistribution of mass and aerodynamic forces. In this study, we develop two-dimensional theoretical models to assess the stability characteristics of snakes in the pitch direction. Previously measured force coefficients are used to simulate aerodynamic forces acting on the models, and undulation is simulated by varying mass. Model 1 is a simple three-airfoil representation of the snake’s body that possesses a passively stable equilibrium solution, whose basin of stability contains initial conditions observed in experimental gliding trajectories. Model 2 is more sophisticated, with more degrees of freedom allowing for postural changes to better represent the snake’s real kinematics; in addition, a restoring moment is added to simulate potential active control. The application of static and dynamic stability criteria show that Model 2 is passively unstable, but can be stabilized with a restoring moment. Overall, these models suggest that undulation does not contribute to stability in pitch, and that flying snakes require a closed-loop control system formed around a passively stable dynamical framework.

THE ROLE OF BILINGUALS IN LANGUAGE COMPETITION

We study the role played by bilinguals in the competition between two languages and in the formation of a bilingual community. To this aim we introduce a simple three-state model that combines the Minett–Wang model, in which the bilinguals do not affect directly the probability of transition of an individual from monolingualism to bilingualism, and the Baggs–Freedman model, in which such a transition probability only depends on bilinguals. The model predicts the possibility for the existence of a stable bilingual community though no particular conditions are assumed for the two competing languages: the asymptotic stability of the bilinguals community is only due to the type of dynamics regulating the transitions between different linguistic groups. The proposed model and the obtained results give some suggestions for the conditions necessary for the formation of a stable bilingual community. First of all, it is important that the bilinguals are valid representatives of the two languages, in the sense that they are regarded by monolinguals also as speakers of the other language. Besides that, the transition from bilinguals to monolinguals must be smaller than the opposite transition. Unless both these conditions are fulfilled, the stable equilibrium solution of the language competition is a monolingual society.

Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: x n+1 = (2ax n)/(1 + x n 2) − x n−1, n = 0,1,…, where x −1, x 0 ∈ (−∞, ∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

Stability and pattern formation for competing populations with asymmetric nonlocal coupling

We consider a model of two competing species with asymmetric nonlocal coupling in a competition for resources. The nonlocal coupling is via convolution integrals and the asymmetry is via convolution kernel functions which are not even functions of their arguments. The nonlocality is due to species mobility, so that at any fixed point in space the competition for resources depends not just on the populations at that point but on a suitably weighted average of the populations. We introduce two parameters, δ, describing the extent of the coupling, with δ=0 corresponding to local coupling, and α, describing the extent of the asymmetry, with α=0 corresponding to symmetric nonlocal interactions. We consider the case where the model admits a stable coexistence equilibrium solution. We perform a linear stability analysis and show that this solution can be destabilized by sufficient nonlocality, i.e., when δ increases beyond a critical value. We consider two specific kernel functions, (i) an asymmetric Gaussian and (ii) an asymmetric stepfunction. We compute the stability boundary as a function of α, and for δ beyond the stability boundary we determine unstable wavenumber bands. We compute nonlinear patterns for δ significantly beyond the stability boundary. Patterns consist of arrays of islands, regions of nonzero population, separated by either near-deadzones where the populations are small, but nonzero, or by deadzones where populations are exponentially small and essentially extinct. We find solutions consisting of propagating traveling waves of islands, solutions exhibiting colony formation, where a colony is formed just ahead of an island and eventually grows as the parent island decays, and modulated traveling waves, where competition between the two species allows propagation and inhibits colony formation. We explain colony formation and the modulated traveling waves as due to a positive feedback mechanism associated with small variations in the amplitude of the parent island.

Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation

We are dealing with a two-component system of reaction–diffusion equations with mass conservation in a bounded domain with the Neumann boundary conditions. We prove the global boundedness of the solution in L∞-norm for t⩾0 under a condition, and then the existence of a Lyapunov function. Moreover, by studying the linearized eigenvalue problem of a nonconstant equilibrium solution, we provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation. As an application of the comparison result we obtain that any stable equilibrium solution must be monotone if the space dimension is one. It is also shown that a modified system with a new parameter, which covers the present model, possesses a Lyapunov function.

Stability of equilibrium solutions in the critical case of even-order resonance in periodic Hamiltonian systems with one degree of freedom

The stability in the Lyapunov sense of an equilibrium position in a periodic Hamiltonian system with one degree of freedom is studied. It is assumed that the equilibrium is stable in the first approximation and that there exists an even resonance of order $$k$$ , arbitrary. The critical case is considered, i.e., when the system of parameters is such that, in order to draw rigorous conclusions about the stability of the equilibrium position in the Lypaunov sense, terms or order higher than three in the series expansion of the Hamiltonian function must be taken into account. Sufficient conditions are derived for stability and instability.

Post-buckling bifurcations and stability of high-speed axially moving beams

The nonlinear forced dynamics of an axially moving beam in the supercritical speed regime is numerically investigated in this paper, with special consideration to the case possessing an internal resonance between the first two modes. At critical speed, the initial equilibrium configuration of the beam becomes unstable and a new stable non-trivial equilibrium solution together with its symmetric counterpart arises. A closed-form solution for the post-buckling configuration is introduced and the system is assumed to be subject to a transverse harmonic excitation load at its buckled state. The equation of motion is cast into new coordinates which gives the equation governing the motion of the beam about the buckled state. This equation is discretized via the Galerkin method which yields a set of nonlinear ordinary differential equations (NODEs) with quadratic and cubic nonlinear terms. The set of NODEs are solved either via the pseudo-arclength continuation technique or by means of direct time integration so as to obtain frequency–response curves as well as bifurcation diagrams of Poincaré sections about the buckled state. Results are shown through time histories, phase-plane diagrams, and Poincaré sections.

A Reaction Diffusion Model for Inter-Species Competition and Intra-Species Cooperation

This thesis deals with a two component reaction-diffusion system (RDS) for competing and cooperating species. We have analyse in detail the stability and bifurcation structure of equilibrium solutions of this system, a natural extension of the Lotka-Volterra system. We find seven topologically different regions separated by bifurcation boundaries depending on the number and stability of equilibrium solutions, with four regions in which the solutions are similar to those in the Lotka-Volterra system. We study RDS in the small parameter of the range $0< \lambda \ll 1 $ (fast diffusion and slow reaction), and in a few cases we assume $\lambda=O(1)$. We consider three types of initial conditions, and we find three types of travelling wave solutions using numerical and asymptotic methods. However, neither numerical nor asymptotic methods were able to find a particular travelling wave solution which connects a coexistence state say, $(u_0,w_0)$ to an extinction state $(0,0)$ when $0< \lambda \ll 1 $. This type can be found when the reaction-diffusion system satisfy the symmetry property and $\lambda=1$.

Falling motion of a circular cylinder interacting dynamically with a point vortex

The dynamical behavior of a heavy circular cylinder and a point vortex in an unbounded volume of ideal liquid is considered. The liquid is assumed to be irrotational and at rest at infinity. The circulation about the cylinder is different from zero. The governing equations are Hamiltonian and admit an evident autonomous integral of motion — the horizontal component of the linear momentum. Using the integral we reduce the order and thereby obtain a system with two degrees of freedom. The stability of equilibrium solutions is investigated and some remarkable types of partial solutions of the system are presented.

A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations

The stability of equilibrium solutions of a deterministic linear system of delay differential equationscan be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based onthe vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providingsufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system andthen compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.

Necking limits of conoid membrane structures with variable stress ratio

Form finding of conoid membrane structures separated by two coaxial circular rings is investigated for different values of fixed stress ratios between meridianal and circumferential stress. Of primary interest is the maximum separation of the rings as a function of stress ratio to achieve a stable equilibrium solution and the corresponding radius at the mid-plane between the rings, termed the necking radius. A rapidly converging computer program was developed to determine these limits quickly for a wide range of stress ratios. Asymptotic limits for the necking radius and separation distance were determined as the stress ratio approaches infinity.

Dynamics and stability of a three-dimensional model of cell signal transduction

In this paper, we consider a three-dimensional model of cell signal transduction. In this model, the deactivation of signalling proteins occur throughout the cytosol and activation is localized to specific sites in the cell. We use matched asymptotic expansions to construct the dynamic solutions of signalling protein concentrations. The result of the asymptotic analysis is a system of ordinary differential equations. This reduced system is compared to numerical simulations of the full three-dimensional system. As well, we consider the stability of equilibrium solutions. We find that the systems under consideration may undergo sustained oscillations, hysteresis and other complex behaviors. The simulations of the full three-dimensional system agree with simulations of the reduced ordinary differential equations.

Helical swimming can provide robust upwards transport for gravitactic single-cell algae; a mechanistic model

In still fluid, many phytoplankton swim in helical paths with an average upwards motion. A new mechanistic model for gravitactic algae subject to an intrinsic torque is developed here, based on Heterosigma akashiwo, which results in upwards helical trajectories in still fluid. The resultant upwards swimming speed is calculated as a function of the gravitactic and intrinsic torques. Helical swimmers have a reduced upwards speed in still fluid compared to cells which swim straight upwards. However a novel result is obtained when the effect of fluid shear is considered. For intermediate values of shear and intrinsic torque, a new stable equilibrium solution for swimming direction is obtained for helical swimmers. This results in positive upwards transport in vertical shear flow, in contrast to the stable equilibrium solution for straight swimmers which results in downwards transport in vertical shear flow. Furthermore, for strong intrinsic torque, when there is no longer a stable orientation equilibrium, we show that the average downwards transport of helical swimmers in vertical shear flow is greatly suppressed compared to straight swimmers. We hypothesise that helical swimming provides robustness for upwards transport in the presence of fluid shearing motions.

Biomass dynamics of the phytoplankton under impact of the nutrient

The phytoplankton production is a basis for a life activity of all species in every aquatic ecological system. Remote sounding methods make it possible to estimate chlorophyll concentration on the ocean surface. The data from remote sounding methods and the data of a nutrient concentration enable us to calculate and model the phytoplankton abundance. These calculations provide a possibility for the description of the phytoplankton production processes. We present here some models for the dynamics of the phytoplankton biomass. The qualitative properties of the solutions of these models are investigated. Strategically activities of the cell are defined by its internal state and by the environmental conditions. The state of the cell depends on nutrients. Our mathematical models describe the abundance dynamics for phytoplankton's community and dynamics of the cellular nutrients content. The models are based on the systems of differential equations. These models may be open or closed relative to the matter. The open models describe the chemostat processes. We investigate the property of the solutions of these models. The closed models have a infinite set of the positive solutions. The open models have a finite set of the non-negative solutions. The algorithm for finding the stable equilibrium solutions in open models is constructed. The theorems about the stable equilibrium solutions are proved with help the methods of structural analysis. These models describe the phytoplankton dynamics in the Black Sea.

The Impact of Process Architecture on Equilibrium Stability in Distributed Design

In distributed design processes, individual design subsystems have local control over design variables and seek to satisfy their own individual objectives, which may also be influenced by some system level objectives. The resulting network of coupled subsystems will either converge to a stable equilibrium or diverge in an unstable manner. In this paper, we study the dependence of system stability on the solution process architecture. The solution process architecture describes how the design subsystems are ordered and can be either sequential, parallel, or a hybrid that incorporates both parallel and sequential elements. In this paper, we demonstrate that the stability of a distributed design system does indeed depend on the solution process architecture chosen, and we create a general process architecture model based on linear systems theory. The model allows the stability of equilibrium solutions to be analyzed for distributed design systems by converting any process architecture into an equivalent parallel representation. Moreover, we show that this approach can accurately predict when the equilibrium is unstable and the system divergent when previous models suggest that the system is convergent.

Time-delay feedback control of lathe cutting tools

For some machining operations, tool vibrations decay with time, the trivial solution is a stable equilibrium solution, and hence the machined surface is smooth. For other operating conditions, the trivial solution is unstable and the tool undergoes large oscillations called chatter, which might be periodic, quasiperiodic, or chaotic, and hence the surface finish will not be smooth. Therefore, chatter must be avoided or mitigated to maintain machining tolerances, preserve surface finish, and prevent tool breakage. Therefore, extensive research has been devoted to the understanding of the mechanisms of, modeling of, and methods of controlling machine–tool chatter. In this paper, we show that the region of linearly stable trivial solution can be widened considerably using time-delay acceleration feedback.