Critical Value Of Time Delay Research Articles

Multistability and gluing bifurcation to butterflies in coupled networks with non-monotonic feedback

Neural networks with a non-monotonic activation function have been proposed to increase their capacity for memory storage and retrieval, but there is still a lack of rigorous mathematical analysis and detailed discussions of the impact of time lag. Here we consider a two-neuron recurrent network. We first show how supercritical pitchfork bifurcations and a saddle-node bifurcation lead to the coexistence of multiple stable equilibria (multistability) in the instantaneous updating network. We then study the effect of time delay on the local stability of these equilibria and show that four equilibria lose their stability at a certain critical value of time delay, and Hopf bifurcations of these equilibria occur simultaneously, leading to multiple coexisting periodic orbits. We apply centre manifold theory and normal form theory to determine the direction of these Hopf bifurcations and the stability of bifurcated periodic orbits. Numerical simulations show very interesting global patterns of periodic solutions as the time delay is varied. In particular, we observe that these four periodic solutions are glued together along the stable and unstable manifolds of saddle points to develop a butterfly structure through a complicated process of gluing bifurcations of periodic solutions.

Dynamic properties of a delayed protein cross talk model

In this paper we investigate how the inclusion of time delay alters the dynamical properties of the Jacob–Monod model, describing the control of the β-galactosidase synthesis by the lac repressor protein in E. coli. The consequences of a time delay on the dynamics of this system are analysed using Hopf's theorem and Lyapunov–Andronov's theory applied to the original mathematical model and to an approximated version. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations. A critical value of time delay, which depends on the values of the model parameters, is analytically established. Around this critical value, the properties of the system change drastically, allowing under certain conditions the emergence of stable limit cycles, that is self-sustained oscillations. In addition, the features of the end product repression play an essential role in the characterisation of these limit cycles: if cooperativity is considered in the end product repression, time delay higher than the mentioned critical value induce differentiated responses during the oscillations, provoking cycles of all-or-nothing response in the concentration of the species.

Active vibration control of flexible structures using delayed position feedback

The paper discusses the use of a simple position control system approach to improve the performance of lightly damped dynamic systems. This approach uses a delayed position feedback signal to actively control the vibrations of flexible structures. A complete analysis of the stability of a single-link flexible manipulator under time delay control is presented and critical values of time delay for a given controller gain have been determined. The paper also presents a short comparison between the delayed feedback signal control and the linear quadratic regulator.

Competition in chemostat-type equations with two habitats

Competition on a model with nutrient recycling is considered. The model is based on a chemostat-type equation which is used to study population dynamics of microorganisms. The model consists of four organisms competing for a limiting nutrient. Nutrient is supplied both from the in-flow of medium and a recycling with delay, the latter is generated from dead organisms by bacterial decomposition. This paper shows that the model undergoes a Hopf bifurcation through a critical value of time delay when the in-flow is small. Coexistence of four organisms competing for one limiting nutrient is indicated by numerical simulation results.

Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system

We consider the following Lotka-Volterra predator-prey system with two delays: x '( t ) = x ( t ) [ r(1) - ax ( t - tau(1) ) - by( t ) ] y '( t ) = y ( t ) [ - r(1) + cx ( t ) - dy( t - tau(2) ) ] ( E ) We show that a positive equilibrium of system ( E ) is globally asymptotically stable for small delays. Critical values of time delay through which system ( E ) undergoes a Hopf bifurcation are analytically determined. Some numerical simulations suggest an existence of subcritical Hopf bifurcation near the critical values of time delay. Further system (E) exhibits some chaotic behavior when tau(2) becomes large.

Bifurcation analysis on a survival red blood cells model

In this paper, we consider a model described the survival of red blood cells in animal. Its dynamics are studied in terms of local and global Hopf bifurcations. We show that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay crosses some critical values. Using the reduced system on the center manifold, we also obtain that the periodic orbits bifurcating from the positive equilibrium are stable in the center manifold, and all Hopf bifurcations are supercritical. Further, particular attention is focused on the continuation of local Hopf bifurcation. We show that global Hopf bifurcations exist after the second critical value of time delay.

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.

GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism—mathematical models and experimental observations

Planktonic blooms and its control is an intriguing problem in ecology. To investigate the oscillatory successions of blooms, three simple phytoplankton–zooplankton systems are proposed. It is observed that if the uptake function is linear and the process of toxin liberation is instantaneous, the oscillatory nature of blooms is not observed. On the other hand, periodic planktonic blooms are observed when toxin liberation process follows discrete time variation. The bloom phenomena described by this mechanism can be controlled through toxin producing phytoplankton (TPP). Introducing environmental fluctuation in the system, a critical value of time delay in terms of correlation time of the fluctuation is worked out. We observed from our mathematical analysis, numerical simulation and field observation that TPP and control of the rapidity of environmental fluctuation are key factors for the termination of planktonic blooms.

On the optimality criterion in structural control

The classical performance index optimization control algorithm is considered in order to check the real optimality of the control procedure; the basic steps for the optimal algorithm are reviewed, and the equation for the optimal control force derived. It is shown that the optimality conditions cannot be met with regard to the performance index, unless one is concerned with simple free oscillations. It is proved that in this case on one side the optimal control turns out to be of the linear closed-loop type, yielding explicit optimal control coefficients, and on the other side that no solution can exist of the optimal problem for a generic forcing function. It is concluded that one is forced to calibrate the control force for free oscillations, and that the reliability of the index procedure mainly rests on some implicit expectation that linear control can be calibrated in the absence of the external disturbance and that it works under forced oscillations as well. Furthermore, the problem of delayed active control, with reference to a s.d.o.f. system controlled by a closed-loop linear algorithm and under the action of a dynamic forcing function is investigated. In particular, the effects produced on the response of the structure by the introduction in the control law of assessed critical values of time delay are analysed and the comparison is proposed between the numerical results that one gets by adopting two different procedures (on one hand the above-mentioned optimal linear control law and on the other hand the constrained minimization of the structural response norm) to compensate for time lag occurring in the actuation of the active control servomechanisms. Copyright © 2000 John Wiley & Sons, Ltd.