Differential Delay Equations

This set of equations is a nonlinear eigenvalue time delay differential equation. Such equations, even for one variable, often have periodic or chaotic solutions and, from the physics of the problem are also certain of having pulse-like solutions in some systems. 

In the above definitions, 9 represents a set of parameters of the system, having constant values. These parameters are also called control parameters. The set of the system s variables forms a representation space called the phase space [32]. A point in the phase space represents a unique state of the dynamic system. Thus, the evolution of the system in time is represented by a curve in the phase space called trajectory or orbit for the flow or the map, respectively. The number of variables needed to describe the system s state, which is the number of initial conditions needed to determine a unique trajectory, is the dimension of the system. There are also dynamic systems that have infinite dimension. In these cases, the processes are usually described by differential equations with partial derivatives or time-delay differential equations, which can be considered as a set of infinite in number ordinary differential equations. The fundamental property of the phase space is that trajectories can never intersect themselves or each other. The phase space is a valuable tool in dynamic systems analysis since it is easier to analyze the properties of a dynamic system by determining... 

Cortisol concentration is described by a nonlinear time-delay differential equation [47,519] with two terms, i.e., a secretion rate term that adheres to the negative feedback mechanism [520, 521] and drives the pulsatile secretion, and a first-order output term with rate constant ka ... 

This illustrates that the proper initial value problem is the one indicated by the initial conditions just listed. The theory for such delay differential equations is much more complicated than that for ordinary differential equations, and is not so widely known among nonspecialists. The basic reference is Hale [HI] see also Kuang [K2j. 

We review the basic definitions and set up the semidynamical system appropriate for systems of the form (D.l). Let A" be a locally compact metric space with metric d, and let be a closed subset of X with boundary dE and interior E. The boundary, dE, corresponds to extinction in the ecological problems. Let tt be a semidynamical system defined on E which leaves dE invariant. (A set B in A" is said to be invariant if n-(B, t) = B.) Dynamical systems and semidynamical systems were discussed in Chapter 1. The principal difficulty for our purposes is that for semidynamical systems, the backward orbit through a point need not exist and, if it does exist, it need not be unique. Hence, in general, the alpha limit set needs to be defined with care (see [H3]) and, for a point x, it may not exist. Those familiar with delay differential equations are aware of the problem. Fortunately, for points in an omega limit set (in general, for a compact invariant set), a backward orbit always exists. The definition of the alpha limit set for a specified backward orbit needs no modification. We will use the notation a.y(x) to denote the alpha limit set for a given orbit 7 through the point x. 

JST] W. Jager, H. Smith, and B. Tang (1991), Some aspects of competitive coexistence and persistence, in S. Busenberg and M. Martelli (eds.), Delay Differential Equations and Dynamical Systems. Berlin Springer, pp. 200-9. 

K2] Y. Kuang (1993), Delay differential equations with applications in population dynamics, in Mathematics in Science and Engineering, vol. 191. Boston Academic Press. 

Inserting Eq. (2.37) into Eq. (2,28) gives the delay differential equation ... 

Time delays also play a significant role in the onset of oscillations in the cascade. Such time delays result here in a natural way from the thresholds in the activation curves of cdc2 kinase and cyclin protease, and are by no means inserted in an ad hoc manner into the kinetic equations (10.1), as is often done for systems governed by time-delay differential equations. 

To model the regeneration of waviness inside the process model, history information about the cut is required. Generally, such relations are described by delayed differential equations (DDEs) in literature and within this article as well. 

Research Studies Press, Taunton Gawronski WK (2004) Advanced structural dynamics and active control of structures. Springer, New York Insperger T, Stepan G (2004) Updated semi-discretization method for periodic delay-differential equations with discrete delay. Int J Numer Meth Eng 61(1) 117-141... 

This is a system of nonlinear autonomous delay-differential equations. Linearization around the steady-state motion (the constant deflection of the tool, Xq and jo) gives... 

If sufficient high frequency components are present, then the mass of the memory kernel is typically found to be localized around 0 in such a situation of rapid decay it is found that one may replace the function by a Dirac delta function scaled by a positive coefficient y we say that the system is Markovian or memory-less in this situation and we replace the delay-differential equation (6.29) by the SDEs... 

New mathematical models and reviews on the lac operon have appeared recently. " Yildirim and Mackey used delay differential equations to account for the transcriptional and translational steps that are missing in their model. An earlier detailed kinetic model was proposed and analyzed by Wong, Gladney and Keasling. Recently, Vilar, Guet, and Leibler" used a four-variable model that captures many of the essential dynamics of the lac operon. Note that the Vilar-Guet-Leibler model is essentially a three-variable model. The bistability exhibited by the model was the explanation for the on-off behavior of the lac operon. 

Chemical reaction network is a typical example of complexity, where the reactants can interact in a variety of ways depending on the nature of interaction (chemical as well as non-chemical). Oscillatory reactions involve a number of steps, including positive and negative feedbacks. The complexity leads to periodic as well as aperiodic oscillations (multi-periodic, bursting/intermittency sequential oscillations separated by a time pause, relaxation and chaotic oscillations). The mechanism is usually determined by non-linear kinetics and computer modelling. Once the reaction mechanism has been postulated, the non-linear time-dependent kinetic equation can be formulated in terms of concentrations of different reactants, which would yield a multi-variable equation. Delay differential equations are sometimes used to characterize oscillatory behaviour as in economics (Chapter 14). 

This equation can be solved by the technique of delay differential equation, or one can use the following transformation... 

The simplest model of time-dependent behavior of a neutron population in a reactor consists of the point kinetics differential equations, where the space-dependence of neutrons is disregarded. The safety of reactors is greatly enhanced inherently by the existence of delayed neutrons, which come from radioactive decay rather than fission. The differential equations for the neutron population, n, and delayed neutron emitters, are... 

This is identical to the first order Fade power series and gives a crude time delay approximation, when transformed back into differential equation form. 

The limitation of transfer function representation becomes plain obvious as we tackle more complex problems. For complex systems with multiple inputs and outputs, transfer function matrices can become very clumsy. In the so-called modem control, the method of choice is state space or state variables in time domain—essentially a matrix representation of the model equations. The formulation allows us to make use of theories in linear algebra and differential equations. It is always a mistake to tackle modem control without a firm background in these mathematical topics. For this reason, we will not overreach by doing both the mathematical background and the control together. Without a formal mathematical framework, we will put the explanation in examples as much as possible. The actual state space control has to be delayed until after tackling classical transfer function feedback systems. 

Since there is no axial mixing, clearly any input is delayed at the outlet by the period of the residence time. This result also can demonstrated formally by solving the pertinent differential equations. 

Dynamic problems expressed in transfer function form are often very easily reformulated back into sets of differential equation and associated time delay functions. An example of this is shown in the simulation example TRANSIM. 

According to Eq. (3.31), the heating power would produce a final static temperature difference, A Tf. Now the dynamics have to be taken into account. A reformulation of the differential equation leads to an expression including a certain delay ... 

The outline of this paper is as follows. First, a theoretical model of unsteady motions in a combustion chamber with feedback control is constructed. The formulation is based on a generalized wave equation which accommodates all influences of acoustic wave motions and combustion responses. Control actions are achieved by injecting secondary fuel into the chamber, with its instantaneous mass flow rate determined by a robust controller. Physically, the reaction of the injected fuel with the primary combustion flow produces a modulated distribution of external forcing to the oscillatory flowfield, and it can be modeled conveniently by an assembly of point actuators. After a procedure equivalent to the Galerkin method, the governing wave equation reduces to a system of ordinary differential equations with time-delayed inputs for the amplitude of each acoustic mode, serving as the basis for the controller design. 

The delay in the tubuloglomerular feedback is represented by means of three first-order coupled differential equations ... 

Time delays can also be handled with the LQP, although the discrete-time formulation (46) of the LQP is better suited to the time delay problem (especially when there are only a few such elements in the differential equations). ... 

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