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In this chapter we will discuss the effects of time delay on the collective states of a model mathematical system composed of a collection of coupled limit cycle oscillators. Such an assembly of coupled nonlinear oscillators serves as a useful paradigm for the study of collective phenomena in many physical, chemical, and biological systems and has therefore led to a great deal of theoretical and experimental...
To those who have ever dealt with delay equations, the instabilities and oscillatory behavior caused by delays are all too well known. What perhaps not so familiar is that the delays could also have the opposite effect, namely that they could suppress oscillations and stabilize equilibria which would be unstable in the absence of delays. Research in this area is relatively sparse, as stabilization...
The existence of time delays at the actuating input in a feedback control system is usually known to cause instability or poor performance for the corresponding closed-loop schemes (see, for instance [10, 18, 20, 23] and the references therein).
Over the past decade control of unstable states has evolved into a central issue in applied nonlinear science [1]. This field has various aspects comprising stabilization of unstable periodic orbits embedded in a deterministic chaotic attractor, which is generally referred to as chaos control, stabilization of unstable fixed points (steady states), or control of the coherence and timescales of stochastic...
In recent decades finite propagation speeds have been observed experimentally in spatially extended systems. For instance, in neural and biological systems they have been found to evoke novel spatio-temporal phenomena [4, 8–10, 12, 15, 21, 22, 24, 26, 28, 29, 37, 39]. This effect may be understood by the similarity of the delay caused by the finite propagation speed and other intrinsic timescales.
This chapter concerns the effect of noise on linear and nonlinear delay-differential equations. Currently there exists no formalism to exactly compute the effects of noise in nonlinear systems with delays. The standard Fokker–Planck approach is not justified because it is meant for Markovian systems. Delay-differential systems are non-Markovian, although various approximations to them might be Markovian.
Artificial neural networks arise from the research of the configuration and function of the brain. As pointed out in [79], the brain can be regarded as a complex nonlinear parallel information processing system with a concept of neuron as a basic functional unit.
This class of equations is widely used in many research fields—it can be obtained through the linearization of different nonlinear problems (see, for example, Sect. 8.5)—such as automatic, economic, and, for our purpose, in biological modeling because it can be associated with problems in which it is important to take into account some history of the state variable (e.g., gestation period, cell cycle...
Research in understanding traffic flow is conducted since 1930s in mathematics, physics, and engineering fields. The main interest is to reveal the characteristics of traffic dynamics and consequently propose ways to reduce undesirable impacts of traffic flow to social and economical life. This can be achieved only if rigorous and reliable mathematical models are constructed. The first part of this...
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