4,531 research outputs found
Delay Equations and Radiation Damping
Starting from delay equations that model field retardation effects, we study
the origin of runaway modes that appear in the solutions of the classical
equations of motion involving the radiation reaction force. When retardation
effects are small, we argue that the physically significant solutions belong to
the so-called slow manifold of the system and we identify this invariant
manifold with the attractor in the state space of the delay equation. We
demonstrate via an example that when retardation effects are no longer small,
the motion could exhibit bifurcation phenomena that are not contained in the
local equations of motion.Comment: 15 pages, 1 figure, a paragraph added on page 5; 3 references adde
Asymptotic methods for delay equations.
Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006
Slowly modulated oscillations in nonlinear diffusion processes
It is shown here that certain systems of nonlinear (parabolic) reaction-diffusion equations have solutions which are approximated by oscillatory functions in the form R(ξ - cτ)P(t^*) where P(t^*) represents a sinusoidal oscillation on a fast time scale t* and R(ξ - cτ) represents a slowly-varying modulating amplitude on slow space (ξ) and slow time (τ) scales. Such solutions describe phenomena in chemical reactors, chemical and biological reactions, and in other media where a stable oscillation at each point (or site) undergoes a slow amplitude change due to diffusion
Multiple solutions and periodic oscillations in nonlinear diffusion processes
We study the oscillatory stationary states in the temperature and concentration fields occurring in tubular chemical reactors. Singular perturbation and multitime scale procedures are combined formally to clearly and simply reveal the mechanism controlling these oscillatory states. Their stability is also studied, and when coupled with previously obtained results on multiple steady states, this information completes the response (bifurcation) diagram in one-parameter range of the tubular reactor. The results apply also to more general nonlinear parabolic problems of which the first order tubular reactor is a special case
Derivation of Delay Equation Climate Models Using the Mori-Zwanzig Formalism
Models incorporating delay have been frequently used to understand climate
variability phenomena, but often the delay is introduced through an ad-hoc
physical reasoning, such as the propagation time of waves. In this paper, the
Mori-Zwanzig formalism is introduced as a way to systematically derive delay
models from systems of partial differential equations and hence provides a
better justification for using these delay-type models. The Mori-Zwanzig
technique gives a formal rewriting of the system using a projection onto a set
of resolved variables, where the rewritten system contains a memory term. The
computation of this memory term requires solving the orthogonal dynamics
equation, which represents the unresolved dynamics. For nonlinear systems, it
is often not possible to obtain an analytical solution to the orthogonal
dynamics and an approximate solution needs to be found. Here, we demonstrate
the Mori-Zwanzig technique for a two-strip model of the El Nino Southern
Oscillation (ENSO) and explore methods to solve the orthogonal dynamics. The
resulting nonlinear delay model contains an additional term compared to
previously proposed ad-hoc conceptual models. This new term leads to a larger
ENSO period, which is closer to that seen in observations.Comment: Submitted to Proceedings of the Royal Society A, 25 pages, 10 figure
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