9,880 research outputs found
Global dynamics of a novel delayed logistic equation arising from cell biology
The delayed logistic equation (also known as Hutchinson's equation or
Wright's equation) was originally introduced to explain oscillatory phenomena
in ecological dynamics. While it motivated the development of a large number of
mathematical tools in the study of nonlinear delay differential equations, it
also received criticism from modellers because of the lack of a mechanistic
biological derivation and interpretation. Here we propose a new delayed
logistic equation, which has clear biological underpinning coming from cell
population modelling. This nonlinear differential equation includes terms with
discrete and distributed delays. The global dynamics is completely described,
and it is proven that all feasible nontrivial solutions converge to the
positive equilibrium. The main tools of the proof rely on persistence theory,
comparison principles and an -perturbation technique. Using local
invariant manifolds, a unique heteroclinic orbit is constructed that connects
the unstable zero and the stable positive equilibrium, and we show that these
three complete orbits constitute the global attractor of the system. Despite
global attractivity, the dynamics is not trivial as we can observe long-lasting
transient oscillatory patterns of various shapes. We also discuss the
biological implications of these findings and their relations to other logistic
type models of growth with delays
Dynamical Systems, Stability, and Chaos
In this expository and resources chapter we review selected aspects of the
mathematics of dynamical systems, stability, and chaos, within a historical
framework that draws together two threads of its early development: celestial
mechanics and control theory, and focussing on qualitative theory. From this
perspective we show how concepts of stability enable us to classify dynamical
equations and their solutions and connect the key issues of nonlinearity,
bifurcation, control, and uncertainty that are common to time-dependent
problems in natural and engineered systems. We discuss stability and
bifurcations in three simple model problems, and conclude with a survey of
recent extensions of stability theory to complex networks.Comment: 28 pages, 10 figures. 26/04/2007: The book title was changed at the
last minute. No other changes have been made. Chapter 1 in: J.P. Denier and
J.S. Frederiksen (editors), Frontiers in Turbulence and Coherent Structures.
World Scientific Singapore 2007 (in press
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
Qualitative analysis of some models of delay differential equations
This thesis concerns the study of the global dynamics of delay differential
equations of the so-called production and destruction type, which find applications to the modelling of several
phenomena in areas such as population growth dynamics, economics, cell production, etc. For instance, by
applying tools coming from discrete dynamics, we provide sufficient conditions for the existence of globally
attracting equilibria for families of scalar or multidimensional equations. Moreover, we extend some known results in
the scalar non-autonomous case by the use of integral inequalities. Finally, the existence of periodic solutions is
analysed in the general context of infinite delay, impulses and periodic coefficients
Approximation Of Continuously Distributed Delay Differential Equations
We establish a theorem on the approximation of the solutions of delay differential equations with continuously distributed delay with solutions of delay differential equations with discrete delays. We present numerical simulations of the trajectories of discrete delay differential equations and the dependence of their behavior for various delay amounts. We further simulate continuously distributed delays by considering discrete approximation of the continuous distribution
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