Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis
McCarthy, Michael J. (2011) Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis. PhD thesis, Dublin City University.
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This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be
replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We
therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends
on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering
this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose.
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