Thesis defence: Mathematics

Mr. M. Rafei | Master of Science in Mechanical Engineering
promotor | Prof.dr.ir. A.W. Heemink EWI
copromotor | Dr.ir. W.T. van Horssen UHD-EWI

On asymptotics for difference equations

In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for nonlinear difference equations are constructed by using the recently developed perturbation method based on invariance vectors. The asymptotic approximations of the solutions of the oscillator equations can be derived from these asymptotic approximations of the first integrals. It is shown that all invariance factors have to satisfy a functional equation. One of the main difficulties in finding first integrals for a system of first order difference equations is solving the aforementioned functional equation.
In this thesis, we consider a functional equation which is related to a system of two first order, linear ordinary difference equations. Linear transformations and an adapted version of the method of separation of variables is used to construct the general solution of this functional equation. A perturbation method based on invariance factors and multiple scales is also presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals are constructed on long iteration-scales, that is, on iteration-scales of order Ɛ⁻¹, where Ɛ is a small parameter.
To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. We also apply an improved version of the multiple scales perturbation method to a general system of weakly nonlinear, regularly perturbed ordinary difference equations including linear, quadratic, and qubic terms. Such systems arise as a result of the discretization of a system of
nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations.
As an example, in the study of the forced vibrations of a (damped) linear sdofo with a time-varying mass a system of two nonlinear ordinary difference equations is obtained to describe the stability properties of the oscillator. In such oscillators the forced vibrations are due to small masses which are periodically
hitting and leaving the oscillator with different velocities. In our procedure, asymptotic approximations of the solutions of the difference equations are constructed which are valid on long iteration scales.
In this thesis it is shown that the presented perturbation method based on invariance vectors can be applied to weakly nonlinear oscillator equations, which are ”close” to integrable equations (that is, are integrable in the unperturbed case).

More information?
For access to theses by the PhD students you can have a look in TU Delft Repository at: repository.tudelft.nl. TU Delft Repository is the digital storage of publications of TU Delft. Theses will be available within a few weeks after the actual thesis defence.