MS03.2: Computational Methods

A novel approach to the practical study of delay equations consists in reformulating them as abstract differential equations, applying a pseudospectral discretization in order to obtain a system of ordinary differential equations (ODEs) and using available numerical methods for ODEs. This technique has been applied in particular to study the stability and bifurcations using MatCont, a MATLAB-based bifurcation package for ODEs. In order to make the approach more accessible to users who are not familiar with numerical methods and programming, we developed an extension of MatCont implementing the technique for delay differential equations and renewal equations with constant finite discrete and distributed delays.

The analysis and control of mixing in fluid flows has numerous applications. We extend a diffusion map approach in trajectory space in order to model the evolution of scalar quantities in a data-based manner. This allows us to study and quantify mixing in closed and open flows. The framework is applied to several example systems.

In the present work we show experimental frequency-response curves in the near resonant regime of dynamic mode atomic force microscopy that are obtained through a control-based continuation method. In dynamic mode atomic force microscopy a microcantilever is monoharmonically forced in proximity (nanometer regime) of a sample. Highly nonlinear interaction forces between tip and sample induce unstable periodic orbits in the displacement motion of the tip, manifesting in nonlinear frequency response curves with fold bifurcations. By utilizing adaptive filters to estimate Fourier coefficients online, we are experimentally able to stabilize formerly unstable periodic orbits in lab experiments of dynamic mode atomic force microscopy by a non-invasive phase-locked loop controller and measure complete frequency-response curves. In addition, we change the distance between the cantilever tip and sample and show the transition from linear to nonlinear frequency-response curves.