Nonlinear Oscillators

Nonlinear oscillators arise frequently in the field of engineering. Of particular interest to me are periodically driven oscillators subject to noise which are often modeled using the Duffing Equation with an added noise term resulting in a Stochastic Differential Equation. Examples of such oscillators include the blades of a jet turbine or the massive turbine blades on windmills. These blades are subject to a periodic driving force and can oscillate in the direction perpendicular to the plane of rotation. There also exists noise inherent to this driving force which can come from the random fluctuations in wind speeds or the small instabilities in jet engine force output. Under certain conditions, the noise can drive the oscillator into a regime where the amplitude of the oscillator grows quickly. This behavior can lead to a loss a efficiency or to catastrophic failure making understanding the dynamics essential to applications.

Typical computational studies of these systems involve running long trajectories from a wide range of initial conditions. This can be computationally prohibitive, especially as the dimensionality grows when working with coupled oscillators. One of the major issues is that numerical stochastic integrators are typically of a much lower order than the typical ordinary differential equation solvers. However, I found that using a modified version of the Runge-Kutta 4^th order method had similar behavior and much improved performance compared to the Euler-Maruyama approach.

I combined this with a new way of studying these systems taken from transition path theory. To do so, I run many short trajectories from a large number of points. After running the trajectories for a single period of the driving force, I generate the transition matrix between all of the sample states. I then find the committor and from that I can compute things like the probability density and the transition rate between attractors.