Parameter space and phase space study of the
non-chaotic Duffing oscillator
Troels Mikkelsen, Center for Ice and Climate, Niels Bohr Institute, Copenhagen
The Duffing equation x + δ.x + x + x^3 = γ cos(ωt) is one of the classical examples of a non-linear system. The model was initially introduced to study problems in mechanics, but due to its rich behavior it has been studied extensively in its own right. Recently, non-linear oscillations have been proposed as a tool for modeling the climate. In this talk I will focus on the behavior of Duffing's equation from a more theoretical standpoint and examine the behavior of at the low dissipation (δ << 1) limit, both in the (γ, ω)-parameter space and in phase space. The goal is to investigate the sensitive dependence on parameters - and initial conditions - of the periodic attractors, which emerge as a fractal structure. The results obtained are compared with the literature. The evolution of the periodic attractors in time are studied in order to establish whether these attractors are transient or not. Finally a connection between the fractal structure of the periodic attractors and the stability of numerical solutions is explored.
Q: Will I find this interesting?
A: If you like to marvel at the intricacies of the Mandelbrot set and consider it interesting that the Lorenz attractor has non-integer dimension then yes, you probably will!