Dynamics and geometry near resonant bifurcations

PhD ceremony: S.J. Holtman, 13.15 uur, Academiegebouw, Broerstraat 5, Groningen

Thesis: Dynamics and geometry near resonant bifurcations

Promotor(s): prof. H.W. Broer, prof. G. Vegter

Faculty: Mathematics and Natural Sciences

 

This work deals with non-linear parameter dependent dynamical systems exhibiting resonance. This phenomenon occurs if oscillatory subsystems interact, while the corresponding frequencies are rationally related. Such a situation appears in many real-world scenarios, e.g., coupled pendula, the earth-moon system and electric circuits. The main goal is to understand the parameter dependence of the qualitative behavior of the dynamics. To this end, we apply dynamical systems and singularity theory, which we both extend as needed. Our contribution is twofold. First, we study the geometry of parameter values for which resonance occurs. It turns out that in the non-degenerate case the geometry is given by the well-known 2-dimensional cusp-shaped Arnol’d resonance tongue. A mildly degenerate case corresponds to a more complicated 4-dimensional geometry, which we describe in detail. Moreover, we present a new algorithmic procedure determining to which of these two cases a given resonant family of dynamical systems belongs. Besides the geometrical study we also provide a numerical study and visualization of the generic dynamics in the neighborhood of resonance.