Fractal Analysis of Planar Slow-Fast Systems

Abstract

This thesis investigates a fractal approach to Hilbert’s 16th problem, which concerns, among other things, determining an upper bound on the number of limit cycles in planar polynomial vector fields. The focus is on planar slow–fast systems, characterized by slow and fast time scales and widely used in fields such as biology, chemistry, and climate modeling. Rather than adopting a global perspective, a local analysis is developed using the Minkowski dimension of point sequences generated by exit-entry relationships to extract information about limit cycles. The thesis provides a fractal classification of slow–fast Hopf points and (un)bounded canard cycles. Furthermore, a connection is established between fractal dimensions for slow–fast Hopf points and the maximal number of limit cycles for slow–fast Hopf points. These results contribute to a deeper understanding of limit cycles in slow–fast systems and fractal classification in slow-fast systems, including (piecewise smooth) Liénard equations.