Fractal Analysis of Pseudo Foci in Piecewise Analytic Systems

TL;DR

This paper investigates the fractal geometry of spiral trajectories near pseudo foci in piecewise analytic systems, extending known results from smooth systems to more complex, non-smooth contexts.

Contribution

It provides a complete fractal analysis of pseudo foci in piecewise analytic systems, linking Minkowski dimension and degeneracy to bifurcation behavior.

Findings

Minkowski dimension characterizes spiral trajectories near pseudo foci.

Results extend classical focus cyclicity concepts to non-smooth systems.

Fractal analysis aids in understanding bifurcations in piecewise systems.

Abstract

It is well known that the Minkowski dimension of spiral trajectories near a non-degenerate focus in analytic (smooth) systems is in one-to-one correspondence with the cyclicity of the focus in generic unfoldings. We give a complete fractal treatment, in terms of the Minkowski dimension and (non-)degeneracy, of spiral trajectories near pseudo foci of piecewise analytic systems. We hope these results will prove useful in the study of bifurcations of such pseudo foci through inherent geometry of associated spiral orbits.