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Fractal Analysis of Pseudo Foci in Piecewise Analytic Systems

Vlatko Crnković

TL;DR

The paper investigates the fractal geometry of spiral trajectories near pseudo foci in piecewise analytic planar systems using Minkowski (box) dimension as a quantitative proxy for cyclicity. By extending the analytic focus framework to piecewise analytic settings, it classifies pseudo foci into FF, PP, and mixed types and shows that the order k of a pseudo focus determined by the first return map P(x)=x+\alpha_k x^k+o(x^k) corresponds directly to distinct fractal signatures of nearby spirals. It derives explicit dimension formulas, notably for PP types where \dim_B = 2-\frac{3}{k+1} (k even) and for FF/mixed types where \dim_B = 2-\frac{2}{k} (k>1) or 1 (k=1), along with non-degeneracy conditions. The reconstruction results (FF_real and mixed_real) establish how the fractal properties of V^+ and V^- can be realized by appropriate piecewise constructions, linking local bifurcation invariants to geometric fractal data. These results provide a rigorous fractal-dynamics bridge for bifurcation analysis in piecewise analytic systems and suggest extensions to broader pseudogroups and probabilistic switching scenarios.

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.

Fractal Analysis of Pseudo Foci in Piecewise Analytic Systems

TL;DR

The paper investigates the fractal geometry of spiral trajectories near pseudo foci in piecewise analytic planar systems using Minkowski (box) dimension as a quantitative proxy for cyclicity. By extending the analytic focus framework to piecewise analytic settings, it classifies pseudo foci into FF, PP, and mixed types and shows that the order k of a pseudo focus determined by the first return map P(x)=x+\alpha_k x^k+o(x^k) corresponds directly to distinct fractal signatures of nearby spirals. It derives explicit dimension formulas, notably for PP types where \dim_B = 2-\frac{3}{k+1} (k even) and for FF/mixed types where \dim_B = 2-\frac{2}{k} (k>1) or 1 (k=1), along with non-degeneracy conditions. The reconstruction results (FF_real and mixed_real) establish how the fractal properties of V^+ and V^- can be realized by appropriate piecewise constructions, linking local bifurcation invariants to geometric fractal data. These results provide a rigorous fractal-dynamics bridge for bifurcation analysis in piecewise analytic systems and suggest extensions to broader pseudogroups and probabilistic switching scenarios.

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.

Paper Structure

This paper contains 18 sections, 16 theorems, 58 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Fractal analysis of non-degenerate foci in analytic systems
  3. Pseudo foci in piecewise analytic systems
  4. Cyclicity of pseudo-foci
  5. Main results of the paper
  6. Fractal analysis of pseudo foci
  7. The reconstructibility question
  8. Conventions and notations
  9. First return maps
  10. First return maps of non-degenerate foci
  11. Semi-monodromy maps of non-degenerate foci
  12. Transition maps near parabolic contacts
  13. Fractal analysis of pseudo foci
  14. Separation of halfplanes and the reduction of \ref{['thm:PPfrac']} and \ref{['thm:FFfrac']}
  15. Proofs of \ref{['prop:dim_focus']} and \ref{['prop:dim_para']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\Gamma$ be a part of a trajectory of the system near the origin. Then

Figures (12)

  • Figure 1: The $\delta$-neighbourhood of a spiral trajectory in \ref{['tm:focus_dim']}, thesis_Crnkovic
  • Figure 2: A pseudo focus of the FF type, thesis_Crnkovic
  • Figure 3: Parabolic contact with $\{y = 0\}$, thesis_Crnkovic
  • Figure 4: A pseudo focus of the PP type, thesis_Crnkovic
  • Figure 5: A pseudo focus of FP type, thesis_Crnkovic
  • ...and 7 more figures

Theorems & Definitions (31)

  • Definition 1: Minkowski dimension, tricot95
  • Theorem 1: Žubrinić, Županović, 2005, zz05
  • Theorem 2: The flow-sector theorem, zz08
  • Theorem 3: Fractal analysis of line diffeomorphisms, ezz07
  • Theorem 4: Fractal properties of pseudo foci of PP type, thesis_Crnkovic
  • Theorem 5: Fractal properties of pseudo foci of FF and mixed type, thesis_Crnkovic
  • Theorem 6: Realization of FF type pseudo foci, thesis_Crnkovic
  • Theorem 7: Realization of mixed type pseudo foci, thesis_Crnkovic
  • Remark 1
  • Proposition 1
  • ...and 21 more