[Paper Review] Box dimension of unit-time map near nilpotent singularity of planar vector field

Lana Horvat Dmitrović, Vesna Županović|arXiv (Cornell University)|May 24, 2012

Advanced Differential Equations and Dynamical Systems14 references3 citations

TL;DR

This paper investigates the box dimension of discrete orbits generated by the unit-time map near nilpotent nonmonodromic singularities in planar vector fields, using normal forms and asymptotic analysis. It derives exact box dimension values—specifically, $\dim_B S(v_1) = 1 - \frac{1}{2k}$—for orbits on separatrices at finite and infinite singular points, linking them to the order $k$ of weak saddle singularities and dual Lyapunov constants.

ABSTRACT

The connection between discrete and continuous dynamical systems through the unit-time map has shown a significant role in bifurcation theory. Recently, it has also been used in fractal analysis of bifurcations. We study fractal properties of the unit-time map near nilpotent nonmonodromic singularities of planar vector fields using normal forms. We are interested in nilpotent singularities because they are nonhyperbolic, and we know that near nonhyperbolic singularities the box dimension is nontrivial. We study discrete orbits generated by the unit-time map, on the separatrices at the bifurcation point, and get results for the box dimensions of these orbits. Box dimension results will be illustrated in details using the examples of Bogdanov-Takens bifurcation and bifurcation of the nilpotent saddle. The study is also been extended to the appropriate singular points at infinity of normal form for the nilpotent singularity. Moreover, we study the unit-time map of the normal form for a saddle, near singular points at infinity.

Motivation & Objective

• To analyze fractal properties of discrete orbits generated by the unit-time map near nilpotent nonmonodromic singularities in planar vector fields.
• To extend the application of box dimension to nonhyperbolic, nilpotent singularities where traditional hyperbolicity fails.
• To investigate the behavior of box dimension at singular points at infinity using Poincaré compactification and normal forms.
• To establish a connection between the box dimension of orbits and the order $k$ of weak saddle singularities via dual Lyapunov constants.
• To demonstrate that box dimension is sensitive to bifurcation structure in nonmonodromic cases, unlike Hausdorff dimension.

Proposed method

• Utilizes normal forms to simplify the vector field near nilpotent singularities, particularly for nonmonodromic cases.
• Applies the unit-time map (time-1 map) to connect continuous dynamics to discrete orbits, enabling fractal analysis.
• Employs asymptotic expansions and Picard iteration to approximate the unit-time map near singular points at infinity.
• Transforms the system into hyperbolic coordinates to compare trajectories to power-law spirals $r = \varphi^{-\alpha}$, for which box dimension is known.
• Uses Tricot’s formula for box dimension of nonrectifiable spirals: $\dim_B \Gamma = \frac{2}{1 + \alpha}$, adapted to hyperbolic trajectories.
• Analyzes the $v$-axis orbits in the compactified system to compute box dimension via the asymptotic behavior $v - v^{2k}$ of the unit-time map.

Experimental results

Research questions

• RQ1 What is the box dimension of the unit-time map orbit on the separatrix near a nilpotent nonmonodromic singularity?
• RQ2 How does the box dimension at infinity of a weak saddle singularity depend on the order $k$ of the first nonvanishing dual Lyapunov constant?
• RQ3 Can the box dimension of discrete orbits distinguish between different bifurcation scenarios in nonhyperbolic planar systems?
• RQ4 How does the unit-time map behave near singular points at infinity, and what fractal properties emerge?
• RQ5 Is there a structural link between the box dimension of orbits and the cyclicity or stability properties of the underlying vector field?

Key findings

• The box dimension of the unit-time map orbit on the $v$-axis near a singularity at infinity is $\dim_B S(v_1) = 1 - \frac{1}{2k}$, where $k$ is the order of the weak saddle.
• For $k=1$, the box dimension is $\frac{1}{2}$; for $k=2$, it increases to $\frac{3}{4}$, showing a direct dependence on the weak saddle order.
• The unit-time map near a weak saddle at infinity is asymptotically comparable to $v - v^{2k}$, which determines the fractal scaling of the orbit.
• The box dimension is nontrivial and sensitive to the dual Lyapunov constants, unlike in hyperbolic cases where it is trivial.
• The results confirm that box dimension can detect bifurcation structure in nonmonodromic, nilpotent singularities, even at infinity.
• The method successfully extends fractal analysis to singular points at infinity via Poincaré compactification and coordinate transformation.