fractal, 2024-2025

Computation of bifurcation fractals for 2D symplectic mappings

Watch supplementary videos on YouTube

Cite as:

@article{fractal2025,
title = {Isochronous and period-doubling diagrams for symplectic maps of the plane},
journal = {Chaos, Solitons & Fractals},
volume = {198},
pages = {116513},
year = {2025},
issn = {0960-0779},
doi = {https://doi.org/10.1016/j.chaos.2025.116513},
url = {https://www.sciencedirect.com/science/article/pii/S0960077925005260},
author = {T. Zolkin and S. Nagaitsev and I. Morozov and S. Kladov and Y.-K. Kim},
keywords = {Chaos, Integrability, Perturbation theory, Stability}
}

For a quadratic mapping:

$$ \begin{aligned} & q \to p \\ & p \to -q + w p + p^2 \end{aligned} $$

two symmetry lines, $p=q$ and $p=1/2(w q + q^2)$ cover full 2D plane. Fractals bellow are computed in $(w, q)$ space with $w$ in [-3.00, 2.00] and $q$ in [-1.25, 1.75] using the second symmetry line. Color corresponds to different indicators.

• Phase space coverage by symmetry lines

• REM

• Frequency

• Frequency & mode locking (tongues)

• FMA

• GALI