Citation :

Neer Dubey, "Box-Counting Fractal Dimension of the Lorenz Attractor Across the Route to Chaos: A Bootstrapped, Multi-Estimator Analysis," International Journal of Applied Physics, vol. 13, no. 3, pp. 1-5, 2026. Crossref, https://doi.org/10.14445/23500301/IJAP-V13I3P101

Abstract

When describing chaotic attractors, the fractal dimension is a quantitative measure of their geometric complexity, in addition to qualitative visualisation. The box-counting dimension of the Lorenz attractor is estimated for the range of the bifurcation parameter, ρ ∈ [0, 30], with the following pipeline: occupied-box counting on a fixed absolute grid ladder, 128-replicate grid-origin bootstrap for confidence interval, and a two-segment piecewise-linear fit for automatic selection of the scaling band. In the steady regime (ρ ≲ 22), the box counting dimension is nearly zero, and it increases rapidly to about 1.8 near the onset of chaos (ρ ≈ 23), and thus the box counting dimension serves as a numerical measure of the onset of chaos. At the classical parameter set (σ = 10, ρ = 28, β = 8/3), the box-counting dimension is 1.86 (95% confidence interval 1.84–1.87) and increases monotonically towards the theoretical value with sample size. The estimate is compared with the Grassberger–Procaccia correlation dimension (2.03) and the Kaplan–Yorke dimension based on the Lyapunov spectrum (2.06). It is found that the capacity dimension is systematically underestimated by box-counting when computed on finite trajectories and that the dimension of a strange attractor is sensitively dependent on the estimator used: a walking-divider dimension gives a dimension around unity, consistent with a trajectory itself being regarded as a curve, not the attractor set.

Keywords

Box-counting dimension, Chaos, Fractal dimension, Kaplan–Yorke dimension, Lorenz attractor.

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