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. 2017 Apr 24;7:1087. doi: 10.1038/s41598-017-01083-x

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© The Author(s) 2017

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Strong chaos (red, for ρ < ρ _p ≈ 180.72), partially predictable chaos (green, for ρ _p < ρ < ρ _C ≈ 180.96) and regular flow (blue, for ρ _C < ρ), for the Lorenz system (1). Shown are (from top to bottom), z _n from the Poincaré section x = 15 (for a window of z _n), the maximal Lyapunov exponent λ _m, the cross-distance scaling exponent ν, see Eq. (2), and the cross-correlation C ₁₂(t = 200), as defined by Eq. (3). PD1 and PD2 denote examples of period doubling bifurcations and SSB a bifurcation spontaneously breaking the symmetry (x, y, z) ↔ (−x, −y, z) of (1).