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Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications
Fractal dimensionality is accepted as a measure of complexity for systems that cannot be described by integer dimensions. However, fractal control mechanisms, physical implications, and relations to nonlinear dynamics have not yet been fully clarified. Herein we explore these issues in a spacetime u...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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Nature Publishing Group UK
2018
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6037749/ https://www.ncbi.nlm.nih.gov/pubmed/29985429 http://dx.doi.org/10.1038/s41598-018-28669-3 |
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author | He, Zonglu |
author_facet | He, Zonglu |
author_sort | He, Zonglu |
collection | PubMed |
description | Fractal dimensionality is accepted as a measure of complexity for systems that cannot be described by integer dimensions. However, fractal control mechanisms, physical implications, and relations to nonlinear dynamics have not yet been fully clarified. Herein we explore these issues in a spacetime using a nonlinear integrated model derived by applying Newton’s second law into self-regulating systems. We discover that (i) a stochastic stable fixed point exhibits self-similarity and long-term memory, while a deterministic stable fixed point usually only exhibits self-similarity, if our observation scale is large enough; (ii) stochastic/deterministic period cycles and chaos only exhibit long-term memory, but also self-similarity for even restorative delays; (iii) fractal level of a stable fixed point is controlled primarily by the wave indicators that reflect the relative strength of extrinsic to intrinsic forces: a larger absolute slope (smaller amplitude) indicator leads to higher positive dependence (self-similarity), and a relatively large amplitude indicator or an even restorative delay could make the dependence oscillate; and (iv) fractal levels of period cycles and chaos rely on the intrinsic resistance, restoration, and regulative delays. Our findings suggest that fractals of self-regulating systems can be measured by integer dimensions. |
format | Online Article Text |
id | pubmed-6037749 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2018 |
publisher | Nature Publishing Group UK |
record_format | MEDLINE/PubMed |
spelling | pubmed-60377492018-07-12 Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications He, Zonglu Sci Rep Article Fractal dimensionality is accepted as a measure of complexity for systems that cannot be described by integer dimensions. However, fractal control mechanisms, physical implications, and relations to nonlinear dynamics have not yet been fully clarified. Herein we explore these issues in a spacetime using a nonlinear integrated model derived by applying Newton’s second law into self-regulating systems. We discover that (i) a stochastic stable fixed point exhibits self-similarity and long-term memory, while a deterministic stable fixed point usually only exhibits self-similarity, if our observation scale is large enough; (ii) stochastic/deterministic period cycles and chaos only exhibit long-term memory, but also self-similarity for even restorative delays; (iii) fractal level of a stable fixed point is controlled primarily by the wave indicators that reflect the relative strength of extrinsic to intrinsic forces: a larger absolute slope (smaller amplitude) indicator leads to higher positive dependence (self-similarity), and a relatively large amplitude indicator or an even restorative delay could make the dependence oscillate; and (iv) fractal levels of period cycles and chaos rely on the intrinsic resistance, restoration, and regulative delays. Our findings suggest that fractals of self-regulating systems can be measured by integer dimensions. Nature Publishing Group UK 2018-07-09 /pmc/articles/PMC6037749/ /pubmed/29985429 http://dx.doi.org/10.1038/s41598-018-28669-3 Text en © The Author(s) 2018 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/. |
spellingShingle | Article He, Zonglu Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
title | Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
title_full | Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
title_fullStr | Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
title_full_unstemmed | Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
title_short | Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
title_sort | integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6037749/ https://www.ncbi.nlm.nih.gov/pubmed/29985429 http://dx.doi.org/10.1038/s41598-018-28669-3 |
work_keys_str_mv | AT hezonglu integerdimensionalfractalsofnonlineardynamicscontrolmechanismsandphysicalimplications |