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. 2022 Oct 21;12:17629. doi: 10.1038/s41598-022-22526-0

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© The Author(s) 2022

Open AccessThis 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 licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence 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 licence, visit http://creativecommons.org/licenses/by/4.0/.

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Dynamical phase transitions driven by the F(r) evolution. As the fitting function s(I) changes with the external current, the cubic function slides through different configurations from (A) to (G). The black and red points indicate the radius of the stable and unstable limit cycle respectively. Configurations from (A) to (D) show the birth of a pair of limit cycles and the unstable limit cycle swallows the stable fixed point at the origin subsequently. Compared with the previous classification scheme, it belongs to the subcritical Hopf transition. Configurations from (E) to (G) illustrate the process of a stable limit cycle shrinks to an unstable fixed point, leading to a stable fixed point subsequently. This dynamical transition belongs to the supercritical Hopf transition.