Creation of hidden $ n $-scroll Lorenz-like attractors

Jun Pan; Haijun Wang; Feiyu Hu; Jun Pan; Haijun Wang; Feiyu Hu

doi:10.3934/era.2025188

Electronic Research Archive

2025, Volume 33, Issue 7: 4167-4183. doi: 10.3934/era.2025188

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Research article

Creation of hidden $ n $-scroll Lorenz-like attractors

Jun Pan ^{1
,
,},
Haijun Wang ^{2
,
,},
Feiyu Hu ³

1.
School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
2.
School of Electronic and Information Engineering (School of Big Data Science), Taizhou University, Taizhou 318000, China
3.
College of Sustainability and Tourism, Ritsumeikan Asia Pacific University, Beppu 874-8577, Japan

Received: 06 April 2025 Revised: 28 June 2025 Accepted: 01 July 2025 Published: 10 July 2025

Compared with the recently reported hidden two-scroll Lorenz-like attractors in symmetric quadratic and sub-quadratic Lorenz-like dynamical systems, little seems to be concerned with the generation of hidden $ n $-scroll ($ n\in\mathbb{N} $) attractors as far as one knows, especially the one whose number of scrolls equals the one of equilibria. To achieve this target, we first constructed a new asymmetric quadratic Lorenz-like analogue and seized hidden single-scroll Lorenz-like attractors, which were also created through the collapse of pseudo singularly degenerate heteroclinic cycles. Then, utilizing the fractal process or rotation symmetry, the proposed system may exhibit hidden $ n $-scroll Lorenz-like attractors coexisting with $ n $ stable equilibria, and two examples of hidden two/three-scroll Lorenz-like attractors coexisting with two/three stable equilibria were given. In addition, the existence of a single heteroclinic orbit was proved with the help of suitable Lyapunov functions. The obtained results may not only generalize the second part of Hilbert's 16th problem (i.e., the degree may determine the geometrical structure of strange attractors), but also provide reference for practical application.
Citation: Jun Pan, Haijun Wang, Feiyu Hu. Creation of hidden $ n $-scroll Lorenz-like attractors[J]. Electronic Research Archive, 2025, 33(7): 4167-4183. doi: 10.3934/era.2025188

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Abstract

Compared with the recently reported hidden two-scroll Lorenz-like attractors in symmetric quadratic and sub-quadratic Lorenz-like dynamical systems, little seems to be concerned with the generation of hidden $ n $-scroll ($ n\in\mathbb{N} $) attractors as far as one knows, especially the one whose number of scrolls equals the one of equilibria. To achieve this target, we first constructed a new asymmetric quadratic Lorenz-like analogue and seized hidden single-scroll Lorenz-like attractors, which were also created through the collapse of pseudo singularly degenerate heteroclinic cycles. Then, utilizing the fractal process or rotation symmetry, the proposed system may exhibit hidden $ n $-scroll Lorenz-like attractors coexisting with $ n $ stable equilibria, and two examples of hidden two/three-scroll Lorenz-like attractors coexisting with two/three stable equilibria were given. In addition, the existence of a single heteroclinic orbit was proved with the help of suitable Lyapunov functions. The obtained results may not only generalize the second part of Hilbert's 16th problem (i.e., the degree may determine the geometrical structure of strange attractors), but also provide reference for practical application.

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