New insights to the Hide-Skeldon-Acheson dynamo

Ximing Li

doi:10.3934/dcdsb.2021315

Article Contents

2022, Volume 27, Issue 11: 6257-6267. Doi: 10.3934/dcdsb.2021315

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New insights to the Hide-Skeldon-Acheson dynamo

Ximing Li

Department of Mathematical Sciences and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Current affiliation: Qiuzhen College, Tsinghua University, Beijing, China

Received: May 31, 2021

Revised: October 31, 2021

Early access: January 2022

Published: October 2022

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Abstract

The present work is devoted to giving new insights into the Hide-Skeldon-Acheson (HSA) dynamo. The paper discusses the locally and globally asymptotical stability of the equilibrium. All orbits of the system are proved to be bounded. The existence of periodic orbits is proved by the generalized Melnikov method. The paper proves rigorously that Hopf bifurcation occurs and gives the formulae to determine the direction, stability and period of bifurcating periodic solutions. Finally, the paper investigates the coexistence of three types of attractors: equilibria, hidden periodic attractors and hidden chaotic attractors.Note: Author's current affiliation "Qiuzhen College, Tsinghua University, Beijing, China" is added.

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Mathematics Subject Classification: Primary: 34D20, 34K18; Secondary: 34D45.

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Figure 1. Waveform diagram and phase diagram of system (1.1) for $ \alpha = 1.9 $

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Figure 2. Waveform diagram and phase diagram of system (1.1) for $ \alpha = 2.1 $

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Figure 3. A hidden chaotic attractor of system (1.1) with no equilibria for $ (\alpha, k, b, \beta) = (19, 0, 0, 2) $ and initial values (0.1190, 0.4984, 0.9597)

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Figure 4. The largest Lyapunov exponents (LLE) and bifurcation diagrams of system (1.1) versus $ \alpha \in [19, 30] $ and two sets of initial points: initial values (0.1190, 0.4984, 0.9597) (blue); initial values (0, 0.1, 0.1)(green)

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Figure 5. (a) A hidden periodic attractor of system (1.1); (b) time series of $ x $ for $ (\alpha, k, b, \beta) = (3.995, 1, 3, 9.1) $ and initial values (0.1656, 6.0198, 0.2630)

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Figure 6. Parameters $ (\alpha, k, b, \beta) = (15, 0, 20, -0.1) $ and initial values (-0.001, 1.01, 0); (a) a hidden chaotic attractor and two stable equilibria of system (1.1); (b) Poincaré map on the $ y $-$ z $ plane

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