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doi: 10.3934/dcdsb.2021315
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New insights to the Hide-Skeldon-Acheson dynamo

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

Received  June 2021 Revised  November 2021 Early access January 2022

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.

Citation: Ximing Li. New insights to the Hide-Skeldon-Acheson dynamo. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021315
References:
[1]

M.-F. Danca, Hidden transient chaotic attractors of Rabinovich-Fabrikant system, Nonlinear Dyn., 86 (2016), 1263-1270.  doi: 10.1007/s11071-016-2962-3.

[2]

D. DudkowskiS. JafariT. KapitaniakN. V. KuznetsovG. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Phys. Rep., 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002.

[3] B. D. HassardN. D. Kazarinoff and Y.-H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge-New York, 1981. 
[4]

R. HideA. C. Skeldon and D. J. Acheson, A study of two novel self-exciting single-disk homopolar dynamos: Theory, Proc. R. Soc. London Ser. A, 452 (1996), 1369-1395. 

[5]

N. V. Kuznetsov, Theory of hidden oscillations and stability of control systems, J. Comput. Syst. Sci. Int., 59 (2020), 647-668.  doi: 10.1134/S1064230720050093.

[6]

N. V. KuznetsovG. A. LeonovM. V. Yuldashev and R. V. Yuldashev, Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE, Commun. Nonlinear Sci. Numer. Simulat., 51 (2017), 39-49. 

[7]

A. Mahdi and C. Valls, Integrability of the Hide-Skeldon-Acheson dynamo, Bull. Sci. Math., 138 (2014), 470-482.  doi: 10.1016/j.bulsci.2013.04.001.

[8]

I. M. Moroz, The Hide, Skeldon, Acheson dynamo revisited, Proc. R. Soc. A, 463 (2007), 113-130.  doi: 10.1098/rspa.2006.1758.

[9]

I. M. Moroz, Template analysis of a Faraday disk dynamo, Eur. Phys. J. Special Topics, 165 (2008), 211-220.  doi: 10.1140/epjst/e2008-00864-x.

[10]

A. C. Skeldon and I. M. Moroz, On a codimension-three bifurcation arising in a simple dynamo model, Phys. D, 117 (1998), 117-127.  doi: 10.1016/S0167-2789(97)00316-3.

[11]

N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov and L. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurc. Chaos, 27 (2017), 1730038, 18 pp. doi: 10.1142/S0218127417300385.

[12]

Z. Wei, I. Moroz, J. C. Sprott, Z. Wang and W. Zhang, Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo, Int. J. Bifurc. Chaos, 27 (2017), 1730008, 19 pp. doi: 10.1142/S0218127417300087.

[13]

Z. Wei, W. Zhang, Z. Wang and M. Yao, Hidden attractors and dynamical behaviors in an extended Rikitake system, Int. J. Bifurc. Chaos, 25 (2015), 1550028, 11 pp. doi: 10.1142/S0218127415500285.

[14]

S. Wiggins and P. Holmes, Periodic orbits in slowly varying oscillators, SIAM J. Math. Anal., 18 (1987), 592-611.  doi: 10.1137/0518046.

show all references

References:
[1]

M.-F. Danca, Hidden transient chaotic attractors of Rabinovich-Fabrikant system, Nonlinear Dyn., 86 (2016), 1263-1270.  doi: 10.1007/s11071-016-2962-3.

[2]

D. DudkowskiS. JafariT. KapitaniakN. V. KuznetsovG. A. Leonov and A. Prasad, Hidden attractors in dynamical systems, Phys. Rep., 637 (2016), 1-50.  doi: 10.1016/j.physrep.2016.05.002.

[3] B. D. HassardN. D. Kazarinoff and Y.-H. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, Cambridge-New York, 1981. 
[4]

R. HideA. C. Skeldon and D. J. Acheson, A study of two novel self-exciting single-disk homopolar dynamos: Theory, Proc. R. Soc. London Ser. A, 452 (1996), 1369-1395. 

[5]

N. V. Kuznetsov, Theory of hidden oscillations and stability of control systems, J. Comput. Syst. Sci. Int., 59 (2020), 647-668.  doi: 10.1134/S1064230720050093.

[6]

N. V. KuznetsovG. A. LeonovM. V. Yuldashev and R. V. Yuldashev, Hidden attractors in dynamical models of phase-locked loop circuits: Limitations of simulation in MATLAB and SPICE, Commun. Nonlinear Sci. Numer. Simulat., 51 (2017), 39-49. 

[7]

A. Mahdi and C. Valls, Integrability of the Hide-Skeldon-Acheson dynamo, Bull. Sci. Math., 138 (2014), 470-482.  doi: 10.1016/j.bulsci.2013.04.001.

[8]

I. M. Moroz, The Hide, Skeldon, Acheson dynamo revisited, Proc. R. Soc. A, 463 (2007), 113-130.  doi: 10.1098/rspa.2006.1758.

[9]

I. M. Moroz, Template analysis of a Faraday disk dynamo, Eur. Phys. J. Special Topics, 165 (2008), 211-220.  doi: 10.1140/epjst/e2008-00864-x.

[10]

A. C. Skeldon and I. M. Moroz, On a codimension-three bifurcation arising in a simple dynamo model, Phys. D, 117 (1998), 117-127.  doi: 10.1016/S0167-2789(97)00316-3.

[11]

N. V. Stankevich, N. V. Kuznetsov, G. A. Leonov and L. Chua, Scenario of the birth of hidden attractors in the Chua circuit, Int. J. Bifurc. Chaos, 27 (2017), 1730038, 18 pp. doi: 10.1142/S0218127417300385.

[12]

Z. Wei, I. Moroz, J. C. Sprott, Z. Wang and W. Zhang, Detecting hidden chaotic regions and complex dynamics in the self-exciting homopolar disc dynamo, Int. J. Bifurc. Chaos, 27 (2017), 1730008, 19 pp. doi: 10.1142/S0218127417300087.

[13]

Z. Wei, W. Zhang, Z. Wang and M. Yao, Hidden attractors and dynamical behaviors in an extended Rikitake system, Int. J. Bifurc. Chaos, 25 (2015), 1550028, 11 pp. doi: 10.1142/S0218127415500285.

[14]

S. Wiggins and P. Holmes, Periodic orbits in slowly varying oscillators, SIAM J. Math. Anal., 18 (1987), 592-611.  doi: 10.1137/0518046.

Figure 1.  Waveform diagram and phase diagram of system (1.1) for $ \alpha = 1.9 $
Figure 2.  Waveform diagram and phase diagram of system (1.1) for $ \alpha = 2.1 $
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)
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)
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)
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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