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December  2016, 21(10): 3301-3314. doi: 10.3934/dcdsb.2016098

Complex dynamics in the segmented disc dynamo

Jianghong Bao 1,

1. 

School of Mathematics, South China University of Technology, Guangzhou, Guangdong, China

Received  June 2015 Revised  July 2016 Published  November 2016

The present work is devoted to giving new insights into the segmented disc dynamo. The integrability of the system is studied. The paper provides its first integrals for the parameter $r=0$. For $r>0$, the system has neither polynomial first integrals nor exponential factors, and it is also further proved not to be Darboux integrable. In addition, by choosing an appropriate bifurcation parameter, the paper proves that Hopf bifurcations occur in the system and presents the formulae for determining the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions.
Keywords: Segmented disc dynamo, Darboux first integrals, Darboux integrability, Hopf bifurcation., exponential factors.
Mathematics Subject Classification: Primary: 37G10, 37J30; Secondary: 34C2.
Citation: Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3301-3314. doi: 10.3934/dcdsb.2016098
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show all references

References:
[1]

Proc. Camb. Phil. Soc., 51 (1955), 744-760. doi: 10.1017/S0305004100030814.  Google Scholar

[2]

Pacific J. Math., 229 (2007), 63-117. doi: 10.2140/pjm.2007.229.63.  Google Scholar

[3]

Cambridge University Press, 1981.  Google Scholar

[4]

Nature, 271 (1978), 640-641. doi: 10.1038/271640a0.  Google Scholar

[5]

J. Comput. Appl. Math., 200 (2007), 193-207. doi: 10.1016/j.cam.2005.12.013.  Google Scholar

[6]

Phys. Lett. A, 82 (1981), 439-440. doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[7]

Springer-Verlag, New York, 1998.  Google Scholar

[8]

Math. Comput. Modelling, 57 (2013), 2473-2493. doi: 10.1016/j.mcm.2012.12.006.  Google Scholar

[9]

J. Differ. Equ., 246 (2009), 541-551. doi: 10.1016/j.jde.2008.07.020.  Google Scholar

[10]

Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166. doi: 10.1080/03091927908244536.  Google Scholar

[11]

Cambridge University Press, 1978. Google Scholar

[12]

Appl. Math. Comput., 218 (2011), 3297-3302. doi: 10.1016/j.amc.2011.08.069.  Google Scholar

[13]

Phys. Lett. A, 376 (2011), 102-108. doi: 10.1016/j.physleta.2011.10.040.  Google Scholar

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