American Institute of Mathematical Sciences

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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
References:
[1]

E. C. Bullard, The stability of a homopolar dynamo, Proc. Camb. Phil. Soc., 51 (1955), 744-760. doi: 10.1017/S0305004100030814.  Google Scholar

[2]

C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229 (2007), 63-117. doi: 10.2140/pjm.2007.229.63.  Google Scholar

[3]

B. Hassard, N. Kazarinoff and Y. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, 1981.  Google Scholar

[4]

R. Hide, How to locate the electrically-conducting fluid core a planet from external magnetic observations, Nature, 271 (1978), 640-641. doi: 10.1038/271640a0.  Google Scholar

[5]

G. Jiang and Q. Lu, Impulsive state feedback control of a predator-prey model, J. Comput. Appl. Math., 200 (2007), 193-207. doi: 10.1016/j.cam.2005.12.013.  Google Scholar

[6]

E. Knobloch, Chaos in the segmented disc dynamo, Phys. Lett. A, 82 (1981), 439-440. doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[7]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998.  Google Scholar

[8]

Y. Liu, S. Pang and D. Chen, An unusual chaotic system and its control, Math. Comput. Modelling, 57 (2013), 2473-2493. doi: 10.1016/j.mcm.2012.12.006.  Google Scholar

[9]

J. Llibre and X. Zhang, Darboux theory of integrability in image taking into account the multiplicity, J. Differ. Equ., 246 (2009), 541-551. doi: 10.1016/j.jde.2008.07.020.  Google Scholar

[10]

H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166. doi: 10.1080/03091927908244536.  Google Scholar

[11]

H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, 1978. Google Scholar

[12]

C. Valls, Darboux integrability of a nonlinear financial system, Appl. Math. Comput., 218 (2011), 3297-3302. doi: 10.1016/j.amc.2011.08.069.  Google Scholar

[13]

Z. Wei, Dynamical behaviors of a chaotic system with no equilibria, Phys. Lett. A, 376 (2011), 102-108. doi: 10.1016/j.physleta.2011.10.040.  Google Scholar

show all references

References:
[1]

E. C. Bullard, The stability of a homopolar dynamo, Proc. Camb. Phil. Soc., 51 (1955), 744-760. doi: 10.1017/S0305004100030814.  Google Scholar

[2]

C. Christopher, J. Llibre and J. V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math., 229 (2007), 63-117. doi: 10.2140/pjm.2007.229.63.  Google Scholar

[3]

B. Hassard, N. Kazarinoff and Y. Wan, Theory and Application of Hopf Bifurcation, Cambridge University Press, 1981.  Google Scholar

[4]

R. Hide, How to locate the electrically-conducting fluid core a planet from external magnetic observations, Nature, 271 (1978), 640-641. doi: 10.1038/271640a0.  Google Scholar

[5]

G. Jiang and Q. Lu, Impulsive state feedback control of a predator-prey model, J. Comput. Appl. Math., 200 (2007), 193-207. doi: 10.1016/j.cam.2005.12.013.  Google Scholar

[6]

E. Knobloch, Chaos in the segmented disc dynamo, Phys. Lett. A, 82 (1981), 439-440. doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[7]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1998.  Google Scholar

[8]

Y. Liu, S. Pang and D. Chen, An unusual chaotic system and its control, Math. Comput. Modelling, 57 (2013), 2473-2493. doi: 10.1016/j.mcm.2012.12.006.  Google Scholar

[9]

J. Llibre and X. Zhang, Darboux theory of integrability in image taking into account the multiplicity, J. Differ. Equ., 246 (2009), 541-551. doi: 10.1016/j.jde.2008.07.020.  Google Scholar

[10]

H. K. Moffatt, A self-consistent treatment of simple dynamo systems, Geophys. Astrophys. Fluid Dyn., 14 (1979), 147-166. doi: 10.1080/03091927908244536.  Google Scholar

[11]

H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, 1978. Google Scholar

[12]

C. Valls, Darboux integrability of a nonlinear financial system, Appl. Math. Comput., 218 (2011), 3297-3302. doi: 10.1016/j.amc.2011.08.069.  Google Scholar

[13]

Z. Wei, Dynamical behaviors of a chaotic system with no equilibria, Phys. Lett. A, 376 (2011), 102-108. doi: 10.1016/j.physleta.2011.10.040.  Google Scholar

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