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

Complex dynamics in the segmented disc dynamo

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.
Citation: Jianghong Bao. Complex dynamics in the segmented disc dynamo. Discrete and 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.

[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.

[3]

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

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

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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.

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.

[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.

[3]

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

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

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