American Institute of Mathematical Sciences

Advanced
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 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.

[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.

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.

[1]

Jianghong Bao, Dandan Chen, Yongjian Liu, Hongbo Deng. Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6053-6069. doi: 10.3934/dcdsb.2019130
[2]

Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451
[3]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001
[4]

Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543
[5]

Zhouchao Wei, Fanrui Wang, Huijuan Li, Wei Zhang. Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5029-5045. doi: 10.3934/dcdsb.2021263
[6]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[7]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[8]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[9]

Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045
[10]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977
[11]

Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2829-2840. doi: 10.3934/dcdss.2020121
[12]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[13]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208
[14]

Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135
[15]

Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079
[16]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[17]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[18]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure and Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[19]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[20]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (181)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy