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 & 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.  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.  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.  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.  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.  doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[7]

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

[8]

Y. Liu, S. Pang and D. Chen, An unusual chaotic system and its control,, Math. Comput. Modelling, 57 (2013), 2473.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1016/0375-9601(81)90274-7.  Google Scholar

[7]

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

[8]

Y. Liu, S. Pang and D. Chen, An unusual chaotic system and its control,, Math. Comput. Modelling, 57 (2013), 2473.  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.  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.  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.  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.  doi: 10.1016/j.physleta.2011.10.040.  Google Scholar

[1]

Jianghong Bao, Dandan Chen, Yongjian Liu, Hongbo Deng. Coexisting hidden attractors in a 5D segmented disc dynamo with three types of equilibria. Discrete & 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]

Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543
[4]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[5]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[6]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[7]

Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045
[8]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977
[9]

Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020121
[10]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[11]

Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135
[12]

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 & Continuous Dynamical Systems - A, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079
[13]

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
[14]

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
[15]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[16]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[17]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[18]

Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531
[19]

Zhaosheng Feng, Guangyue Gao, Jing Cui. Duffing--van der Pol--type oscillator system and its first integrals. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1377-1391. doi: 10.3934/cpaa.2011.10.1377
[20]

Rehana Naz. On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020170

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences