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 & Continuous Dynamical Systems - B, 2016, 21 (10) : 3301-3314. doi: 10.3934/dcdsb.2016098
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

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

[1]

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

[2]

Zemer Kosloff, Terry Soo. The orbital equivalence of Bernoulli actions and their Sinai factors. Journal of Modern Dynamics, 2021, 17: 145-182. doi: 10.3934/jmd.2021005

[3]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017

[4]

Cheng-Kai Hu, Fung-Bao Liu, Hong-Ming Chen, Cheng-Feng Hu. Network data envelopment analysis with fuzzy non-discretionary factors. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1795-1807. doi: 10.3934/jimo.2020046

[5]

Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025

[6]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[7]

Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021026

[8]

Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021019

[9]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206

[10]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[11]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[12]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021039

[13]

Hao Li, Honglin Chen, Matt Haberland, Andrea L. Bertozzi, P. Jeffrey Brantingham. PDEs on graphs for semi-supervised learning applied to first-person activity recognition in body-worn video. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021039

[14]

Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069

[15]

Mehmet Duran Toksari, Emel Kizilkaya Aydogan, Berrin Atalay, Saziye Sari. Some scheduling problems with sum of logarithm processing times based learning effect and exponential past sequence dependent delivery times. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021044

[16]

Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021058

[17]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[18]

Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089

[19]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023

[20]

Anastasiia Panchuk, Frank Westerhoff. Speculative behavior and chaotic asset price dynamics: On the emergence of a bandcount accretion bifurcation structure. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021117

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]