Abstract.
In this article, the complex dynamics of a modified dynamo is studied in detail. A modified disc dynamo is a self-existing disc dynamo that is supposed to be the cause of the magnetic field of Earth, Sun and stars. The stability analysis and chaoticity of the attractor of the modified disc dynamo is discussed in detail. More precisely, using the Lyapunov coefficient, it is proved that the modified disc dynamo system has a subcritical Hopf bifurcation for a specific set of parameters. To strengthen the analytical results, numerical continuation is used to investigate subcritical criteria by computing a Hopf bifurcation diagram. In the process of numerical continuation, some interesting features of the modified disc dynamo are revealed. Subcritical Hopf bifurcation occurs at two points due to the symmetry of equilibrium points.
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References
S.H. Strogatz, Nonlinear Dynamics and Chaos: With Application to Physics, Biology, Chemistry, and Engineering (Westview, New York, 2000)
X.F. Li, Y.D. Chu, J.G. Zhang, Y.X. Chang, Chaos, Solitons Fractals 49, 2360 (2009)
S.T. Kingni, S. Jafari, H. Simo, P. Woafo, Eur. Phys. J. Plus 129, 76 (2014)
K.T. Alligood, T.D. Sauer, J.A. Yorke, An Introduction to Dynamical Systems (Springer Verlag, New York, 1997)
X. Wang, Chaos, Solitons Fractals 42, 2208 (2009)
Q. Gao, J.H. Ma, Nonlinear Dyn. 58, 209 (2009)
Mauro Bologna, Eur. Phys. J. Plus 131, 386 (2016)
Z. Zhao, L. Yang, L. Chen, Nonlinear Dyn. 63, 521 (2010)
B.Z. Yue, M. Aqeel, Chin. Sci. Bull. 58, 1655 (2013)
K.A. Robbins, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 82 (1977)
K.A. Robbins, PhD Thesis, Massachusetts Institute of Technology (1975)
H.W. Li, M. Wang, Nonlinear Dyn. 71, 235 (2012)
K.J. Zhuang, J. Comput. Nonlinear Dyn. 8, 014501 (2012)
Z.Y. Yan, Chaos, Solitons Fractals 31, 1135 (2007)
Y.G. Yu, S.C. Zhang, Chaos, Solitons Fractals 21, 1215 (2004)
M. Sun, L.X. Tian, J. Yin, Int. J. Nonlinear Sci. 1, 49 (2006)
M. Aqeel, S. Ahmad, Nonlinear Dyn. 84, 755 (2016)
E.C. Bullard, Proc. Cambridge Philos. Soc. 51, 744 (1955)
W.V.R. Malkus, EOS, Trans. Am. Geophys. Union 53, 617 (1972)
J.E. Marsden, M. McCracken, The Hopf Bifurcation and its Application (Springer, New York, 1976)
Y.A. Kuznetsov, Elements of Applied Bifurcation Theory, second edition (Springer-Verlag, New York, 1998)
L.M. Pecora, T.L. Carrol, Phys. Rev. Lett. 64, 821 (1990)
R. Seydel, Practical Bifurcation and Stability Analysis from Equilibrium to Chaos (Springer, New York, 1994)
E.J. Doedel, H.B. Keller, J.P. Kernevez, Int. J. Bifurc. Chaos 1, 493 (1991)
W.J. Beyn, A. Champneys, E.J. Doedel, Y.A. Kuznetsov, W. Govaerts, B. Sandstede, Numerical continuation and computation of normal forms, in Handbook of Dynamical Systems, edited by B. Fiedler (Elsevier, Amsterdam, 2001)
M.J. Friedman, E.J. Doel, SIAM J. Numer. Anal. 28, 789 (1991)
T.S. Parker, L.O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer, Berlin, 1989)
B. Ermentrout, Simulating Analyzing and Animating Dynamical Systems (SIAM, Pennsylvania, 2002)
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Aqeel, M., Masood, H., Azam, A. et al. Complex dynamics in a modified disc dynamo: A nonlinear approach. Eur. Phys. J. Plus 132, 282 (2017). https://doi.org/10.1140/epjp/i2017-11552-3
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DOI: https://doi.org/10.1140/epjp/i2017-11552-3