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September  2013, 18(7): 1755-1776. doi: 10.3934/dcdsb.2013.18.1755

A Rikitake type system with one control

Tudor Bînzar 1, and  Cristian Lăzureanu 1,

1. 

"Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei nr. 2, 300006 - Timişoara, Romania, Romania

Received  May 2011 Revised  March 2013 Published  May 2013

A Rikitake type system with one control is defined and some of this geometrical and dynamical properties are pointed out.
Keywords: stability analysis, Hamiltonian dynamics, periodic orbits, homoclinic and heteroclinic orbits., Rikitake two-dynamo system.
Mathematics Subject Classification: Primary: 70H05; Secondary: 70H14, 70K4.
Citation: Tudor Bînzar, Cristian Lăzureanu. A Rikitake type system with one control. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1755-1776. doi: 10.3934/dcdsb.2013.18.1755
References:
[1]

V. Arnold, On conditions for non-linear stability of stationary plane curvilinear flows on an ideal fluid, Akad. Nauk. Doklady SSSR, 162 (1965), 975-978.  Google Scholar

[2]

M. Barge, Invariant manifolds and the onset of reversal in the Rikitake two-disk dynamo, SIAM J. Math. Anal., 15 (1984), 514-529. doi: 10.1137/0515039.  Google Scholar

[3]

P. Birtea, M. Puta and R. M. Tudoran, Periodic orbits in the case of a zero eigenvalue, C. R. Acad. Sci. Paris, Ser. I, 344 (2007), 779-784. doi: 10.1016/j.crma.2007.05.003.  Google Scholar

[4]

Y. X. Chang, X. J. Liu and X. F. Li, Chaos and chaos control of the Rikitake two-disk dynamo, J. Liaoning Norm. Univ. Nat. Sci., 29 (2006), 422-426.  Google Scholar

[5]

G. Chen and X. Dong, From chaos to order-Perspectives and methodologies in controlling chaotic nonlinear dynamic systems, Int. J. of Bifurcation and Chaos, 3 (1993), 1363-1409. doi: 10.1142/S0218127493001112.  Google Scholar

[6]

G. Chen and X. Dong, "From Chaos to Order. Methodologies, Perspectives and Applications," World Scientific Pub. Co., Singapore, 1998.  Google Scholar

[7]

A. E. Cook, Two-disc dynamo with viscous friction and time delay, Proc. Camb. Phil. Soc., 71 (1972), 135-153. doi: 10.1017/S0305004100050374.  Google Scholar

[8]

R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[9]

A. Figueiredo, T. M. Rocha Filho and L. Brenig, Algebraic structures and invariant manifolds of differential systems, J. Math. Phys., 39 (1998), 2929-2946. doi: 10.1063/1.532429.  Google Scholar

[10]

Y. Hardy and W.-H. Steeb, The Rikitake two-disc dynamo system and domains with periodic orbits, Internat. J. Theoret. Phys., 38 (1999), 2413-2417. doi: 10.1023/A:1026640221874.  Google Scholar

[11]

R. Hide, Structural instability of the Rikitake disk dynamo, Geophys. Res. Lett., 22 (1995), 1057-1059. doi: 10.1029/95GL00779.  Google Scholar

[12]

D. Holm, J. Marsden, T. Raţiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123 (1985), 1-116. doi: 10.1016/0370-1573(85)90028-6.  Google Scholar

[13]

K. Ito, Chaos in the Rikitake two-disc dynamo system, Earth Planet. Sci. Lett., 51 (1980), 451-456. doi: 10.1016/0012-821X(80)90224-1.  Google Scholar

[14]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Applied Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.  Google Scholar

[15]

D. F. Lawden, "Elliptic Functions and Applications," Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, 1989.  Google Scholar

[16]

J. Llibre and X. Zhang, Invariant algebraic surfaces of the Rikitake system, J. Phys. A, 33 (2000), 7613-7635. doi: 10.1088/0305-4470/33/42/310.  Google Scholar

[17]

A. M. Lyapunov, The general problem of the stability of motion, Translated by A. T. Fuller from douard Davaux's French translation (1907) of the 1892 Russian original, Internat. J. Control, 55 (1992), 521-790. doi: 10.1080/00207179208934253.  Google Scholar

[18]

J. Marsden and T. S. Raţiu, "Introduction to Mechanics and Symmetry," 2nd ed., Text and Appl. Math., 17, Springer, Berlin, 1999.  Google Scholar

[19]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747.  Google Scholar

[20]

M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces," Lecture Notes in Math., 1768, Springer-Verlag, Berlin, 2001.  Google Scholar

[21]

F. Plunian, Ph. Marty and A. Alemany, Chaotic behaviour of the Rikitake dynamo with symmetric mechanical friction and azimuthal currents, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 1835-1842. doi: 10.1098/rspa.1998.0235.  Google Scholar

[22]

M. Puta, "Hamiltonian Mechanical System and Geometric Quantization," Kluwer Academic Publishers, Dordrecht, Boston, London, 1993. doi: 10.1007/978-94-011-1992-4.  Google Scholar

[23]

T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., 54 (1958), 89-105. doi: 10.1017/S0305004100033223.  Google Scholar

[24]

W.-H. Steeb, Continuous symmetries of the Lorenz model and the Rikitake two-disc dynamo system, J. Phys. A: Math. Gen., 15 (1982), L389-L390. doi: 10.1088/0305-4470/15/8/002.  Google Scholar

[25]

R. M. Tudoran, A. Aron and Ş. Nicoară, On a Hamiltonian version of the Rikitake system, SIAM J. Applied Dynamical Systems, 8 (2009), 454-479. doi: 10.1137/080728822.  Google Scholar

[26]

D. L. Turcotte, "Fractals and Chaos in Geology and Geophysics," 2nd ed., Cambridge University Press, Cambridge, UK, 1997.  Google Scholar

[27]

C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309-1326. doi: 10.1017/S030821050000439X.  Google Scholar

[28]

T. Yajima and H. Nagahama, KCC-theory and geometry of the Rikitake system, J. Phys. A, 40 (2007), 2755-2772. doi: 10.1088/1751-8113/40/11/011.  Google Scholar

[29]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57. doi: 10.1007/BF01405263.  Google Scholar

[30]

H. Whitney, Tangents to an analytic variety, Ann. of Math (2), 81 (1965), 496-549. doi: 10.2307/1970400.  Google Scholar

show all references

References:
[1]

V. Arnold, On conditions for non-linear stability of stationary plane curvilinear flows on an ideal fluid, Akad. Nauk. Doklady SSSR, 162 (1965), 975-978.  Google Scholar

[2]

M. Barge, Invariant manifolds and the onset of reversal in the Rikitake two-disk dynamo, SIAM J. Math. Anal., 15 (1984), 514-529. doi: 10.1137/0515039.  Google Scholar

[3]

P. Birtea, M. Puta and R. M. Tudoran, Periodic orbits in the case of a zero eigenvalue, C. R. Acad. Sci. Paris, Ser. I, 344 (2007), 779-784. doi: 10.1016/j.crma.2007.05.003.  Google Scholar

[4]

Y. X. Chang, X. J. Liu and X. F. Li, Chaos and chaos control of the Rikitake two-disk dynamo, J. Liaoning Norm. Univ. Nat. Sci., 29 (2006), 422-426.  Google Scholar

[5]

G. Chen and X. Dong, From chaos to order-Perspectives and methodologies in controlling chaotic nonlinear dynamic systems, Int. J. of Bifurcation and Chaos, 3 (1993), 1363-1409. doi: 10.1142/S0218127493001112.  Google Scholar

[6]

G. Chen and X. Dong, "From Chaos to Order. Methodologies, Perspectives and Applications," World Scientific Pub. Co., Singapore, 1998.  Google Scholar

[7]

A. E. Cook, Two-disc dynamo with viscous friction and time delay, Proc. Camb. Phil. Soc., 71 (1972), 135-153. doi: 10.1017/S0305004100050374.  Google Scholar

[8]

R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser, Basel, 1997. doi: 10.1007/978-3-0348-8891-2.  Google Scholar

[9]

A. Figueiredo, T. M. Rocha Filho and L. Brenig, Algebraic structures and invariant manifolds of differential systems, J. Math. Phys., 39 (1998), 2929-2946. doi: 10.1063/1.532429.  Google Scholar

[10]

Y. Hardy and W.-H. Steeb, The Rikitake two-disc dynamo system and domains with periodic orbits, Internat. J. Theoret. Phys., 38 (1999), 2413-2417. doi: 10.1023/A:1026640221874.  Google Scholar

[11]

R. Hide, Structural instability of the Rikitake disk dynamo, Geophys. Res. Lett., 22 (1995), 1057-1059. doi: 10.1029/95GL00779.  Google Scholar

[12]

D. Holm, J. Marsden, T. Raţiu and A. Weinstein, Nonlinear stability of fluid and plasma equilibria, Physics Reports, 123 (1985), 1-116. doi: 10.1016/0370-1573(85)90028-6.  Google Scholar

[13]

K. Ito, Chaos in the Rikitake two-disc dynamo system, Earth Planet. Sci. Lett., 51 (1980), 451-456. doi: 10.1016/0012-821X(80)90224-1.  Google Scholar

[14]

P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Applied Math., 21 (1968), 467-490. doi: 10.1002/cpa.3160210503.  Google Scholar

[15]

D. F. Lawden, "Elliptic Functions and Applications," Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, 1989.  Google Scholar

[16]

J. Llibre and X. Zhang, Invariant algebraic surfaces of the Rikitake system, J. Phys. A, 33 (2000), 7613-7635. doi: 10.1088/0305-4470/33/42/310.  Google Scholar

[17]

A. M. Lyapunov, The general problem of the stability of motion, Translated by A. T. Fuller from douard Davaux's French translation (1907) of the 1892 Russian original, Internat. J. Control, 55 (1992), 521-790. doi: 10.1080/00207179208934253.  Google Scholar

[18]

J. Marsden and T. S. Raţiu, "Introduction to Mechanics and Symmetry," 2nd ed., Text and Appl. Math., 17, Springer, Berlin, 1999.  Google Scholar

[19]

J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747.  Google Scholar

[20]

M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces," Lecture Notes in Math., 1768, Springer-Verlag, Berlin, 2001.  Google Scholar

[21]

F. Plunian, Ph. Marty and A. Alemany, Chaotic behaviour of the Rikitake dynamo with symmetric mechanical friction and azimuthal currents, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 1835-1842. doi: 10.1098/rspa.1998.0235.  Google Scholar

[22]

M. Puta, "Hamiltonian Mechanical System and Geometric Quantization," Kluwer Academic Publishers, Dordrecht, Boston, London, 1993. doi: 10.1007/978-94-011-1992-4.  Google Scholar

[23]

T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., 54 (1958), 89-105. doi: 10.1017/S0305004100033223.  Google Scholar

[24]

W.-H. Steeb, Continuous symmetries of the Lorenz model and the Rikitake two-disc dynamo system, J. Phys. A: Math. Gen., 15 (1982), L389-L390. doi: 10.1088/0305-4470/15/8/002.  Google Scholar

[25]

R. M. Tudoran, A. Aron and Ş. Nicoară, On a Hamiltonian version of the Rikitake system, SIAM J. Applied Dynamical Systems, 8 (2009), 454-479. doi: 10.1137/080728822.  Google Scholar

[26]

D. L. Turcotte, "Fractals and Chaos in Geology and Geophysics," 2nd ed., Cambridge University Press, Cambridge, UK, 1997.  Google Scholar

[27]

C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309-1326. doi: 10.1017/S030821050000439X.  Google Scholar

[28]

T. Yajima and H. Nagahama, KCC-theory and geometry of the Rikitake system, J. Phys. A, 40 (2007), 2755-2772. doi: 10.1088/1751-8113/40/11/011.  Google Scholar

[29]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57. doi: 10.1007/BF01405263.  Google Scholar

[30]

H. Whitney, Tangents to an analytic variety, Ann. of Math (2), 81 (1965), 496-549. doi: 10.2307/1970400.  Google Scholar

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