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

A Rikitake type system with one control

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.
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. 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. 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, 344 (2007), 779. 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. 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. doi: 10.1142/S0218127493001112. Google Scholar

[6]

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

[7]

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

[8]

R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems,", Birkhäuser, (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. 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. doi: 10.1023/A:1026640221874. Google Scholar

[11]

R. Hide, Structural instability of the Rikitake disk dynamo,, Geophys. Res. Lett., 22 (1995), 1057. 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. 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. 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. doi: 10.1002/cpa.3160210503. Google Scholar

[15]

D. F. Lawden, "Elliptic Functions and Applications,", Springer-Verlag, (1989). Google Scholar

[16]

J. Llibre and X. Zhang, Invariant algebraic surfaces of the Rikitake system,, J. Phys. A, 33 (2000), 7613. 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, 55 (1907), 521. doi: 10.1080/00207179208934253. Google Scholar

[18]

J. Marsden and T. S. Raţiu, "Introduction to Mechanics and Symmetry,", 2nd ed., 17 (1999). Google Scholar

[19]

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

[20]

M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces,", Lecture Notes in Math., 1768 (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. doi: 10.1098/rspa.1998.0235. Google Scholar

[22]

M. Puta, "Hamiltonian Mechanical System and Geometric Quantization,", Kluwer Academic Publishers, (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. 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). 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. doi: 10.1137/080728822. Google Scholar

[26]

D. L. Turcotte, "Fractals and Chaos in Geology and Geophysics,", 2nd ed., (1997). Google Scholar

[27]

C. Valls, Rikitake system: Analytic and Darbouxian integrals,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309. 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. doi: 10.1088/1751-8113/40/11/011. Google Scholar

[29]

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

[30]

H. Whitney, Tangents to an analytic variety,, Ann. of Math (2), 81 (1965), 496. 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. 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. 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, 344 (2007), 779. 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. 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. doi: 10.1142/S0218127493001112. Google Scholar

[6]

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

[7]

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

[8]

R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems,", Birkhäuser, (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. 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. doi: 10.1023/A:1026640221874. Google Scholar

[11]

R. Hide, Structural instability of the Rikitake disk dynamo,, Geophys. Res. Lett., 22 (1995), 1057. 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. 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. 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. doi: 10.1002/cpa.3160210503. Google Scholar

[15]

D. F. Lawden, "Elliptic Functions and Applications,", Springer-Verlag, (1989). Google Scholar

[16]

J. Llibre and X. Zhang, Invariant algebraic surfaces of the Rikitake system,, J. Phys. A, 33 (2000), 7613. 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, 55 (1907), 521. doi: 10.1080/00207179208934253. Google Scholar

[18]

J. Marsden and T. S. Raţiu, "Introduction to Mechanics and Symmetry,", 2nd ed., 17 (1999). Google Scholar

[19]

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

[20]

M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces,", Lecture Notes in Math., 1768 (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. doi: 10.1098/rspa.1998.0235. Google Scholar

[22]

M. Puta, "Hamiltonian Mechanical System and Geometric Quantization,", Kluwer Academic Publishers, (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. 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). 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. doi: 10.1137/080728822. Google Scholar

[26]

D. L. Turcotte, "Fractals and Chaos in Geology and Geophysics,", 2nd ed., (1997). Google Scholar

[27]

C. Valls, Rikitake system: Analytic and Darbouxian integrals,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309. 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. doi: 10.1088/1751-8113/40/11/011. Google Scholar

[29]

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

[30]

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

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