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Interaction of waves in a one dimensional stochastic PDE model of excitable media
A Rikitake type system with one control
1. | "Politehnica" University of Timişoara, Department of Mathematics, Piaţa Victoriei nr. 2, 300006 - Timişoara, Romania, Romania |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
G. Chen and X. Dong, "From Chaos to Order. Methodologies, Perspectives and Applications," World Scientific Pub. Co., Singapore, 1998. |
[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. |
[8] |
R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[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. |
[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. |
[11] |
R. Hide, Structural instability of the Rikitake disk dynamo, Geophys. Res. Lett., 22 (1995), 1057-1059.
doi: 10.1029/95GL00779. |
[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. |
[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. |
[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. |
[15] |
D. F. Lawden, "Elliptic Functions and Applications," Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, 1989. |
[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. |
[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. |
[18] |
J. Marsden and T. S. Raţiu, "Introduction to Mechanics and Symmetry," 2nd ed., Text and Appl. Math., 17, Springer, Berlin, 1999. |
[19] |
J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747. |
[20] |
M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces," Lecture Notes in Math., 1768, Springer-Verlag, Berlin, 2001. |
[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. |
[22] |
M. Puta, "Hamiltonian Mechanical System and Geometric Quantization," Kluwer Academic Publishers, Dordrecht, Boston, London, 1993.
doi: 10.1007/978-94-011-1992-4. |
[23] |
T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., 54 (1958), 89-105.
doi: 10.1017/S0305004100033223. |
[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. |
[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. |
[26] |
D. L. Turcotte, "Fractals and Chaos in Geology and Geophysics," 2nd ed., Cambridge University Press, Cambridge, UK, 1997. |
[27] |
C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309-1326.
doi: 10.1017/S030821050000439X. |
[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. |
[29] |
A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[30] |
H. Whitney, Tangents to an analytic variety, Ann. of Math (2), 81 (1965), 496-549.
doi: 10.2307/1970400. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
G. Chen and X. Dong, "From Chaos to Order. Methodologies, Perspectives and Applications," World Scientific Pub. Co., Singapore, 1998. |
[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. |
[8] |
R. H. Cushman and L. Bates, "Global Aspects of Classical Integrable Systems," Birkhäuser, Basel, 1997.
doi: 10.1007/978-3-0348-8891-2. |
[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. |
[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. |
[11] |
R. Hide, Structural instability of the Rikitake disk dynamo, Geophys. Res. Lett., 22 (1995), 1057-1059.
doi: 10.1029/95GL00779. |
[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. |
[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. |
[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. |
[15] |
D. F. Lawden, "Elliptic Functions and Applications," Springer-Verlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong, 1989. |
[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. |
[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. |
[18] |
J. Marsden and T. S. Raţiu, "Introduction to Mechanics and Symmetry," 2nd ed., Text and Appl. Math., 17, Springer, Berlin, 1999. |
[19] |
J. Moser, Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747. |
[20] |
M. J. Pflaum, "Analytic and Geometric Study of Stratified Spaces," Lecture Notes in Math., 1768, Springer-Verlag, Berlin, 2001. |
[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. |
[22] |
M. Puta, "Hamiltonian Mechanical System and Geometric Quantization," Kluwer Academic Publishers, Dordrecht, Boston, London, 1993.
doi: 10.1007/978-94-011-1992-4. |
[23] |
T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., 54 (1958), 89-105.
doi: 10.1017/S0305004100033223. |
[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. |
[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. |
[26] |
D. L. Turcotte, "Fractals and Chaos in Geology and Geophysics," 2nd ed., Cambridge University Press, Cambridge, UK, 1997. |
[27] |
C. Valls, Rikitake system: Analytic and Darbouxian integrals, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 1309-1326.
doi: 10.1017/S030821050000439X. |
[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. |
[29] |
A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
[30] |
H. Whitney, Tangents to an analytic variety, Ann. of Math (2), 81 (1965), 496-549.
doi: 10.2307/1970400. |
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