Article
MSC:
70H05,
70H06,
70H12,
70H14,
70K20,
70K44 | MR 4263155 | Zbl 07361059 | DOI: 10.21136/AM.2021.0303-19
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Keywords:
integrable deformation; Hamilton-Poisson system; stability; energy-Casimir mapping; periodic orbit; heteroclinic orbit; mid-point rule
integrable deformation; Hamilton-Poisson system; stability; energy-Casimir mapping; periodic orbit; heteroclinic orbit; mid-point rule
Summary:
We give some deformations of the Rikitake two-disk dynamo system. Particularly, we consider an integrable deformation of an integrable version of the Rikitake system. The deformed system is a three-dimensional Hamilton-Poisson system. We present two Lie-Poisson structures and also symplectic realizations. Furthermore, we give a prequantization result of one of the Poisson manifold. We study the stability of the equilibrium states and we prove the existence of periodic orbits. We analyze some properties of the energy-Casimir mapping $\mathcal {EC}$ associated to our system. In many cases the dynamical behavior of such systems is related with some geometric properties of the image of the energy-Casimir mapping. These connections were observed in the cases when the image of $\mathcal {EC}$ is a convex proper subset of $\mathbb {R}^2$. In order to point out new connections, we choose deformation functions such that Im$(\mathcal {EC})=\mathbb {R}^2.$ Using the images of the equilibrium states through the energy-Casimir mapping we give parametric equations of some special orbits, namely heteroclinic orbits, split-heteroclinic orbits, and split-homoclinic orbits. Finally, we implement the mid-point rule to perform some numerical integrations of the considered system.
References:
[1] Adams, R. M., Biggs, R., Holderbaum, W., Remsing, C. C.: On the stability and integration of Hamilton-Poisson systems on $\frak{s}\frak{o}(3)_{-}^*$. Rend. Mat. Appl., VII. Ser. 37 (2016), 1-42. MR 3622303 | Zbl 1416.17022
[2] Arnol'd, V. I.: Conditions for nonlinear stability of stationary plane curvilinear flows on an ideal fluid. Sov. Math., Dokl. 6 (1965), 773-777 translation from Dokl. Akad. Nauk SSSR 162 1965 975-978. DOI 10.1007/978-3-642-31031-7_4 | MR 0180051 | Zbl 0141.43901
[3] Austin, M. A., Krishnaprasad, P. S., Wang, L.-S.: Almost Poisson integration of rigid body systems. J. Comput. Phys. 107 (1993), 105-117. DOI 10.1006/jcph.1993.1128 | MR 1226376 | Zbl 0782.70001
[4] Ballesteros, Á., Blasco, A., Musso, F.: Integrable deformations of Rössler and Lorenz systems from Poisson-Lie groups. J. Differ. Equations 260 (2016), 8207-8228. DOI 10.1016/j.jde.2016.02.014 | MR 3479208 | Zbl 1382.37057
[5] Barrett, D. I., Biggs, R., Remsing, C. C.: Quadratic Hamilton-Poisson systems on $\frak{s}\frak{e}(1, 1)^*$: The inhomogeneous case. Acta Appl. Math. 154 (2018), 189-230. DOI 10.1007/s10440-017-0140-3 | MR 3770248 | Zbl 1411.53069
[6] Bînzar, T., Lăzureanu, C.: A Rikitake type system with one control. Discrete Contin. Dyn. Syst, Ser. B. 18 (2013), 1755-1776. DOI 10.3934/dcdsb.2013.18.1755 | MR 3066320 | Zbl 1391.70049
[7] Bînzar, T., Lăzureanu, C.: On some dynamical and geometrical properties of the MaxwellBloch equations with a quadratic control. J. Geom. Phys. 70 (2013), 1-8. DOI 10.1016/j.geomphys.2013.03.016 | MR 3054279 | Zbl 1332.70026
[8] Bolsinov, A. V., Borisov, A. V.: Compatible Poisson brackets on Lie algebras. Math. Notes 72 (2002), 10-30 translation from Mat. Zametki 72 2002 11-34. DOI 10.1023/A:1019856702638 | MR 1942578 | Zbl 1042.37041
[9] Chillingworth, D. R. J., Holmes, P. J.: Dynamical systems and models for reversals of the earth's magnetic field. J. Internat. Assoc. Math. Geol. 12 (1980), 41-59. DOI 10.1007/BF01039903 | MR 0594011
[10] Cook, A. E., Roberts, P. H.: The Rikitake two-disc dynamo system. Proc. Camb. Philos. Soc. 68 (1970), 547-569. DOI 10.1017/S0305004100046338
[11] Dirac, P. A. M.: The Principles of Quantum Mechanics. Oxford University Press, Oxford (1947). MR 0023198 | Zbl 0030.04801
[12] Evripidou, C. A., Kassotakis, P., Vanhaecke, P.: Integrable deformations of the Bogoyavlenskij-Itoh Lotka-Volterra systems. Regul. Chaotic Dyn. 22 (2017), 721-739. DOI 10.1134/S1560354717060090 | MR 3736470 | Zbl 06845459
[13] Galajinsky, A.: Remark on integrable deformations of the Euler top. J. Math. Anal. Appl. 416 (2014), 995-997. DOI 10.1016/j.jmaa.2014.03.008 | MR 3188751 | Zbl 1362.70006
[14] Glatzmaier, G. A., Roberts, P. H.: A three-dimensional self-consistent computer simulation of a geomagnetic field reversal. Nature 377 (1995), 203-209. DOI 10.1038/377203a0
[15] Hardy, Y., Steeb, W.-H.: The Rikitake two-disc dynamo system and domains with periodic orbits. Int. J. Theor. Phys. 38 (1999), 2413-2417. DOI 10.1023/A:1026640221874 | MR 1722023 | Zbl 0980.86005
[16] Holm, D. D., Marsden, J. E.: The rotor and the pendulum. Symplectic Geometry and Mathematical Physics Birkhäuser, Boston (1991), 189-203. DOI 10.1007/978-1-4757-2140-9_9 | MR 1156540 | Zbl 0744.70011
[17] Huang, K., Shi, S., Xu, Z.: Integrable deformations, bi-Hamiltonian structures and nonintegrability of a generalized Rikitake system. Int. J. Geom. Methods Mod. Phys. 16 (2019), Article ID 1950059, 17 pages. DOI 10.1142/S0219887819500592 | MR 3940519 | Zbl 1421.37026
[18] Ito, K.: Chaos in the Rikitake two-disc dynamo system. Earth Planet. Sci. Lett. 51 (1980), 451-456. DOI 10.1016/0012-821X(80)90224-1
[19] Ivan, M., Ivan, G.: On the fractional Euler top system with two parameters. Int. J. Modern Eng. Research 8 (2018), 10-22.
[20] Jian, X.: Anti-synchronization of uncertain Rikitake systems via active sliding mode control. Int. J. Phys. Sci. 6 (2011), 2478-2482. DOI 10.5897/IJPS11.221
[21] Kostant, B.: Quantization and unitary representations I. Prequantization. Lectures in Modern Analysis and Applications III Lecture Notes in Mathematics 170. Springer, Berlin (1970), 87-208. DOI 10.1007/BFb0079068 | MR 0294568 | Zbl 0223.53028
[22] Lăzureanu, C.: Hamilton-Poisson realizations of the integrable deformations of the Rikitake system. Adv. Math. Phys. 2017 (2017), Article ID 4596951, 9 pages. DOI 10.1155/2017/4596951 | MR 3696014 | Zbl 1401.37063
[23] Lăzureanu, C.: On a Hamilton-Poisson approach of the Maxwell-Bloch equations with a control. Math. Phys. Anal. Geom. 20 (2017), Article ID 20, 22 pages. DOI 10.1007/s11040-017-9251-3 | MR 3683715 | Zbl 1413.37043
[24] Lăzureanu, C.: On the Hamilton-Poisson realizations of the integrable deformations of the Maxwell-Bloch equations. C. R., Math., Acad. Sci. Paris 355 (2017), 596-600. DOI 10.1016/j.crma.2017.04.002 | MR 3650389 | Zbl 1366.37125
[25] Lăzureanu, C.: Integrable deformations of three-dimensional chaotic systems. Int. J. Bifurcation Chaos Appl. Sci. Eng. 28 (2018), Article ID 1850066, 7 pages. DOI 10.1142/S0218127418500669 | MR 3807430 | Zbl 1392.34039
[26] Lăzureanu, C., Bînzar, T.: A Rikitake type system with quadratic control. Int. J. Bifurcation Chaos Appl. Sci. Eng. 22 (2012), Article ID 1250274, 14 pages. DOI 10.1142/S0218127412502744 | MR 3006345 | Zbl 1258.34100
[27] Lăzureanu, C., Bînzar, T.: On the symmetries of a Rikitake type system. C. R., Math., Acad. Sci. Paris 350 (2012), 529-533. DOI 10.1016/j.crma.2012.04.016 | MR 2929062 | Zbl 1253.34041
[28] Lăzureanu, C., Petrişor, C.: Stability and energy-Casimir mapping for integrable deformations of the Kermack-McKendrick system. Adv. Math. Phys. 2018 (2018), Article ID 5398768, 9 pages. DOI 10.1155/2018/5398768 | MR 3816095 | Zbl 1419.37056
[29] Libermann, P., Marle, C.-M.: Symplectic Geometry and Analytical Mechanics. Mathematics and Its Applications 35. D. Reidel, Dordrecht (1987). DOI 10.1007/978-94-009-3807-6 | MR 0882548 | Zbl 0643.53002
[30] Llibre, J., Zhang, X.: Invariant algebraic surfaces of the Rikitake system. J. Phys. A, Math. Gen. 33 (2000), 7613-7635. DOI 10.1088/0305-4470/33/42/310 | MR 1802113 | Zbl 0967.34002
[31] McLachlan, R. I.: On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16 (1995), 151-168. DOI 10.1137/0916010 | MR 1311683 | Zbl 0821.65048
[32] McMillen, T.: The shape and dynamics of the Rikitake attractor. Nonlinear J. 1 (1999), 1-10.
[33] Moser, J.: Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29 (1976), 727-747. DOI 10.1002/cpa.3160290613 | MR 0426052 | Zbl 0346.34024
[34] Pehlivan, I., Uyaroglu, Y.: Rikitake attractor and it's synchronization application for secure communication systems. J. Appl. Sci. 7 (2007), 232-236. DOI 10.3923/jas.2007.232.236
[35] Puta, M.: Hamiltonian Mechanical Systems and Geometric Quantization. Mathematics and Its Applications (Dordrecht) 260. Kluwer Academic, Dordrecht (1993). DOI 10.1007/978-94-011-1992-4 | MR 1247960 | Zbl 0795.70001
[36] Puta, M.: Lie-Trotter formula and Poisson dynamics. Int. J. Bifurcation Chaos Appl. Sci. Eng. 9 (1999), 555-559. DOI 10.1142/S0218127499000390 | MR 1702129 | Zbl 0970.37063
[37] Rikitake, T.: Oscillations of a system of disk dynamos. Proc. Camb. Philos. Soc. 54 (1958), 89-105. DOI 10.1017/S0305004100033223 | MR 0092682 | Zbl 0087.23703
[38] Tudoran, R. M., Aron, A., Nicoară, Ş.: On a Hamiltonian version of the Rikitake system. SIAM J. Appl. Dyn. Sys. 8 (2009), 454-479. DOI 10.1137/080728822 | MR 2496764 | Zbl 1159.70356
[39] Tudoran, R. M., Gîrban, A.: On a Hamiltonian version of a three-dimensional LotkaVolterra system. Nonlinear Anal., Real World Appl. 13 (2012), 2304-2312. DOI 10.1016/j.nonrwa.2012.01.025 | MR 2911917 | Zbl 1257.34037
[40] Turcotte, D. L.: Fractals and Chaos in Geology and Geophysics. Cambridge University Press, Cambridge (1997). DOI 10.1017/CBO9781139174695 | MR 1458893 | Zbl 0785.58005
[41] Valls, C.: Rikitake system: Analytic and Darbouxian integrals. Proc. R. Soc. Edinb., Sect. A, Math. 135 (2005), 1309-1326. DOI 10.1017/S030821050000439X | MR 2191901 | Zbl 1098.34028
[42] Balasubramaniam, V. Vembarasan P.: Chaotic synchronization of Rikitake system based on T-S fuzzy control techniques. Nonlinear Dyn. 74 (2013), 31-44. DOI 10.1007/s11071-013-0946-0 | MR 3105173 | Zbl 1281.34097
[43] Vincent, U. E.: Synchronization of Rikitake chaotic attractor using active control. Phys. Lett., A 343 (2005), 133-138. DOI 10.1016/j.physleta.2005.06.003 | Zbl 1194.34091
[44] Wei, Z., Zhang, W., Wang, Z., Yao, M.: Hidden attractors and dynamical behaviors in an extended Rikitake system. Int. J. Bifurcation Chaos Appl. Sci. Eng. 25 (2015), Article ID 1550028, 11 pages. DOI 10.1142/S0218127415500285 | MR 3316322 | Zbl 1309.34009
[45] Wei, Z., Zhu, B., Yang, J., Perc, M., Slavinec, M.: Bifurcation analysis of two disc dynamos with viscous friction and multiple time delays. Appl. Math. Comput. 347 (2019), 265-281. DOI 10.1016/j.amc.2018.10.090 | MR 3880151 | Zbl 1428.34055
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