Abstract
We present a technique to control chaos in Hamiltonian systems which are close to integrable. By adding a small and simple control term to the perturbation, the system becomes more regular than the original one. We apply this technique to a forced pendulum model and show numerically that the control is able to drastically reduce chaos.
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References
C. Chandre H. R. Jauslin (2002) ArticleTitle‘Renormalization-group analysis for the transition to chaos in Hamiltonian systems’ Physics Reports 365 1–64 Occurrence Handle10.1016/S0370-1573(01)00094-1
Chandre, C., Vittot, M., Elskens, Y., Ciraolo, G. and Pettini, M.: 2004, ‘Controlling chaos in area-preserving maps’, submitted and archived in http://arXiv.org/nlin.CD/0405008.
Chen, G. and Dong, X.: 1998, From Chaos to Order, World Scientific, Singapore.
G. Ciraolo C. Chandre R. Lima M. Vittot M. Pettini C. Figarella Ph. Ghendrih (2004a) ArticleTitle‘Controlling chaotic transport in a Hamiltonian model of interest to magnetized plasmas’ J.Phys. A: Math. Gen. 37 3589–3597 Occurrence Handle10.1088/0305-4470/37/11/004
Ciraolo, G., Briolle, F., Chandre, C., Floriani, E., Lima, R., Vittot, M., Pettini, M., Figarella, C. and Ghendrih, Ph.: 2004b, ‘Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas’, Phys. Rev. E 69(4), 056213.
G. Gallavotti (1982) ArticleTitle‘A criterion of integrability for perturbed nonresonant harmonic oscillators. ‘‘Wick ordering’’ of the perturbations in classical mechanics and invariance of the frequency spectrum’ Commun. Math. Phys 87 365–383 Occurrence Handle10.1007/BF01206029
G. Gallavotti (1985) Classical mechanics and renormalization-group G. Velo A.S. Wightman (Eds) Regular and Chaotic Motions in Dynamical Systems Plenum New York 185–231
D. J. Gauthier (2003) ArticleTitle‘Controlling chaos’ Am. J. Phys 71 750–759 Occurrence Handle10.1119/1.1572488
G. Gentile V. Mastropietro (1996) ArticleTitle‘Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics’ Rev. Math. Phys 8 393–444 Occurrence Handle10.1142/S0129055X96000135
J. Laskar (1999) Introduction to frequency map analysis C. Simò (Eds) Hamiltonian Systems with Three or More Degrees of Freedom, NATO ASI Series Kluwer Academic Publishers Dordrecht 134
M. Vittot (2004) ArticleTitle‘Perturbation theory and control in classical or quantum mechanics by an inversion formula’ J. Phys. A: Math. Gen 37 6337–6357 Occurrence Handle10.1088/0305-4470/37/24/011
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Ciraolo, G., Chandre, C., Lima, R. et al. Control Of Chaos In Hamiltonian Systems. Celestial Mech Dyn Astr 90, 3–12 (2004). https://doi.org/10.1007/s10569-004-6445-3
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DOI: https://doi.org/10.1007/s10569-004-6445-3