American Institute of Mathematical Sciences

March  2016, 36(3): 1677-1692. doi: 10.3934/dcds.2016.36.1677

On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems

 1 Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China 2 Department of Mathematics, Southeast University, Nanjing 210096

Received  October 2014 Revised  June 2015 Published  August 2015

This work focuses on the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. By KAM method and the special structure of unperturbed nonlinear terms, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of [22].
Citation: Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1677-1692. doi: 10.3934/dcds.2016.36.1677
References:
 [1] V. I. Arnold, Reversible systems,, in Nonlinear and Turbulent Processes in Physics, (1983), 1161. Google Scholar [2] H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos,, Lecture Notes in Math., (1645). Google Scholar [3] H. W. Broer and G. B. Huitema, Unfoldings of quasi-periodic tori in reversible systems,, J. Dynam. Differ. Equations, 7 (1995), 191. doi: 10.1007/BF02218818. Google Scholar [4] H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori,, J. Differ. Equations, 232 (2007), 355. doi: 10.1016/j.jde.2006.08.022. Google Scholar [5] H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori,, Physica D, 238 (2009), 309. doi: 10.1016/j.physd.2008.10.004. Google Scholar [6] H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605. doi: 10.1142/S021812740701866X. Google Scholar [7] H. Hanßmann, Quasi-periodic bifurcations in reversible systems,, Regular and Chaotic Dynamics, 16 (2011), 51. doi: 10.1134/S1560354710520059. Google Scholar [8] B. Liu, On lower dimensional invariant tori in reversible systems,, J. Differ. Equations, 176 (2001), 158. doi: 10.1006/jdeq.2000.3960. Google Scholar [9] J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136. doi: 10.1007/BF01399536. Google Scholar [10] J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics,, Annals Mathematics Studies, (1973). Google Scholar [11] I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems,, Dokl. Akad. Nauk Ukrain. SSR. Ser. A, 9 (1982), 19. Google Scholar [12] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559. doi: 10.1007/BF01221590. Google Scholar [13] J. Pöschel, A lecture on the classical KAM theorem,, in Smooth Ergodic Theory and Its Applications, (1999), 707. doi: 10.1090/pspum/069/1858551. Google Scholar [14] M. B. Sevryuk, Reversible Systems,, Lecture Notes in Math., (1211). Google Scholar [15] M. B. Sevryuk, Invariant m-dimensional tori of reversible systems with phase space of dimension greater than 2m,, J. Soviet. Math., 51 (1990), 2374. doi: 10.1007/BF01094996. Google Scholar [16] M. B. Sevryuk, New results in the reversible KAM theory,, in Seminar on Dynamical Systems (eds. S. B. Kuksin, (1994), 184. doi: 10.1007/978-3-0348-7515-8_14. Google Scholar [17] M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory,, Chaos, 5 (1995), 552. doi: 10.1063/1.166125. Google Scholar [18] M. B. Sevryuk, Partial preservation of frequencies in KAM theory,, Nonlinearity, 19 (2006), 1099. doi: 10.1088/0951-7715/19/5/005. Google Scholar [19] V. N. Tkhai, Reversibility of mechanical systems,, J. Appl. Math. Mech., 55 (1991), 461. doi: 10.1016/0021-8928(91)90007-H. Google Scholar [20] X. Wang and J. Xu, Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition,, Discrete and Continuous Dynamical Systems series A, 25 (2009), 701. doi: 10.3934/dcds.2009.25.701. Google Scholar [21] X. Wang, J. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems,, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1237. doi: 10.3934/dcdsb.2010.14.1237. Google Scholar [22] X. Wang, D. Zhang and J. Xu, Persistence of lower dimensional tori for a class of nearly integrable reversible systems,, Acta Applicanda Mathematicae, 115 (2011), 193. doi: 10.1007/s10440-011-9615-9. Google Scholar [23] X. Wang, J. Xu and D. Zhang, Degenerate lower dimensional tori in reversible systems,, J. Math. Anal. Appl., 387 (2012), 776. doi: 10.1016/j.jmaa.2011.09.030. Google Scholar [24] X. Wang, J. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems,, Ergodic Theory and Dynamical Systems, (). Google Scholar [25] B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems,, J. Math. Anal. Appl., 253 (2001), 558. doi: 10.1006/jmaa.2000.7165. Google Scholar [26] J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions,, SIAM J. Math. Anal., 36 (2004), 233. doi: 10.1137/S0036141003421923. Google Scholar

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References:
 [1] V. I. Arnold, Reversible systems,, in Nonlinear and Turbulent Processes in Physics, (1983), 1161. Google Scholar [2] H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos,, Lecture Notes in Math., (1645). Google Scholar [3] H. W. Broer and G. B. Huitema, Unfoldings of quasi-periodic tori in reversible systems,, J. Dynam. Differ. Equations, 7 (1995), 191. doi: 10.1007/BF02218818. Google Scholar [4] H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori,, J. Differ. Equations, 232 (2007), 355. doi: 10.1016/j.jde.2006.08.022. Google Scholar [5] H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori,, Physica D, 238 (2009), 309. doi: 10.1016/j.physd.2008.10.004. Google Scholar [6] H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605. doi: 10.1142/S021812740701866X. Google Scholar [7] H. Hanßmann, Quasi-periodic bifurcations in reversible systems,, Regular and Chaotic Dynamics, 16 (2011), 51. doi: 10.1134/S1560354710520059. Google Scholar [8] B. Liu, On lower dimensional invariant tori in reversible systems,, J. Differ. Equations, 176 (2001), 158. doi: 10.1006/jdeq.2000.3960. Google Scholar [9] J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1967), 136. doi: 10.1007/BF01399536. Google Scholar [10] J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics,, Annals Mathematics Studies, (1973). Google Scholar [11] I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems,, Dokl. Akad. Nauk Ukrain. SSR. Ser. A, 9 (1982), 19. Google Scholar [12] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559. doi: 10.1007/BF01221590. Google Scholar [13] J. Pöschel, A lecture on the classical KAM theorem,, in Smooth Ergodic Theory and Its Applications, (1999), 707. doi: 10.1090/pspum/069/1858551. Google Scholar [14] M. B. Sevryuk, Reversible Systems,, Lecture Notes in Math., (1211). Google Scholar [15] M. B. Sevryuk, Invariant m-dimensional tori of reversible systems with phase space of dimension greater than 2m,, J. Soviet. Math., 51 (1990), 2374. doi: 10.1007/BF01094996. Google Scholar [16] M. B. Sevryuk, New results in the reversible KAM theory,, in Seminar on Dynamical Systems (eds. S. B. Kuksin, (1994), 184. doi: 10.1007/978-3-0348-7515-8_14. Google Scholar [17] M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory,, Chaos, 5 (1995), 552. doi: 10.1063/1.166125. Google Scholar [18] M. B. Sevryuk, Partial preservation of frequencies in KAM theory,, Nonlinearity, 19 (2006), 1099. doi: 10.1088/0951-7715/19/5/005. Google Scholar [19] V. N. Tkhai, Reversibility of mechanical systems,, J. Appl. Math. Mech., 55 (1991), 461. doi: 10.1016/0021-8928(91)90007-H. Google Scholar [20] X. Wang and J. Xu, Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition,, Discrete and Continuous Dynamical Systems series A, 25 (2009), 701. doi: 10.3934/dcds.2009.25.701. Google Scholar [21] X. Wang, J. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems,, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1237. doi: 10.3934/dcdsb.2010.14.1237. Google Scholar [22] X. Wang, D. Zhang and J. Xu, Persistence of lower dimensional tori for a class of nearly integrable reversible systems,, Acta Applicanda Mathematicae, 115 (2011), 193. doi: 10.1007/s10440-011-9615-9. Google Scholar [23] X. Wang, J. Xu and D. Zhang, Degenerate lower dimensional tori in reversible systems,, J. Math. Anal. Appl., 387 (2012), 776. doi: 10.1016/j.jmaa.2011.09.030. Google Scholar [24] X. Wang, J. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems,, Ergodic Theory and Dynamical Systems, (). Google Scholar [25] B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems,, J. Math. Anal. Appl., 253 (2001), 558. doi: 10.1006/jmaa.2000.7165. Google Scholar [26] J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions,, SIAM J. Math. Anal., 36 (2004), 233. doi: 10.1137/S0036141003421923. Google Scholar
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