Existence of invariant curves for area-preserving mappings under weaker non-degeneracy conditions

Abstract

We consider a class of analytic area-preserving mappings C^m-smoothly depending on a parameter. Without imposing on any non-degeneracy assumption, we prove a formal KAM theorem for the mappings, which implies many previous KAM-type results under some non-degeneracy conditions. Moreover, by this formal KAM theorem, we can also obtain some new interesting results under some weaker non-degeneracy conditions. Thus, the formal KAM theorem can be regarded as a general KAM theorem for areapreserving mappings.

Keywords

Area-preserving mapping / invariant curve / KAM iteration / nondegeneracy condition

References

[1] Aubry S, Abramovici G. Chaotic trajectories in the standard map. The concept of anti-integrability. Phys D, 1990, 43(2{3): 199–219

[2] Bi Q Y, Xu J X. Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings. Qual Theory Dyn Syst, 2014, 13(2): 269–288

[3] Cheng C Q, Sun Y S. Existence of invariant tori in three-dimensional measure preserving mappings. Celestial Mech Dynam Astronom, 1990, 47(3): 275–292

[4] Duarte P. Plenty of elliptic islands for the standard family of area preserving maps. Ann Inst H Poincaré Anal Non Linéaire, 1994, 11(4): 359–409

[5] Dullin H R, Meiss J D. Resonances and twist in volume preserving mappings. SIAM J Appl Dyn Syst, 2012, 11(1): 319–349

[6] Fox A M, Meiss J D. Greene's residue criterion for the breakup of invariant tori of volume-preserving maps. Phys D, 2013, 243: 45–63

[7] Gelfreich V,Simó C, Vieiro A. Dynamics of 4D symplectic maps near a double resonance. Phys D, 2013, 243: 92–110

[8] Herman M. Topological stability of the Hamiltonian and volume-preserving dynamical systems. Lecture at the International Conference on Dynamical Systems, Evanston, Illinois, March, 1991

[9] Llave R D L, James J D M. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete Contin Dyn Syst, 2012, 32(12): 4321–4360

[10] Lu X Z, Li J, Xu J X. A KAM theorem for a class of nearly integrable symplectic mappings. J Dynam Differential Equations, 2017, 29(1): 131–154

[11] Moser J. On invariant curves of area preserving mappings of an annulus. Nachr Akad Wiss Göttingen II Math Phys Kl, 1962, 1962: 1–20

[12] Moser J. A rapidly convergent iteration method and nonlinear differential equations. Ann Sc Norm Super Pisa Cl Sci (5), 1966, 20: 499–535

[13] Moser J. Convergent series expansions for quasi-periodic motions. Math Ann, 1967, 169: 136–176

[14] Moser J. Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics. Ann of Math Stud, No 77. Princeton: Princeton Univ Press, 1973

[15] Rüssmann H. Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nachr Akad Wiss Göttingen II Math Phys Kl, 1970, 1970: 67–105

[16] Rüssmann H. On a new proof of Moser's twist mapping theorem. Celestial Mech Dynam Astronom, 1976, 14: 19–31

[17] Rüssmann H. On the existence of invariant curves of twist mappings of an annulus. In: Palis Jr J, ed. Geometric Dynamics. Lecture Notes in Math, Vol 1007. Berlin: Springer, 1983, 677–718

[18] Rüssmann H. Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition. Ergodic Theory Dynam Systems, 2002, 22(5): 1551–1573

[19] Shang Z J. A note on the KAM theorem for symplectic mappings. J Dynam Differential Equations, 2000, 12(2): 357–383

[20] Siegel C L, Moser J K. Lectures on Celestial Mechanics. Grundlehren Math Wiss, Vol 187. Berlin: Springer-Verlag, 1971

[21] Xia Z H. Existence of invariant tori in volume-preserving diffeomorphisms. Ergodic Theory Dynam Systems, 1992, 12(3): 621–631

[22] Xu J X, Lu X Z. General KAM theorems and their applications to invariant tori with prescribed frequencies. Regul Chaotic Dyn, 2016, 21(1): 107–125

[23] Xu J X, Wang K, Zhu M. On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters . Proc Amer Math Soc, 2016, 144(11): 4793–4805

[24] Xu J X, You J G. Persistence of the non-twist torus in nearly integrable Hamiltonian systems. Proc Amer Math Soc, 2010, 138(7): 2385–2395

[25] Zhu W Z, Liu B F, Liu Z X. The hyperbolic invariant tori of symplectic mappings. Nonlinear Anal, 2008, 68(1): 109–126

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Higher Education Press