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December  2012, 32(12): 4321-4360. doi: 10.3934/dcds.2012.32.4321

Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps

Rafael de la Llave 1, and  Jason D. Mireles James 2,

1. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry St, Atlanta, GA 30332-0160, United States
2. 

Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, United States

Received  June 2011 Revised  March 2012 Published  August 2012

We prove the existence of certain analytic invariant manifolds associated with fixed points of analytic symplectic and volume preserving diffeomorphisms. The manifolds we discuss are not defined in terms of either forward or backward asymptotic convergence to the fixed point, and are not required to be stable or unstable. Rather, the manifolds we consider are defined as being tangent to certain "mixed-stable" linear invariant subspaces of the differential (i.e linear subspace which are spanned by some combination of stable and unstable eigenvectors). Our method is constructive, but has to face small divisors. The small divisors are overcome via a quadratic convergent scheme which relies heavily on the geometry of the problem as well as assuming some Diophantine properties of the linearization restricted to the invariant subspace. The theorem proved has an a-posteriori format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.
Keywords: quadratic convergence, Invariant manifolds, KAM theory, volume preserving mappings., symplectic mappings.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
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show all references

References:
[1]

Trudy Moskov. Mat. Obšč., 25 (1971), 199-239; 26 (1972), 119-262.  Google Scholar

[2]

Dover Publications Inc., New York, 1995. Translated from the French, Reprint of the 1973 edition.  Google Scholar

[3]

Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245.  Google Scholar

[4]

Indiana Univ. Math. J., 52 (2003), 329-360. doi: 10.1512/iumj.2003.52.2407.  Google Scholar

[5]

J. Differential Equations, 218 (2005), 444-515. doi: 10.1016/j.jde.2004.12.003.  Google Scholar

[6]

J. Statist. Phys., 87 (1997), 211-249. doi: 10.1007/BF02181486.  Google Scholar

[7]

In "Smooth Ergodic Theory and Its Applications (Seattle, WA , 1999)," volume 69 of "Proc. Sympos. Pure Math.," 175-292, Amer. Math. Soc., Providence, RI, 2001. Google Scholar

[8]

Math. Phys. Electron. J., 9 (2003), Paper 3, 35 pp. (electronic).  Google Scholar

[9]

Nonlinearity, 18 (2005), 855-895. doi: 10.1088/0951-7715/18/2/020.  Google Scholar

[10]

Math. Z., 236 (2001), 717-777. doi: 10.1007/PL00004849.  Google Scholar

[11]

Electron. Res. Announc. Math. Sci., 16 (2009), 9-22.  Google Scholar

[12]

In "Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968)," 133-163, Amer. Math. Soc., Providence, R.I., 1970.  Google Scholar

[13]

L. D. Landau and E. M. Lifschitz, "Course of Theoretical Physics Vol II,", Classical Theory of Fields., ().   Google Scholar

[14]

SIAM J. Appl. Dyn. Syst., 9 (2010), 919-953.  Google Scholar

[15]

J. D. Mireles James, Quadratic Volume-Preserving Maps: (Un)Stable Manifolds, Hyperbolic Dynamics. and Vortex Bubble Bifurcations,, (Submitted). www.math.rutgers.edu/ jmireles/papers.html, ().   Google Scholar

[16]

Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 1824-1831. doi: 10.1073/pnas.47.11.1824.  Google Scholar

[17]

Trans. Amer. Math. Soc., 120 (1965), 286-294. doi: 10.1090/S0002-9947-1965-0182927-5.  Google Scholar

[18]

Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 265-315.  Google Scholar

[19]

Mathematische Zeitschrift, 29 (1928), 129-160.  Google Scholar

[20]

Exposition. Math., 4 (1986), 97-109.  Google Scholar

[21]

In "Proceedings of the Fifth Conference on Mathematical Methods in Celestial Mechanics (Oberwolfach, 1975), Part I," volume 14 (1976), 33-37.  Google Scholar

[22]

In "Nonlinear Dynamics (Internat. Conf., New York, 1979)," volume 357 of "Ann. New York Acad. Sci.," 90-107, New York Acad. Sci., New York, 1980.  Google Scholar

[23]

SIAM J. Math. Anal., 43 (2011), 1557-1594. doi: 10.1137/100812008.  Google Scholar

[24]

Comm. Pure Appl. Math., 28 (1975), 91-140. doi: 10.1002/cpa.3160280104.  Google Scholar

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