# American Institute of Mathematical Sciences

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

 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.
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 - A, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
##### References:
 [1] A. D. Brjuno, Analytic form of differential equations. I, II,, Trudy Moskov. Mat. Obšč., 25 (1971), 199. [2] Henri Cartan, "Elementary Theory of Analytic Functions of One Or Several Complex Variables,", Dover Publications Inc., (1995). [3] Xavier Cabré, Ernest Fontich and Rafael de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces,, Indiana Univ. Math. J., 52 (2003), 283. doi: 10.1512/iumj.2003.52.2245. [4] Xavier Cabré, Ernest Fontich and Rafael de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters,, Indiana Univ. Math. J., 52 (2003), 329. doi: 10.1512/iumj.2003.52.2407. [5] Xavier Cabré, Ernest Fontich and Rafael de la Llave, The parameterization method for invariant manifolds. III. Overview and applications,, J. Differential Equations, 218 (2005), 444. doi: 10.1016/j.jde.2004.12.003. [6] Rafael de la Llave, Invariant manifolds associated to nonresonant spectral subspaces,, J. Statist. Phys., 87 (1997), 211. doi: 10.1007/BF02181486. [7] Rafael de la Llave, A tutorial on KAM theory,, In, 69 (1999), 175. [8] R. de la Llave, Invariant manifolds associated to invariant subspaces without invariant complements: a graph transform approach,, Math. Phys. Electron. J., 9 (2003). [9] R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855. doi: 10.1088/0951-7715/18/2/020. [10] Mohamed S. ElBialy, Sub-stable and weak-stable manifolds associated with finitely non-resonant spectral subspaces,, Math. Z., 236 (2001), 717. doi: 10.1007/PL00004849. [11] Ernest Fontich, Rafael de la Llave and Yannick Sire, A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems,, Electron. Res. Announc. Math. Sci., 16 (2009), 9. [12] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets,, In, (1968), 133. [13] L. D. Landau and E. M. Lifschitz, "Course of Theoretical Physics Vol II,", Classical Theory of Fields., (). [14] J. D. Mireles James and Hector Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family,, SIAM J. Appl. Dyn. Syst., 9 (2010), 919. [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, (). [16] Jürgen Moser, A new technique for the construction of solutions of nonlinear differential equations,, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 1824. doi: 10.1073/pnas.47.11.1824. [17] Jürgen Moser, On the volume elements on a manifold,, Trans. Amer. Math. Soc., 120 (1965), 286. doi: 10.1090/S0002-9947-1965-0182927-5. [18] Jürgen Moser, A rapidly convergent iteration method and non-linear partial differential equations. I,, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 265. [19] O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen,, Mathematische Zeitschrift, 29 (1928), 129. [20] Jürgen Pöschel, On invariant manifolds of complex analytic mappings near fixed points,, Exposition. Math., 4 (1986), 97. [21] Helmut Rüssmann, On optimal estimates for the solutions of linear difference equations on the circle,, In, 14 (1976), 33. [22] Helmut Rüssmann, On the one-dimensional Schrödinger equation with a quasiperiodic potential,, In, 357 (1979), 90. [23] Jan Bouwe van den Berg, Jason D. Mireles-James, Jean-Philippe Lessard and Konstantin Mischaikow, Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation,, SIAM J. Math. Anal., 43 (2011), 1557. doi: 10.1137/100812008. [24] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91. doi: 10.1002/cpa.3160280104.

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##### References:
 [1] A. D. Brjuno, Analytic form of differential equations. I, II,, Trudy Moskov. Mat. Obšč., 25 (1971), 199. [2] Henri Cartan, "Elementary Theory of Analytic Functions of One Or Several Complex Variables,", Dover Publications Inc., (1995). [3] Xavier Cabré, Ernest Fontich and Rafael de la Llave, The parameterization method for invariant manifolds. I. Manifolds associated to non-resonant subspaces,, Indiana Univ. Math. J., 52 (2003), 283. doi: 10.1512/iumj.2003.52.2245. [4] Xavier Cabré, Ernest Fontich and Rafael de la Llave, The parameterization method for invariant manifolds. II. Regularity with respect to parameters,, Indiana Univ. Math. J., 52 (2003), 329. doi: 10.1512/iumj.2003.52.2407. [5] Xavier Cabré, Ernest Fontich and Rafael de la Llave, The parameterization method for invariant manifolds. III. Overview and applications,, J. Differential Equations, 218 (2005), 444. doi: 10.1016/j.jde.2004.12.003. [6] Rafael de la Llave, Invariant manifolds associated to nonresonant spectral subspaces,, J. Statist. Phys., 87 (1997), 211. doi: 10.1007/BF02181486. [7] Rafael de la Llave, A tutorial on KAM theory,, In, 69 (1999), 175. [8] R. de la Llave, Invariant manifolds associated to invariant subspaces without invariant complements: a graph transform approach,, Math. Phys. Electron. J., 9 (2003). [9] R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855. doi: 10.1088/0951-7715/18/2/020. [10] Mohamed S. ElBialy, Sub-stable and weak-stable manifolds associated with finitely non-resonant spectral subspaces,, Math. Z., 236 (2001), 717. doi: 10.1007/PL00004849. [11] Ernest Fontich, Rafael de la Llave and Yannick Sire, A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems,, Electron. Res. Announc. Math. Sci., 16 (2009), 9. [12] Morris W. Hirsch and Charles C. Pugh, Stable manifolds and hyperbolic sets,, In, (1968), 133. [13] L. D. Landau and E. M. Lifschitz, "Course of Theoretical Physics Vol II,", Classical Theory of Fields., (). [14] J. D. Mireles James and Hector Lomelí, Computation of heteroclinic arcs with application to the volume preserving Hénon family,, SIAM J. Appl. Dyn. Syst., 9 (2010), 919. [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, (). [16] Jürgen Moser, A new technique for the construction of solutions of nonlinear differential equations,, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 1824. doi: 10.1073/pnas.47.11.1824. [17] Jürgen Moser, On the volume elements on a manifold,, Trans. Amer. Math. Soc., 120 (1965), 286. doi: 10.1090/S0002-9947-1965-0182927-5. [18] Jürgen Moser, A rapidly convergent iteration method and non-linear partial differential equations. I,, Ann. Scuola Norm. Sup. Pisa (3), 20 (1966), 265. [19] O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen,, Mathematische Zeitschrift, 29 (1928), 129. [20] Jürgen Pöschel, On invariant manifolds of complex analytic mappings near fixed points,, Exposition. Math., 4 (1986), 97. [21] Helmut Rüssmann, On optimal estimates for the solutions of linear difference equations on the circle,, In, 14 (1976), 33. [22] Helmut Rüssmann, On the one-dimensional Schrödinger equation with a quasiperiodic potential,, In, 357 (1979), 90. [23] Jan Bouwe van den Berg, Jason D. Mireles-James, Jean-Philippe Lessard and Konstantin Mischaikow, Rigorous numerics for symmetric connecting orbits: even homoclinics of the Gray-Scott equation,, SIAM J. Math. Anal., 43 (2011), 1557. doi: 10.1137/100812008. [24] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91. doi: 10.1002/cpa.3160280104.
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