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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 |
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. Google Scholar |
[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, (). Google Scholar |
[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. |
show all references
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. Google Scholar |
[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, (). Google Scholar |
[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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