December  2020, 40(12): 6709-6745. doi: 10.3934/dcds.2020213

Time dependent center manifold in PDEs

1. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

2. 

School of Mathematics, Georgia Inst. of Technology, Atlanta GA, 30332, USA

Received  January 2019 Revised  February 2020 Published  December 2020 Early access  May 2020

Fund Project: H. C. supported by CSC by the National Natural Science Foundation of China (Grant Nos. 11171185, 10871117), H.C. thanks G.T. for hospitality 2015-2016.
R. L. supported in part by NSF grant DMS 1800241.
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester

We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).

We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a "time-dependent invariant manifold" (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.

Secondly, we construct the center manifold for skew systems driven by the external forcing.

Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.

Citation: Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213
References:
[1]

R. Abraham and J. Robbin, Transversal Mappings and Flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967.

[2]

A. Afendikov and A. Mielke, A spatial center manifold approach to a hydrodynamical problem with O(2) symmetry, in Dynamics, Bifurcation and Symmetry (Cargèse, 1993), vol. 437 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1994, 1–10. doi: 10.1007/978-94-011-0956-7_1.

[3]

L. F. A. Arbogast, Du Calcul Des Derivations, Levraut, Strasbourg, 1800, Available freely from Google Books.

[4]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[5]

T. BartschJ. M. Moix and S. Kawai, Time-dependent transition state theory, Advance in Chemical Physis, 140 (2008), 189-238.  doi: 10.1002/9780470371572.ch4.

[6]

J. Bass, Les Fonctions Pseudo-aléatoires, Mémor. Sci. Math., Fasc. CLIII, Gauthier-Villars, Éditeur-Imprimeur-Libraire, Paris, 1962.

[7]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), ⅷ+129pp. doi: 10.1090/memo/0645.

[8]

M. Berti, KAM theory for partial differential equations, Anal. Theory Appl., 35 (2019), 235-267.  doi: 10.4208/ata.OA-0013.

[9]

P. Boxler, How to construct stochastic center manifolds on the level of vector fields, in Lyapunov Exponents (Oberwolfach, 1990), vol. 1486 of Lecture Notes in Math., Springer, Berlin, 1991,141–158. doi: 10.1007/BFb0086664.

[10]

M. J. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1997-2026.  doi: 10.1088/0951-7715/25/7/1997.

[11]

J. Carr, Applications of Centre Manifold Theory, vol. 35 of Applied Mathematical Sciences, Springer-Verlag, New York-Berlin, 1981.

[12]

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.  doi: 10.1080/00036817408839081.

[13]

H. Cheng and R. de la Llave, Stable manifolds to bounded solutions in possibly ill-posed PDEs., J. Differ. Equations, 268 (2020), 4830-4899.  doi: 10.1016/j.jde.2019.10.042.

[14]

H. Cheng and J. Si, Quasi-periodic solutions for the quasi-periodically forced cubic complex Ginzburg-Landau equation on ${\mathbb T}^d$, J. Math. Phys., 54 (2013), 082702, 27pp. doi: 10.1063/1.4817864.

[15]

C. Chicone and Y. Latushkin, Center manifolds for infinite-dimensional nonautonomous differential equations, J. Differential Equations, 141 (1997), 356-399.  doi: 10.1006/jdeq.1997.3343.

[16]

S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets, J. Differential Equations, 168 (2000), 355–385, Special issue in celebration of Jack K. Hale's 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998). doi: 10.1006/jdeq.2000.3890.

[17]

S.-N. ChowW. Liu and Y. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc., 352 (2000), 5179-5211.  doi: 10.1090/S0002-9947-00-02443-0.

[18]

D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, John Wiley & Sons, Inc., New York, 1965.

[19]

S. L. Day, A Rigorous Numerical Method in Infinite Dimensions, ProQuest LLC, Ann Arbor, MI, 2003, Thesis (Ph.D.)–Georgia Institute of Technology.

[20]

R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320.  doi: 10.1007/BF02096662.

[21]

R. de la LlaveJ. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2), 123 (1986), 537-611.  doi: 10.2307/1971334.

[22]

R. de la Llave and J. D. Mireles James, Connecting orbits for compact infinite dimensional maps: Computer assisted proofs of existence, SIAM J. Appl. Dyn. Syst., 15 (2016), 1268-1323.  doi: 10.1137/15M1053608.

[23]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.  doi: 10.3934/dcds.1999.5.157.

[24]

R. de la Llave, A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities, J. Dynam. Differential Equations, 21 (2009), 371-415.  doi: 10.1007/s10884-009-9140-y.

[25]

R. de la Llave and Y. Sire, An a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill-posed equations, Arch Rational Mech. Anal., 231 (2019), 971-1044.  doi: 10.1007/s00205-018-1293-6.

[26]

R. de la Llave and A. Windsor, Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, Ergodic Theory Dynam. Systems, 30 (2010), 1055-1100.  doi: 10.1017/S014338570900039X.

[27] J. Duan, An Introduction to Stochastic Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2015. 
[28]

J.-P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Comm. Math. Phys., 136 (1991), 285–307, http://projecteuclid.org/euclid.cmp/1104202352. doi: 10.1007/BF02100026.

[29]

G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885.  doi: 10.1090/tran/7190.

[30]

J. A. Goldstein, Semigroups of Linear Operators & Applications, Dover Publications, Inc., Mineola, NY, 2017, Second edition of [MR0790497], Including transcriptions of five lectures from the 1989 workshop at Blaubeuren, Germany.

[31]

J. Hadamard, Sur le module maximum d'une fonction et de ses derives, Bull. Soc. Math. France, 42 (1898), 68-72. 

[32]

J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.

[33]

M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. doi: 10.1007/978-0-85729-112-7.

[34]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.

[35]

D. A. Jones and S. Shkoller, Persistence of invariant manifolds for nonlinear PDEs, Stud. Appl. Math., 102 (1999), 27-67.  doi: 10.1111/1467-9590.00103.

[36]

J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.  doi: 10.4171/RMI/69.

[37]

K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Differential Equations, 32 (1979), 119-148.  doi: 10.1016/0022-0396(79)90055-X.

[38]

S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math., 1 (1983), 193-260. 

[39]

S. Krantz, Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1992 edition. doi: 10.1090/chel/340.

[40]

O. E. Lanford III, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, in Nonlinear Problems in the Physical Sciences and Biology: P roceedings of a Battelle Summer Institute (eds. I. Stakgold, D. D. Joseph and D. H. Sattinger), Springer-Verlag, Berlin, Lecture Notes in Mathematics, 322 (1973), 159–192.

[41]

J. LiK. Lu and P. Bates, Normally hyperbolic invariant manifolds for random dynamical systems: Part Ⅰ–Persistence, Trans. Amer. Math. Soc., 365 (2013), 5933-5966.  doi: 10.1090/S0002-9947-2013-05825-4.

[42]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66.  doi: 10.1002/mma.1670100105.

[43]

A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, vol. 1489 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991, With applications to elliptic variational problems. doi: 10.1007/BFb0097544.

[44]

A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations, 110 (1994), 322-355.  doi: 10.1006/jdeq.1994.1070.

[45]

A. Pazy, Semigroups of operators in Banach spaces, in Equadiff 82 (Würzburg, 1982), vol. 1017 of Lecture Notes in Math., Springer, Berlin, 1983,508–524. doi: 10.1007/BFb0103275.

[46]

O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29 (1929), 129-160.  doi: 10.1007/BF01180524.

[47]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[48]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[49]

J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289 (1985), 431-469.  doi: 10.1090/S0002-9947-1985-0783998-8.

[50] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. 
[51]

M. E. Taylor, Partial Differential Equations Ⅲ. Nonlinear Equations, vol. 117 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[52]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, vol. 888 of Lecture Notes in Mathematics, North-Holland Publishing Co., Amsterdam-New York, 1981.

[53]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported: Expositions in Dynamical Systems, vol. 1 of Dynam. Report. Expositions Dynam. Systems (N.S.), Springer, Berlin, 1992,125–163.

[54]

L. Zhang and R. de la Llave, Transition state theory with quasi-periodic forcing, Commun. Nonlinear Sci. Numer. Simul., 62 (2018), 229-243.  doi: 10.1016/j.cnsns.2018.02.014.

show all references

References:
[1]

R. Abraham and J. Robbin, Transversal Mappings and Flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967.

[2]

A. Afendikov and A. Mielke, A spatial center manifold approach to a hydrodynamical problem with O(2) symmetry, in Dynamics, Bifurcation and Symmetry (Cargèse, 1993), vol. 437 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1994, 1–10. doi: 10.1007/978-94-011-0956-7_1.

[3]

L. F. A. Arbogast, Du Calcul Des Derivations, Levraut, Strasbourg, 1800, Available freely from Google Books.

[4]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[5]

T. BartschJ. M. Moix and S. Kawai, Time-dependent transition state theory, Advance in Chemical Physis, 140 (2008), 189-238.  doi: 10.1002/9780470371572.ch4.

[6]

J. Bass, Les Fonctions Pseudo-aléatoires, Mémor. Sci. Math., Fasc. CLIII, Gauthier-Villars, Éditeur-Imprimeur-Libraire, Paris, 1962.

[7]

P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), ⅷ+129pp. doi: 10.1090/memo/0645.

[8]

M. Berti, KAM theory for partial differential equations, Anal. Theory Appl., 35 (2019), 235-267.  doi: 10.4208/ata.OA-0013.

[9]

P. Boxler, How to construct stochastic center manifolds on the level of vector fields, in Lyapunov Exponents (Oberwolfach, 1990), vol. 1486 of Lecture Notes in Math., Springer, Berlin, 1991,141–158. doi: 10.1007/BFb0086664.

[10]

M. J. Capiński and C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25 (2012), 1997-2026.  doi: 10.1088/0951-7715/25/7/1997.

[11]

J. Carr, Applications of Centre Manifold Theory, vol. 35 of Applied Mathematical Sciences, Springer-Verlag, New York-Berlin, 1981.

[12]

N. Chafee and E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4 (1974/75), 17-37.  doi: 10.1080/00036817408839081.

[13]

H. Cheng and R. de la Llave, Stable manifolds to bounded solutions in possibly ill-posed PDEs., J. Differ. Equations, 268 (2020), 4830-4899.  doi: 10.1016/j.jde.2019.10.042.

[14]

H. Cheng and J. Si, Quasi-periodic solutions for the quasi-periodically forced cubic complex Ginzburg-Landau equation on ${\mathbb T}^d$, J. Math. Phys., 54 (2013), 082702, 27pp. doi: 10.1063/1.4817864.

[15]

C. Chicone and Y. Latushkin, Center manifolds for infinite-dimensional nonautonomous differential equations, J. Differential Equations, 141 (1997), 356-399.  doi: 10.1006/jdeq.1997.3343.

[16]

S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets, J. Differential Equations, 168 (2000), 355–385, Special issue in celebration of Jack K. Hale's 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998). doi: 10.1006/jdeq.2000.3890.

[17]

S.-N. ChowW. Liu and Y. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc., 352 (2000), 5179-5211.  doi: 10.1090/S0002-9947-00-02443-0.

[18]

D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, John Wiley & Sons, Inc., New York, 1965.

[19]

S. L. Day, A Rigorous Numerical Method in Infinite Dimensions, ProQuest LLC, Ann Arbor, MI, 2003, Thesis (Ph.D.)–Georgia Institute of Technology.

[20]

R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320.  doi: 10.1007/BF02096662.

[21]

R. de la LlaveJ. M. Marco and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2), 123 (1986), 537-611.  doi: 10.2307/1971334.

[22]

R. de la Llave and J. D. Mireles James, Connecting orbits for compact infinite dimensional maps: Computer assisted proofs of existence, SIAM J. Appl. Dyn. Syst., 15 (2016), 1268-1323.  doi: 10.1137/15M1053608.

[23]

R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.  doi: 10.3934/dcds.1999.5.157.

[24]

R. de la Llave, A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities, J. Dynam. Differential Equations, 21 (2009), 371-415.  doi: 10.1007/s10884-009-9140-y.

[25]

R. de la Llave and Y. Sire, An a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill-posed equations, Arch Rational Mech. Anal., 231 (2019), 971-1044.  doi: 10.1007/s00205-018-1293-6.

[26]

R. de la Llave and A. Windsor, Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, Ergodic Theory Dynam. Systems, 30 (2010), 1055-1100.  doi: 10.1017/S014338570900039X.

[27] J. Duan, An Introduction to Stochastic Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2015. 
[28]

J.-P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Comm. Math. Phys., 136 (1991), 285–307, http://projecteuclid.org/euclid.cmp/1104202352. doi: 10.1007/BF02100026.

[29]

G. Faye and A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370 (2018), 5843-5885.  doi: 10.1090/tran/7190.

[30]

J. A. Goldstein, Semigroups of Linear Operators & Applications, Dover Publications, Inc., Mineola, NY, 2017, Second edition of [MR0790497], Including transcriptions of five lectures from the 1989 workshop at Blaubeuren, Germany.

[31]

J. Hadamard, Sur le module maximum d'une fonction et de ses derives, Bull. Soc. Math. France, 42 (1898), 68-72. 

[32]

J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980.

[33]

M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. doi: 10.1007/978-0-85729-112-7.

[34]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981.

[35]

D. A. Jones and S. Shkoller, Persistence of invariant manifolds for nonlinear PDEs, Stud. Appl. Math., 102 (1999), 27-67.  doi: 10.1111/1467-9590.00103.

[36]

J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4 (1988), 187-193.  doi: 10.4171/RMI/69.

[37]

K. Kirchgässner and J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Differential Equations, 32 (1979), 119-148.  doi: 10.1016/0022-0396(79)90055-X.

[38]

S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math., 1 (1983), 193-260. 

[39]

S. Krantz, Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1992 edition. doi: 10.1090/chel/340.

[40]

O. E. Lanford III, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, in Nonlinear Problems in the Physical Sciences and Biology: P roceedings of a Battelle Summer Institute (eds. I. Stakgold, D. D. Joseph and D. H. Sattinger), Springer-Verlag, Berlin, Lecture Notes in Mathematics, 322 (1973), 159–192.

[41]

J. LiK. Lu and P. Bates, Normally hyperbolic invariant manifolds for random dynamical systems: Part Ⅰ–Persistence, Trans. Amer. Math. Soc., 365 (2013), 5933-5966.  doi: 10.1090/S0002-9947-2013-05825-4.

[42]

A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10 (1988), 51-66.  doi: 10.1002/mma.1670100105.

[43]

A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, vol. 1489 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991, With applications to elliptic variational problems. doi: 10.1007/BFb0097544.

[44]

A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations, 110 (1994), 322-355.  doi: 10.1006/jdeq.1994.1070.

[45]

A. Pazy, Semigroups of operators in Banach spaces, in Equadiff 82 (Würzburg, 1982), vol. 1017 of Lecture Notes in Math., Springer, Berlin, 1983,508–524. doi: 10.1007/BFb0103275.

[46]

O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29 (1929), 129-160.  doi: 10.1007/BF01180524.

[47]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.

[48]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.

[49]

J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289 (1985), 431-469.  doi: 10.1090/S0002-9947-1985-0783998-8.

[50] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. 
[51]

M. E. Taylor, Partial Differential Equations Ⅲ. Nonlinear Equations, vol. 117 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2011. doi: 10.1007/978-1-4419-7049-7.

[52]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, vol. 888 of Lecture Notes in Mathematics, North-Holland Publishing Co., Amsterdam-New York, 1981.

[53]

A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported: Expositions in Dynamical Systems, vol. 1 of Dynam. Report. Expositions Dynam. Systems (N.S.), Springer, Berlin, 1992,125–163.

[54]

L. Zhang and R. de la Llave, Transition state theory with quasi-periodic forcing, Commun. Nonlinear Sci. Numer. Simul., 62 (2018), 229-243.  doi: 10.1016/j.cnsns.2018.02.014.

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