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Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium
On the nonautonomous Hopf bifurcation problem
1. | Università Politecnica delle Marche, Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via Brecce Bianche, I-60131 Ancona, Italy |
2. | Dipartimento di Sistemi e Informatica, Università di Firenze, Facolta' di Ingegneria, Via di Santa Marta 3, 50139 Firenze |
3. | Universidad de Valladolid, Departamento de Matemática Aplicada, Esecuela de Ingegnerías Industriales, Paseo del Cauce 59, 47011 Valladolid, Spain |
References:
[1] |
J. Aliste-Prieto and T. Jäger, Almost periodic structures and the semi-conjugacy problem, J. Differential Equations, 252 (2012), 4988-5001.
doi: 10.1016/j.jde.2012.01.030. |
[2] |
V. Anagnostopoulou, T. Jäger and G. Keller, A model for the non-autonomous Hopf bifurcation,, preprint, ().
|
[3] |
L. Arnold, Random Dynamical Systems, in Dynamical Systems, Fondazione C.I.M.E. 1994, ed. R. Johnson, Lecture Notes in Math., Springer-Verlag, Berlin, 1609 (1995), 1-43.
doi: 10.1007/BFb0095238. |
[4] |
M. Bebutov, On dynamical systems in the space of continuous functions, Bull. Moscow Univ. Matematica, (1941), 1-52. |
[5] |
K. Bjerkäv and R. Johnson, Minimal subsets of projective flows, Discrete Contin. Dyn. Syst., 9 (2008), 493-516.
doi: 10.3934/dcdsb.2008.9.493. |
[6] |
R. Botts, A. Homburg and T. Young, The Hopf bifurcation with bounded noise, Discrete Contin. Dyn. Syst., 32 (2012), 2997-3007.
doi: 10.3934/dcds.2012.32.2997. |
[7] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.
doi: 10.1090/S0002-9947-1971-0274707-X. |
[8] |
B. Braaksma and H. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincare Anal. Non Lineaire, 4 (1987), 115-168. |
[9] |
B. Braaksma, H. Broer and G. Huitema, Toward a quasi-periodic bifurcation theory, Memoirs A.M.S., 83 (1990), 83-167. |
[10] |
H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos, Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996. |
[11] |
H. Broer, KAM-theory: Multiperiodicity in conservative and dissipative systems, Nieuw Archief v. Wiskunde, 14 (1996), 65-79. |
[12] |
L. Chierchia and C. Falcolini, Compensations in small divisors problems, Comm. Math. Phys., 175 (1996), 135-160.
doi: 10.1007/BF02101627. |
[13] |
E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill, New York, 1955. |
[14] |
W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 377, Springer-Verlag, Berlin, 1978. |
[15] |
S. Diliberto, Perturbation theorems for periodic surfaces I, Rend. Circ. Math. Palermo, 9 (1960), 265-299.
doi: 10.1007/BF02851248. |
[16] |
S. Diliberto, New results in periodic surfaces and the averaging principle, U.S.-Japanese seminar on Differential Equations, W.A. Benjamin Co., New York, 1967, 49-87. |
[17] |
R. Ellis, Lectures on Topological Dynamics, W.A. Benjamin Co., New York, 1969. |
[18] |
B. Fayad, Weak mixing for reparametrized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201.
doi: 10.1017/S0143385702000081. |
[19] |
B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468.
doi: 10.1017/S0143385702000214. |
[20] |
H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. Jour. Math., 83 (1961), 573-601.
doi: 10.2307/2372899. |
[21] |
A. Gonzalez-Enriquez, A non-perturbative theorem on conjugation of torus diffeomorphisms to rigid rotations, preprint, 2005. |
[22] |
A. Gonzalez-Enriquez and J. Vano, Estimate of smoothing and composition with applications to conjugation problems, J. Dynam. Differential Equations, 20 (2008), 239-270.
doi: 10.1007/s10884-006-9060-z. |
[23] |
W. Gottschalk and G. Hedlund, Topological Dynamics, AMS Colloquium Publications 36, Amer. Math. Soc., Providence USA, 1955. |
[24] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583 Springer-Verlag, New York, 1977. |
[25] |
W. Huang and Y. Yi, Almost periodically forced circle flows, J. Funct. Anal., 257 (2009), 832-902.
doi: 10.1016/j.jfa.2008.12.005. |
[26] |
G. Iooss, Bifurcation of Maps and Applications, North Holland Math. Studies 36, Amsterdam, 1979. |
[27] |
R. Johnson, Concerning a theorem of Sell, J. Differential Equations, 30 (1978), 324-339.
doi: 10.1016/0022-0396(78)90004-9. |
[28] |
R. Johnson, P. Kloeden and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation, Stoch. Dyn., 2 (2002), 67-92.
doi: 10.1142/S0219493702000297. |
[29] |
R. Johnson and Y. Yi, Hopf bifurcation from non-periodic solutions of differential equations II, J. Differential Equations, 107 (1994), 310-340.
doi: 10.1006/jdeq.1994.1015. |
[30] |
J. Hale and H. Ko\ccak, Dynamics and Bifurcations, Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4426-4. |
[31] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge UK, 1995.
doi: 10.1017/CBO9780511809187. |
[32] |
N. Krylov and N. Bogoliubov, La théorie générale de la measure dans son application à l'étude des systémes dynamiques de la méchanique non linéare, Ann. Math., 38 (1937), 65-113. |
[33] |
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-1-4757-2421-9. |
[34] |
N. Levinson, Small periodic perturbations of an autonomous system with a stable orbit, Ann. Math., 52 (1950), 727-738.
doi: 10.2307/1969445. |
[35] |
Y. Neimark, On some cases of periodic motions depending on parameters, Dokl. Akad. Nank. S.S.S.R., 129 (1959), 736-739. |
[36] |
V. Nemytskii and V. Stepanov, Qualitative Theory of Ordinary Differential Equations, Princeton Univ. Press, Princeton USA, 1960. |
[37] |
D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192.
doi: 10.1007/BF01646553. |
[38] |
R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, Ph. D. thesis, IMM-NYU 333, Courant Istitute, New York University, 1964. |
[39] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. |
[40] |
Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187.
doi: 10.1006/jdeq.1993.1026. |
show all references
References:
[1] |
J. Aliste-Prieto and T. Jäger, Almost periodic structures and the semi-conjugacy problem, J. Differential Equations, 252 (2012), 4988-5001.
doi: 10.1016/j.jde.2012.01.030. |
[2] |
V. Anagnostopoulou, T. Jäger and G. Keller, A model for the non-autonomous Hopf bifurcation,, preprint, ().
|
[3] |
L. Arnold, Random Dynamical Systems, in Dynamical Systems, Fondazione C.I.M.E. 1994, ed. R. Johnson, Lecture Notes in Math., Springer-Verlag, Berlin, 1609 (1995), 1-43.
doi: 10.1007/BFb0095238. |
[4] |
M. Bebutov, On dynamical systems in the space of continuous functions, Bull. Moscow Univ. Matematica, (1941), 1-52. |
[5] |
K. Bjerkäv and R. Johnson, Minimal subsets of projective flows, Discrete Contin. Dyn. Syst., 9 (2008), 493-516.
doi: 10.3934/dcdsb.2008.9.493. |
[6] |
R. Botts, A. Homburg and T. Young, The Hopf bifurcation with bounded noise, Discrete Contin. Dyn. Syst., 32 (2012), 2997-3007.
doi: 10.3934/dcds.2012.32.2997. |
[7] |
R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.
doi: 10.1090/S0002-9947-1971-0274707-X. |
[8] |
B. Braaksma and H. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. H. Poincare Anal. Non Lineaire, 4 (1987), 115-168. |
[9] |
B. Braaksma, H. Broer and G. Huitema, Toward a quasi-periodic bifurcation theory, Memoirs A.M.S., 83 (1990), 83-167. |
[10] |
H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos, Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996. |
[11] |
H. Broer, KAM-theory: Multiperiodicity in conservative and dissipative systems, Nieuw Archief v. Wiskunde, 14 (1996), 65-79. |
[12] |
L. Chierchia and C. Falcolini, Compensations in small divisors problems, Comm. Math. Phys., 175 (1996), 135-160.
doi: 10.1007/BF02101627. |
[13] |
E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw Hill, New York, 1955. |
[14] |
W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 377, Springer-Verlag, Berlin, 1978. |
[15] |
S. Diliberto, Perturbation theorems for periodic surfaces I, Rend. Circ. Math. Palermo, 9 (1960), 265-299.
doi: 10.1007/BF02851248. |
[16] |
S. Diliberto, New results in periodic surfaces and the averaging principle, U.S.-Japanese seminar on Differential Equations, W.A. Benjamin Co., New York, 1967, 49-87. |
[17] |
R. Ellis, Lectures on Topological Dynamics, W.A. Benjamin Co., New York, 1969. |
[18] |
B. Fayad, Weak mixing for reparametrized linear flows on the torus, Ergodic Theory Dynam. Systems, 22 (2002), 187-201.
doi: 10.1017/S0143385702000081. |
[19] |
B. Fayad, Analytic mixing reparametrizations of irrational flows, Ergodic Theory Dynam. Systems, 22 (2002), 437-468.
doi: 10.1017/S0143385702000214. |
[20] |
H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. Jour. Math., 83 (1961), 573-601.
doi: 10.2307/2372899. |
[21] |
A. Gonzalez-Enriquez, A non-perturbative theorem on conjugation of torus diffeomorphisms to rigid rotations, preprint, 2005. |
[22] |
A. Gonzalez-Enriquez and J. Vano, Estimate of smoothing and composition with applications to conjugation problems, J. Dynam. Differential Equations, 20 (2008), 239-270.
doi: 10.1007/s10884-006-9060-z. |
[23] |
W. Gottschalk and G. Hedlund, Topological Dynamics, AMS Colloquium Publications 36, Amer. Math. Soc., Providence USA, 1955. |
[24] |
M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583 Springer-Verlag, New York, 1977. |
[25] |
W. Huang and Y. Yi, Almost periodically forced circle flows, J. Funct. Anal., 257 (2009), 832-902.
doi: 10.1016/j.jfa.2008.12.005. |
[26] |
G. Iooss, Bifurcation of Maps and Applications, North Holland Math. Studies 36, Amsterdam, 1979. |
[27] |
R. Johnson, Concerning a theorem of Sell, J. Differential Equations, 30 (1978), 324-339.
doi: 10.1016/0022-0396(78)90004-9. |
[28] |
R. Johnson, P. Kloeden and R. Pavani, Two-step transition in nonautonomous bifurcations: an explanation, Stoch. Dyn., 2 (2002), 67-92.
doi: 10.1142/S0219493702000297. |
[29] |
R. Johnson and Y. Yi, Hopf bifurcation from non-periodic solutions of differential equations II, J. Differential Equations, 107 (1994), 310-340.
doi: 10.1006/jdeq.1994.1015. |
[30] |
J. Hale and H. Ko\ccak, Dynamics and Bifurcations, Texts in Applied Mathematics, 3, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4426-4. |
[31] |
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press, Cambridge UK, 1995.
doi: 10.1017/CBO9780511809187. |
[32] |
N. Krylov and N. Bogoliubov, La théorie générale de la measure dans son application à l'étude des systémes dynamiques de la méchanique non linéare, Ann. Math., 38 (1937), 65-113. |
[33] |
Y. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, Berlin, 1995.
doi: 10.1007/978-1-4757-2421-9. |
[34] |
N. Levinson, Small periodic perturbations of an autonomous system with a stable orbit, Ann. Math., 52 (1950), 727-738.
doi: 10.2307/1969445. |
[35] |
Y. Neimark, On some cases of periodic motions depending on parameters, Dokl. Akad. Nank. S.S.S.R., 129 (1959), 736-739. |
[36] |
V. Nemytskii and V. Stepanov, Qualitative Theory of Ordinary Differential Equations, Princeton Univ. Press, Princeton USA, 1960. |
[37] |
D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys., 20 (1971), 167-192.
doi: 10.1007/BF01646553. |
[38] |
R. Sacker, On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations, Ph. D. thesis, IMM-NYU 333, Courant Istitute, New York University, 1964. |
[39] |
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970. |
[40] |
Y. Yi, A generalized integral manifold theorem, J. Differential Equations, 102 (1993), 153-187.
doi: 10.1006/jdeq.1993.1026. |
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