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On cyclicity-one elliptic islands of the standard map
1. | Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON M5S 2E4, Canada |
References:
[1] |
S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Physica D. Nonlinear Phenomena, 8 (1983), 381-422.
doi: 10.1016/0167-2789(83)90233-6. |
[2] |
Lennart Carleson, Stochastic models of some dynamical systems, in "Geometric Aspects of Functional Analysis (1989-90),'' Lecture Notes in Math., 1469, Springer, Berlin, (1991), 1-12.
doi: 10.1007/BFb0089212. |
[3] |
Boris V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52 (1979), 264-379.
doi: 10.1016/0370-1573(79)90023-1. |
[4] |
R. de la Llave, J. 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. |
[5] |
P. Duarte, Elliptic isles in families of area-preserving maps, Ergodic Theory Dynam. Systems, 28 (2008), 1781-1813.
doi: 10.1017/S0143385707000983. |
[6] |
Pedro Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 359-409. |
[7] |
Pedro Duarte, Abundance of elliptic isles at conservative bifurcations, Dynam. Stability Systems, 14 (1999), 339-356.
doi: 10.1080/026811199281930. |
[8] |
Kenneth J. Falconer, "The Geometry of Fractal Sets,'' Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986. |
[9] |
E. Fontich, Transversal homoclinic points of a class of conservative diffeomorphisms, J. Differential Equations, 87 (1990), 1-27.
doi: 10.1016/0022-0396(90)90012-E. |
[10] |
J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137-149. |
[11] |
V. G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Communications in Mathematical Physics, 201 (1999), 155-216.
doi: 10.1007/s002200050553. |
[12] |
V. G. Gelfreich and V. F. Lazutkin, Splitting of separatrices: Perturbation theory and exponential smallness, Uspekhi Mat. Nauk, 56 (2001), 79-142; translation in Russian Math. Surveys, 56 (2001), 499-558.
doi: 10.1070/RM2001v056n03ABEH000394. |
[13] |
S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, On models with nonrough Poincaré homoclinic curves, Homoclinic chaos (Brussels, 1991), Physica D. Nonlinear Phenomena, 62 (1993), 1-14.
doi: 10.1016/0167-2789(93)90268-6. |
[14] |
S. V. Gonchenko, D. V. Turaev, P. Gaspard and G. Nicolis, Complexity in the bifurcation structure of homoclinic loops to a saddle-focus, Nonlinearity, 10 (1997), 409-423.
doi: 10.1088/0951-7715/10/2/006. |
[15] |
A. Gorodetski, On stochastic sea of the standard map, Communications in Mathematical Physics, 309 (2010), 155-192.
doi: 10.1007/s00220-011-1365-z. |
[16] |
A. Gorodetski and V. Kaloshin, How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency, Adv. Math., 208 (2007), 710-797.
doi: 10.1016/j.aim.2006.03.012. |
[17] |
Daniel L. Goroff, Hyperbolic sets for twist maps, Ergodic Theory Dynam. Systems, 5 (1985), 337-339.
doi: 10.1017/S0143385700002996. |
[18] |
F. M. Izraelev, Nearly linear mappings and their applications, Phys. D, 1 (1980), 243-266.
doi: 10.1016/0167-2789(80)90025-1. |
[19] |
Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. |
[20] |
Oliver Knill, Topological entropy of standard type monotone twist maps, Trans. Amer. Math. Soc., 348 (1996), 2999-3013.
doi: 10.1090/S0002-9947-96-01728-X. |
[21] |
Vladimir F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' With an addendum by A. I. Shnirel'man, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 24, Springer-Verlag, Berlin, 1993. |
[22] |
Carlangelo Liverani, Birth of an elliptic island in a chaotic sea, Math. Phys. Electron. J., 10 (2004), 13 pp. (electronic). |
[23] |
Leonardo Mora and Neptalí Romero, Moser's invariant curves and homoclinic bifurcations, Dynamic Systems and Applications, 6 (1997), 29-41. |
[24] |
Ya. B. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112, 287. |
[25] |
T. Y. Petrosky, Chaos and cometary clouds in the solar system, Physics Letters A, 117 (1986), 328-332.
doi: 10.1016/0375-9601(86)90673-0. |
[26] |
Feliks Przytycki, Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour, Ergodic Theory Dynam. Systems, 2 (1982), 439-463 (1983).
doi: 10.1017/S0143385700001711. |
[27] |
L. D. Pustyl'nikov, Stable and oscillating motions in nonautonomous dynamical systems. A generalization of C. L. Siegel's theorem to the nonautonomous case, Mat. Sb. (N. S.), 94(136) (1974), 407-429, 495. |
[28] |
Ya. G. Sinaĭ, "Topics in Ergodic Theory,'' Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994. |
[29] |
Laura Tedeschini-Lalli and James A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635-657.
doi: 10.1007/BF01463400. |
[30] |
Maciej Wojtkowski, A model problem with the coexistence of stochastic and integrable behaviour, Comm. Math. Phys., 80 (1981), 453-464.
doi: 10.1007/BF01941656. |
show all references
References:
[1] |
S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Physica D. Nonlinear Phenomena, 8 (1983), 381-422.
doi: 10.1016/0167-2789(83)90233-6. |
[2] |
Lennart Carleson, Stochastic models of some dynamical systems, in "Geometric Aspects of Functional Analysis (1989-90),'' Lecture Notes in Math., 1469, Springer, Berlin, (1991), 1-12.
doi: 10.1007/BFb0089212. |
[3] |
Boris V. Chirikov, A universal instability of many-dimensional oscillator systems, Phys. Rep., 52 (1979), 264-379.
doi: 10.1016/0370-1573(79)90023-1. |
[4] |
R. de la Llave, J. 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. |
[5] |
P. Duarte, Elliptic isles in families of area-preserving maps, Ergodic Theory Dynam. Systems, 28 (2008), 1781-1813.
doi: 10.1017/S0143385707000983. |
[6] |
Pedro Duarte, Plenty of elliptic islands for the standard family of area preserving maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 359-409. |
[7] |
Pedro Duarte, Abundance of elliptic isles at conservative bifurcations, Dynam. Stability Systems, 14 (1999), 339-356.
doi: 10.1080/026811199281930. |
[8] |
Kenneth J. Falconer, "The Geometry of Fractal Sets,'' Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986. |
[9] |
E. Fontich, Transversal homoclinic points of a class of conservative diffeomorphisms, J. Differential Equations, 87 (1990), 1-27.
doi: 10.1016/0022-0396(90)90012-E. |
[10] |
J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning, Acad. Sci. U.S.S.R. J. Phys., 1 (1939), 137-149. |
[11] |
V. G. Gelfreich, A proof of the exponentially small transversality of the separatrices for the standard map, Communications in Mathematical Physics, 201 (1999), 155-216.
doi: 10.1007/s002200050553. |
[12] |
V. G. Gelfreich and V. F. Lazutkin, Splitting of separatrices: Perturbation theory and exponential smallness, Uspekhi Mat. Nauk, 56 (2001), 79-142; translation in Russian Math. Surveys, 56 (2001), 499-558.
doi: 10.1070/RM2001v056n03ABEH000394. |
[13] |
S. V. Gonchenko, L. P. Shil'nikov and D. V. Turaev, On models with nonrough Poincaré homoclinic curves, Homoclinic chaos (Brussels, 1991), Physica D. Nonlinear Phenomena, 62 (1993), 1-14.
doi: 10.1016/0167-2789(93)90268-6. |
[14] |
S. V. Gonchenko, D. V. Turaev, P. Gaspard and G. Nicolis, Complexity in the bifurcation structure of homoclinic loops to a saddle-focus, Nonlinearity, 10 (1997), 409-423.
doi: 10.1088/0951-7715/10/2/006. |
[15] |
A. Gorodetski, On stochastic sea of the standard map, Communications in Mathematical Physics, 309 (2010), 155-192.
doi: 10.1007/s00220-011-1365-z. |
[16] |
A. Gorodetski and V. Kaloshin, How often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency, Adv. Math., 208 (2007), 710-797.
doi: 10.1016/j.aim.2006.03.012. |
[17] |
Daniel L. Goroff, Hyperbolic sets for twist maps, Ergodic Theory Dynam. Systems, 5 (1985), 337-339.
doi: 10.1017/S0143385700002996. |
[18] |
F. M. Izraelev, Nearly linear mappings and their applications, Phys. D, 1 (1980), 243-266.
doi: 10.1016/0167-2789(80)90025-1. |
[19] |
Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. |
[20] |
Oliver Knill, Topological entropy of standard type monotone twist maps, Trans. Amer. Math. Soc., 348 (1996), 2999-3013.
doi: 10.1090/S0002-9947-96-01728-X. |
[21] |
Vladimir F. Lazutkin, "KAM Theory and Semiclassical Approximations to Eigenfunctions,'' With an addendum by A. I. Shnirel'man, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 24, Springer-Verlag, Berlin, 1993. |
[22] |
Carlangelo Liverani, Birth of an elliptic island in a chaotic sea, Math. Phys. Electron. J., 10 (2004), 13 pp. (electronic). |
[23] |
Leonardo Mora and Neptalí Romero, Moser's invariant curves and homoclinic bifurcations, Dynamic Systems and Applications, 6 (1997), 29-41. |
[24] |
Ya. B. Pesin, Characteristic Lyapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk, 32 (1977), 55-112, 287. |
[25] |
T. Y. Petrosky, Chaos and cometary clouds in the solar system, Physics Letters A, 117 (1986), 328-332.
doi: 10.1016/0375-9601(86)90673-0. |
[26] |
Feliks Przytycki, Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour, Ergodic Theory Dynam. Systems, 2 (1982), 439-463 (1983).
doi: 10.1017/S0143385700001711. |
[27] |
L. D. Pustyl'nikov, Stable and oscillating motions in nonautonomous dynamical systems. A generalization of C. L. Siegel's theorem to the nonautonomous case, Mat. Sb. (N. S.), 94(136) (1974), 407-429, 495. |
[28] |
Ya. G. Sinaĭ, "Topics in Ergodic Theory,'' Princeton Mathematical Series, 44, Princeton University Press, Princeton, NJ, 1994. |
[29] |
Laura Tedeschini-Lalli and James A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635-657.
doi: 10.1007/BF01463400. |
[30] |
Maciej Wojtkowski, A model problem with the coexistence of stochastic and integrable behaviour, Comm. Math. Phys., 80 (1981), 453-464.
doi: 10.1007/BF01941656. |
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