On cyclicity-one elliptic islands of the standard map

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April 2013, 7(2): 153-208. doi: 10.3934/jmd.2013.7.153

On cyclicity-one elliptic islands of the standard map

Jacopo De Simoi ^1,

1.	Department of Mathematics, University of Toronto, 40 St George St., Toronto, ON M5S 2E4, Canada

Received March 2012 Published September 2013

Abstract
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We study the abundance of a special class of elliptic islands for the standard family of area-preserving diffeomorphism for large parameter values, i.e., far from the KAM regime. Outside a bounded set of parameter values, we prove that the measure of the set of parameter values for which an infinite number of such elliptic islands coexist is zero. On the other hand, we construct a positive Hausdorff dimension set of arbitrarily large parameter values for which the associated standard map admits infinitely many elliptic islands whose centers accumulate on a locally maximal hyperbolic set.

Keywords: Standard Map, elliptic islands, conservative Newhouse phenomenon..

Mathematics Subject Classification: Primary: 37D25; Secondary: 37E40, 37D4.

Citation: Jacopo De Simoi. On cyclicity-one elliptic islands of the standard map. Journal of Modern Dynamics, 2013, 7 (2) : 153-208. doi: 10.3934/jmd.2013.7.153

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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