
-
Previous Article
Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics
- DCDS Home
- This Issue
-
Next Article
Breathers as metastable states for the discrete NLS equation
Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case
Dipartimento di Architettura, Università Roma Tre, Via della Madonna dei Monti 40, I-00184 Rome, Italy |
Hénon map is a well-studied classical example of area-contracting maps, modelling dissipative dynamics. The rich phenomena of coexistence of stable islands and their separatrices is typical of area-preserving maps, modelling conservative dynamics. In this paper we use the Hénon map to ascertain that coexistence of sinks is greater and greater approaching the conservative case, and that part of it can be organized following a renormalization argument. Using a numerical continuation that we devised, and called "dribbling method" [
References:
[1] |
K. Banerjee,
On the widths of the Arnol'd tongues, Ergodic Theory and Dynamical Systems, 34 (2014), 1451-1463.
doi: 10.1017/etds.2013.11. |
[2] |
R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative Standard map, Chaos, 20 (2010), 013121, 9pp.
doi: 10.1063/1.3335408. |
[3] |
R. Calleja, A. Celletti, C. Falcolini and R. de la Llave,
An extension of Greene's criterion for conformally symplectic systems and a partial justification, SIAM Journal on Mathematical Analysis, 46 (2014), 2350-2384.
doi: 10.1137/130929369. |
[4] |
C. Falcolini and R. de la Llave,
A rigorous partial justification of Greene's criterion, Journal of Statistical Physics, 67 (1992), 609-643.
doi: 10.1007/BF01049722. |
[5] |
C. Falcolini and L. Tedeschini-Lalli, Hénon map: simple sinks gaining coexistence as $b \to 1$, International Journal of Bifurcation and Chaos, 23 (2013), 1330030, 13 pp.
doi: 10.1142/S0218127413300309. |
[6] |
C. Falcolini and L. Tedeschini-Lalli, Backbones in the parameter plane of the Hénon map, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 013104, 9pp.
doi: 10.1063/1.4939862. |
[7] |
S. V. Gonchenko, Yu. A. Kuznetsov and H. G. E. Meijer,
Generalized Hénon map and bifurcation of homoclinic tangencies, SIAM Journal on Applied Dynamical Systems, 4 (2005), 407-436.
doi: 10.1137/04060487X. |
[8] |
S.V. Gonchenko, I. Ovsyannikov and D. Turaev,
On the effect of invisibility of stable periodic orbits at homoclinic bifurcations, Physica D Nonlinear Phenomena, 241 (2012), 1115-1122.
doi: 10.1016/j.physd.2012.03.002. |
[9] |
M. Hénon,
A two dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77.
doi: 10.1007/BF01608556. |
[10] |
P. Holmes and D. Whitley, Bifurcations of one- and two-dimensional maps, Phil. Trans. Roy. Soc. Lond. Ser. A, 311 (1984), 43–102. Erratum: Bifurcations of one- and two-dimensional maps, Phil. Trans. Roy. Soc. Lond., 312 (1984), 601–602.
doi: 10.1098/rsta.1984.0020. |
[11] |
C. A. Jousseph, A. Abdulack, C. Manchein and M. W. Beims, Hierarchical collapse of regular islands via dissipation, J. Phys. A: Math. Theor., 51 (2018), 105101, 17pp. |
[12] |
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2010. |
[13] |
N. Miguel, C. Simó and A. Vieiro,
From the Hénon conservative map to the Chirikov standard map for large parameter values, Regul. Chaot. Dyn., 18 (2013), 469-489.
doi: 10.1134/S1560354713050018. |
[14] |
G. Schmidt and B. W. Wang,
Dissipative standard map, Phys. Rev. A, 32 (1985), 2994-2999.
doi: 10.1103/PhysRevA.32.2994. |
[15] |
W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map, Phys. Rev. A, 43 (1991), 6550.
doi: 10.1103/PhysRevA.43.6550. |
[16] |
J. A. Yorke and L. Tedeschini-Lalli,
How often do simple dynamical systems have infinitely many coexisting sinks, Comm. Math. Phys., 106 (1986), 635-657.
doi: 10.1007/BF01463400. |
[17] |
J. A. Yorke and K. T. Alligood,
Period doubling cascades of attractors: A prerequisite for horseshoes, Comm. Math. Phys., 101 (1985), 305-321.
doi: 10.1007/BF01216092. |
show all references
References:
[1] |
K. Banerjee,
On the widths of the Arnol'd tongues, Ergodic Theory and Dynamical Systems, 34 (2014), 1451-1463.
doi: 10.1017/etds.2013.11. |
[2] |
R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative Standard map, Chaos, 20 (2010), 013121, 9pp.
doi: 10.1063/1.3335408. |
[3] |
R. Calleja, A. Celletti, C. Falcolini and R. de la Llave,
An extension of Greene's criterion for conformally symplectic systems and a partial justification, SIAM Journal on Mathematical Analysis, 46 (2014), 2350-2384.
doi: 10.1137/130929369. |
[4] |
C. Falcolini and R. de la Llave,
A rigorous partial justification of Greene's criterion, Journal of Statistical Physics, 67 (1992), 609-643.
doi: 10.1007/BF01049722. |
[5] |
C. Falcolini and L. Tedeschini-Lalli, Hénon map: simple sinks gaining coexistence as $b \to 1$, International Journal of Bifurcation and Chaos, 23 (2013), 1330030, 13 pp.
doi: 10.1142/S0218127413300309. |
[6] |
C. Falcolini and L. Tedeschini-Lalli, Backbones in the parameter plane of the Hénon map, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 013104, 9pp.
doi: 10.1063/1.4939862. |
[7] |
S. V. Gonchenko, Yu. A. Kuznetsov and H. G. E. Meijer,
Generalized Hénon map and bifurcation of homoclinic tangencies, SIAM Journal on Applied Dynamical Systems, 4 (2005), 407-436.
doi: 10.1137/04060487X. |
[8] |
S.V. Gonchenko, I. Ovsyannikov and D. Turaev,
On the effect of invisibility of stable periodic orbits at homoclinic bifurcations, Physica D Nonlinear Phenomena, 241 (2012), 1115-1122.
doi: 10.1016/j.physd.2012.03.002. |
[9] |
M. Hénon,
A two dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77.
doi: 10.1007/BF01608556. |
[10] |
P. Holmes and D. Whitley, Bifurcations of one- and two-dimensional maps, Phil. Trans. Roy. Soc. Lond. Ser. A, 311 (1984), 43–102. Erratum: Bifurcations of one- and two-dimensional maps, Phil. Trans. Roy. Soc. Lond., 312 (1984), 601–602.
doi: 10.1098/rsta.1984.0020. |
[11] |
C. A. Jousseph, A. Abdulack, C. Manchein and M. W. Beims, Hierarchical collapse of regular islands via dissipation, J. Phys. A: Math. Theor., 51 (2018), 105101, 17pp. |
[12] |
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2010. |
[13] |
N. Miguel, C. Simó and A. Vieiro,
From the Hénon conservative map to the Chirikov standard map for large parameter values, Regul. Chaot. Dyn., 18 (2013), 469-489.
doi: 10.1134/S1560354713050018. |
[14] |
G. Schmidt and B. W. Wang,
Dissipative standard map, Phys. Rev. A, 32 (1985), 2994-2999.
doi: 10.1103/PhysRevA.32.2994. |
[15] |
W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map, Phys. Rev. A, 43 (1991), 6550.
doi: 10.1103/PhysRevA.43.6550. |
[16] |
J. A. Yorke and L. Tedeschini-Lalli,
How often do simple dynamical systems have infinitely many coexisting sinks, Comm. Math. Phys., 106 (1986), 635-657.
doi: 10.1007/BF01463400. |
[17] |
J. A. Yorke and K. T. Alligood,
Period doubling cascades of attractors: A prerequisite for horseshoes, Comm. Math. Phys., 101 (1985), 305-321.
doi: 10.1007/BF01216092. |











k | |||||
5 | -0.99999 | 68 | 86 | 144 | |
6 | -0.999999 | 96 | 114 | 188 | |
7 | -0.9999999 | 128 | 146 | 236 | |
8 | -0.99999999 | 164 | 182 | 288 |
k | |||||
5 | -0.99999 | 68 | 86 | 144 | |
6 | -0.999999 | 96 | 114 | 188 | |
7 | -0.9999999 | 128 | 146 | 236 | |
8 | -0.99999999 | 164 | 182 | 288 |
k | |||||
5 | 0.99999 | 20 | 25 | 38 | |
6 | 0.999999 | 25 | 30 | 46 | |
7 | 0.9999999 | 31 | 36 | 55 | |
8 | 0.99999999 | 38 | 43 | 65 |
k | |||||
5 | 0.99999 | 20 | 25 | 38 | |
6 | 0.999999 | 25 | 30 | 46 | |
7 | 0.9999999 | 31 | 36 | 55 | |
8 | 0.99999999 | 38 | 43 | 65 |
-0.999 | 4 | 0.151600 | 20 - 30 | |
-0.9999 | 9 | 0.098750 | 28 - 46 | |
-0.99999 | 16 | 0.076900 | 30 - 74 | |
-0.999999 | 27 | 0.054900 | 42 - 98 | |
-0.9999999 | 39 | 0.045000 | 46 - 134 | |
-0.99999999 | 54 | 0.035794 | 54 - 166 |
-0.999 | 4 | 0.151600 | 20 - 30 | |
-0.9999 | 9 | 0.098750 | 28 - 46 | |
-0.99999 | 16 | 0.076900 | 30 - 74 | |
-0.999999 | 27 | 0.054900 | 42 - 98 | |
-0.9999999 | 39 | 0.045000 | 46 - 134 | |
-0.99999999 | 54 | 0.035794 | 54 - 166 |
0.99999 | 10 | -0.93294 | 11 - 20 | |
0.999999 | 14 | -0.95400 | 12 - 25 | |
0.9999999 | 18 | -0.96490 | 12 - 29 | |
0.99999999 | 23 | -0.97420 | 13 - 35 |
0.99999 | 10 | -0.93294 | 11 - 20 | |
0.999999 | 14 | -0.95400 | 12 - 25 | |
0.9999999 | 18 | -0.96490 | 12 - 29 | |
0.99999999 | 23 | -0.97420 | 13 - 35 |
first eigenvalue | accuracy | |||
42 | -0.02471212033 | -0.004353351966 | 0.999666+0.025007 i | 980 |
44 | 0.003893968151 | 0.01400370243 | 0.998106+0.061164 i | 981 |
46 | 0.03287883220 | 0.03630238615 | 0.996571+0.082467 i | 981 |
48 | 0.06028145342 | 0.05973990689 | 0.994655+0.103021 i | 980 |
50 | 0.08552343060 | 0.08300646446 | 0.992221+0.124292 i | 980 |
52 | 0.1082722485 | 0.1051821971 | 0.989124+0.146904 i | 979 |
54 | 0.1283734421 | 0.1256447760 | 0.985195+0.171279 i | 979 |
56 | 0.1458237800 | 0.1440274072 | 0.980218+0.197778 i | 979 |
58 | 0.1607392037 | 0.1601751773 | 0.973924+0.226748 i | 979 |
60 | 0.1733182271 | 0.1740957680 | 0.965971+0.258534 i | 979 |
62 | 0.1838069083 | 0.1859098749 | 0.955930+0.293489 i | 979 |
64 | 0.1924697936 | 0.1958070451 | 0.943258+0.331965 i | 978 |
66 | 0.1995685673 | 0.2040103236 | 0.927268+0.374309 i | 978 |
68 | 0.2053481559 | 0.2107505750 | 0.907097+0.420841 i | 978 |
70 | 0.2100289888 | 0.2162495981 | 0.881651+0.471827 i | 978 |
72 | 0.2138038071 | 0.2207103044 | 0.849555+0.527432 i | 978 |
74 | 0.2168375173 | 0.2243120570 | 0.809070+0.587650 i | 979 |
76 | 0.2192688855 | 0.2272094757 | 0.758004+0.652192 i | 979 |
78 | 0.2212131915 | 0.2295333706 | 0.693593+0.720313 i | 979 |
80 | 0.2227652522 | 0.2313928385 | 0.612348+0.790538 i | 979 |
82 | 0.2240024479 | 0.2328778729 | 0.509870+0.860204 i | 978 |
84 | 0.2249875438 | 0.2340620792 | 0.380610+0.924690 i | 977 |
86 | 0.2257712060 | 0.2350052607 | 0.217565+0.976002 i | 977 |
88 | 0.2263941791 | 0.2357557547 | 0.011907+0.999885 i | 976 |
90 | 0.2268891314 | 0.2363524720 | -0.247504+0.968841 i | 975 |
92 | 0.2272821942 | 0.2368266348 | -0.574717+0.818296 i | 975 |
94 | 0.2275942304 | 0.2372032327 | -0.987457+0.157593 i | 974 |
96 | 0.2278418727 | 0.2375022270 | -0.379187 | 974 |
98 | 0.2280383657 | 0.2377395373 | -0.244788 | 973 |
first eigenvalue | accuracy | |||
42 | -0.02471212033 | -0.004353351966 | 0.999666+0.025007 i | 980 |
44 | 0.003893968151 | 0.01400370243 | 0.998106+0.061164 i | 981 |
46 | 0.03287883220 | 0.03630238615 | 0.996571+0.082467 i | 981 |
48 | 0.06028145342 | 0.05973990689 | 0.994655+0.103021 i | 980 |
50 | 0.08552343060 | 0.08300646446 | 0.992221+0.124292 i | 980 |
52 | 0.1082722485 | 0.1051821971 | 0.989124+0.146904 i | 979 |
54 | 0.1283734421 | 0.1256447760 | 0.985195+0.171279 i | 979 |
56 | 0.1458237800 | 0.1440274072 | 0.980218+0.197778 i | 979 |
58 | 0.1607392037 | 0.1601751773 | 0.973924+0.226748 i | 979 |
60 | 0.1733182271 | 0.1740957680 | 0.965971+0.258534 i | 979 |
62 | 0.1838069083 | 0.1859098749 | 0.955930+0.293489 i | 979 |
64 | 0.1924697936 | 0.1958070451 | 0.943258+0.331965 i | 978 |
66 | 0.1995685673 | 0.2040103236 | 0.927268+0.374309 i | 978 |
68 | 0.2053481559 | 0.2107505750 | 0.907097+0.420841 i | 978 |
70 | 0.2100289888 | 0.2162495981 | 0.881651+0.471827 i | 978 |
72 | 0.2138038071 | 0.2207103044 | 0.849555+0.527432 i | 978 |
74 | 0.2168375173 | 0.2243120570 | 0.809070+0.587650 i | 979 |
76 | 0.2192688855 | 0.2272094757 | 0.758004+0.652192 i | 979 |
78 | 0.2212131915 | 0.2295333706 | 0.693593+0.720313 i | 979 |
80 | 0.2227652522 | 0.2313928385 | 0.612348+0.790538 i | 979 |
82 | 0.2240024479 | 0.2328778729 | 0.509870+0.860204 i | 978 |
84 | 0.2249875438 | 0.2340620792 | 0.380610+0.924690 i | 977 |
86 | 0.2257712060 | 0.2350052607 | 0.217565+0.976002 i | 977 |
88 | 0.2263941791 | 0.2357557547 | 0.011907+0.999885 i | 976 |
90 | 0.2268891314 | 0.2363524720 | -0.247504+0.968841 i | 975 |
92 | 0.2272821942 | 0.2368266348 | -0.574717+0.818296 i | 975 |
94 | 0.2275942304 | 0.2372032327 | -0.987457+0.157593 i | 974 |
96 | 0.2278418727 | 0.2375022270 | -0.379187 | 974 |
98 | 0.2280383657 | 0.2377395373 | -0.244788 | 973 |
[1] |
Alessandra Celletti, Sara Di Ruzza. Periodic and quasi--periodic orbits of the dissipative standard map. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 151-171. doi: 10.3934/dcdsb.2011.16.151 |
[2] |
Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031 |
[3] |
Fernando Lenarduzzi. Recoding the classical Hénon-Devaney map. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4073-4092. doi: 10.3934/dcds.2020172 |
[4] |
Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1 |
[5] |
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 |
[6] |
Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1263-1284. doi: 10.3934/dcdsb.2021089 |
[7] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[8] |
Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032 |
[9] |
Claire Chavaudret, Stefano Marmi. Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3077-3101. doi: 10.3934/dcds.2022009 |
[10] |
Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212 |
[11] |
Wacław Marzantowicz, Piotr Maciej Przygodzki. Finding periodic points of a map by use of a k-adic expansion. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 495-514. doi: 10.3934/dcds.1999.5.495 |
[12] |
Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102 |
[13] |
Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 |
[14] |
Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403 |
[15] |
Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255 |
[16] |
Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1 |
[17] |
John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723 |
[18] |
Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927 |
[19] |
Roberto De Leo, James A. Yorke. The graph of the logistic map is a tower. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5243-5269. doi: 10.3934/dcds.2021075 |
[20] |
Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]