
-
Previous Article
Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother
- JCD Home
- This Issue
-
Next Article
Computer-assisted estimates for Birkhoff normal forms
A numerical renormalization method for quasi–conservative periodic attractors
Dipartimento di Architettura, Università degli Studi Roma Tre, Roma, Italy |
We describe a renormalization method in maps of the plane $ (x, y) $, with constant Jacobian $ b $ and a second parameter $ a $ acting as a bifurcation parameter. The method enables one to organize high period periodic attractors and thus find hordes of them in quasi-conservative maps (i.e. $ |b| = 1-\varepsilon $), when sharing the same rotation number. Numerical challenges are the high period, and the necessary extreme vicinity of many such different points, which accumulate on a hyperbolic periodic saddle. The periodic points are organized, in the $ (x, y, a) $ space, in sequences of diverging period, that we call "branches". We define a renormalization approach, by "hopping" among branches to maximize numerical convergence. Our numerical renormalization has met two kinds of numerical instabilities, well localized in certain ranges of the period for the parameter $ a $ (see [
The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [
References:
[1] |
R. C. 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 J. Math. Anal., 46 (2014), 2350-2384.
doi: 10.1137/130929369. |
[2] |
C. Falcolini and L. Tedeschini-Lalli, Hénon map: Simple sinks gaining coexistence as $b\rightarrow1$, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 13pp.
doi: 10.1142/S0218127413300309. |
[3] |
C. Falcolini and L. Tedeschini-Lalli,
Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case, Discrete Contin. Dyn. Syst., 38 (2018), 6105-6122.
doi: 10.3934/dcds.2018263. |
[4] |
C. Falcolini and L. Tedeschini-Lalli, Quasi-conservative Hénon: Coexisting sequences of coexisting sinks organized by their rotation number, in progress. |
[5] |
N. K. Gavrilov and L. P. Šil'nikov,
Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve, Mat. Sb. (N.S.), 88 (1972), 475-492.
|
[6] |
J. M. Greene,
Method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201.
doi: 10.2172/6191337. |
[7] |
M. Lyubich and M. Martens,
Renormalization in the Hénon family, II: The heteroclinic web, Invent. Math., 186 (2011), 115-189.
doi: 10.1007/s00222-011-0316-9. |
[8] |
L. Tedeschini-Lalli and J. A. Yorke,
How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635-657.
doi: 10.1007/BF01463400. |
show all references
References:
[1] |
R. C. 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 J. Math. Anal., 46 (2014), 2350-2384.
doi: 10.1137/130929369. |
[2] |
C. Falcolini and L. Tedeschini-Lalli, Hénon map: Simple sinks gaining coexistence as $b\rightarrow1$, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 13pp.
doi: 10.1142/S0218127413300309. |
[3] |
C. Falcolini and L. Tedeschini-Lalli,
Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case, Discrete Contin. Dyn. Syst., 38 (2018), 6105-6122.
doi: 10.3934/dcds.2018263. |
[4] |
C. Falcolini and L. Tedeschini-Lalli, Quasi-conservative Hénon: Coexisting sequences of coexisting sinks organized by their rotation number, in progress. |
[5] |
N. K. Gavrilov and L. P. Šil'nikov,
Three-dimensional dynamical systems that are close to systems with a structurally unstable homoclinic curve, Mat. Sb. (N.S.), 88 (1972), 475-492.
|
[6] |
J. M. Greene,
Method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201.
doi: 10.2172/6191337. |
[7] |
M. Lyubich and M. Martens,
Renormalization in the Hénon family, II: The heteroclinic web, Invent. Math., 186 (2011), 115-189.
doi: 10.1007/s00222-011-0316-9. |
[8] |
L. Tedeschini-Lalli and J. A. Yorke,
How often do simple dynamical processes have infinitely many coexisting sinks?, Comm. Math. Phys., 106 (1986), 635-657.
doi: 10.1007/BF01463400. |

6 | -0.443941901199160 | -0.459628709385692 | -0.748750159238484 | ||
7 | -0.520100870902215 | -0.542645306108413 | -0.853977607768101 | ||
8 | -0.556536885257237 | -0.583659098382858 | -0.905284259156612 | 0.058729274542161 | -0.026306068852380 |
9 | -0.573518924659181 | -0.603467164338756 | -0.930955353820444 | 0.080344362906829 | -0.013649730593885 |
10 | -0.581223006893546 | -0.612863160691578 | -0.943971107021642 | 0.128671768726403 | -0.007465151139187 |
11 | -0.584595134560611 | -0.617236330531766 | -0.950616341128393 | 0.241624760699613 | -0.004174918547764 |
12 | -0.585992440180766 | -0.619223030479677 | -0.954022066928013 | 0.610528974899303 | -0.002190468901347 |
13 | -0.586518745810595 | -0.620095336445413 | -0.955771266316709 | ![]() |
-0.000850216542803 |
14 | -0.586679919269898 | -0.620459027321466 | -0.956670676554674 | 0.000121840268831 | |
15 | -0.586700746506923 | -0.620597972499906 | -0.957133341118791 | 0.000852932852074 | |
16 | -0.586676484989993 | -0.620642321779810 | -0.957371324937994 | 0.413586106013437 | 0.001415295083768 |
17 | -0.586644418330550 | -0.620649931914695 | -0.957493684559891 | 0.190305901218847 | 0.001859614403370 |
18 | -0.586617015202294 | -0.620645368625226 | -0.957556550137796 | 0.109832951493248 | 0.002225549011656 |
19 | -0.586596873075783 | -0.620638314972537 | -0.957588818205496 | 0.071664256512660 | 0.002544802357486 |
20 | -0.586583173943061 | -0.620632083909781 | -0.957605362078047 | 0.050499711583506 | 0.002842453889702 |
21 | -0.586574289856734 | -0.620627443753603 | -0.957613833086740 | 0.037509951787786 | 0.003138215394092 |
22 | -0.586568711100965 | -0.620624267704828 | -0.957618164219074 | 0.028949923195199 | 0.003447850331833 |
23 | -0.586565288541358 | -0.620622200775421 | -0.957620375131279 | 0.023005809098210 | 0.003784631126657 |
24 | -0.586563225447445 | -0.620620900146398 | -0.957621501752285 | 0.018709540891981 | 0.004160716244030 |
25 | -0.586561998839785 | -0.620620101158850 | -0.957622074742812 | 0.015504530869589 | 0.004588418968925 |
26 | -0.586561277586088 | -0.620619619108263 | -0.957622365545888 | 0.013051435125914 | 0.005081420311428 |
27 | -0.586560857315490 | -0.620619332326706 | -0.957622512790829 | 0.011133367899055 | 0.005656036360843 |
28 | -0.586560614274431 | -0.620619163618962 | -0.957622587155352 | 0.009606268344908 | 0.006332699555547 |
29 | -0.586560474623738 | -0.620619065278693 | -0.957622624605403 | 0.008371347061941 | 0.007137875304418 |
30 | -0.586560394821339 | -0.620619008392593 | -0.957622643405331 | 0.007359019822839 | 0.008106737165843 |
31 | -0.586560349435936 | -0.620618975698324 | -0.957622652809200 | 0.006519140961599 | 0.009287105370122 |
32 | -0.586560323731824 | -0.620618957011726 | -0.957622657494091 | 0.005814851668130 | 0.010745485981220 |
33 | -0.586560309227800 | -0.620618946382443 | -0.957622659817296 | 0.005218585512918 | 0.012576672031927 |
34 | -0.586560301070400 | -0.620618940361643 | -0.957622660963252 | 0.004709405818352 | 0.014919580168340 |
35 | -0.586560296495915 | -0.620618936963855 | -0.957622661525026 | 0.004271189936352 | 0.017984458981884 |
36 | -0.586560293937405 | -0.620618935052642 | -0.957622661798422 | 0.003891366100283 | 0.022101890826913 |
37 | -0.586560292509839 | -0.620618933980766 | -0.957622661930319 | 0.003560019024118 | 0.027816130104866 |
38 | -0.586560291715030 | -0.620618933381211 | -0.957622661993280 | 0.003269246510245 | 0.036075439595772 |
39 | -0.586560291273389 | -0.620618933046650 | -0.957622662022941 | 0.003012689961575 | 0.048654717261140 |
40 | -0.586560291028433 | -0.620618932860367 | -0.957622662036681 | 0.002785187299874 | 0.069202823204726 |
41 | -0.586560290892795 | -0.620618932756850 | -0.957622662042905 | 0.002582513285418 | 0.106252407643879 |
42 | -0.586560290817804 | -0.620618932699431 | -0.957622662045639 | 0.002401183065501 | 0.183931206227178 |
43 | -0.586560290776403 | -0.620618932667634 | -0.957622662046787 | 0.002238302019191 | 0.395871858811726 |
44 | -0.586560290753577 | -0.620618932650054 | -0.957622662047233 | 0.002091449881007 | ![]() |
45 | -0.586560290741007 | -0.620618932640348 | -0.957622662047384 | 0.001958590511266 | |
46 | -0.586560290734093 | -0.620618932634996 | -0.957622662047418 | 0.001838001042586 | |
47 | -0.586560290730294 | -0.620618932632049 | -0.957622662047412 | 0.001728215799801 | 0.656599165556123 |
48 | -0.586560290728209 | -0.620618932630427 | -0.957622662047395 | 0.001627981581417 | 0.254309960732174 |
49 | -0.586560290727065 | -0.620618932629537 | -0.957622662047380 | 0.001536221750036 | 0.135036401821108 |
50 | -0.586560290726439 | -0.620618932629048 | -0.957622662047367 | 0.001452007205293 | 0.083733839431576 |
51 | -0.586560290726096 | -0.620618932628780 | -0.957622662047359 | 0.001374532773379 | 0.057000237694008 |
52 | -0.586560290725908 | -0.620618932628633 | -0.957622662047354 | 0.001303097888832 | 0.041307018277047 |
53 | -0.586560290725806 | -0.620618932628552 | -0.957622662047350 | 0.001237090699874 | 0.031310539999293 |
54 | -0.586560290725750 | -0.620618932628508 | -0.957622662047348 | 0.001175974921344 | 0.024551153473680 |
55 | -0.586560290725719 | -0.620618932628484 | -0.957622662047347 | 0.001119278905643 | 0.019767365058109 |
56 | -0.586560290725702 | -0.620618932628471 | -0.957622662047346 | 0.001066586514221 | 0.016257632487938 |
57 | -0.586560290725693 | -0.620618932628464 | -0.957622662047345 | 0.001017529458438 | 0.013606423077801 |
58 | -0.586560290725688 | -0.620618932628460 | -0.957622662047345 | 0.000971780845602 | 0.011554839714071 |
59 | -0.586560290725686 | -0.620618932628458 | -0.957622662047345 | 0.000929049718241 | 0.009934735217316 |
60 | -0.586560290725684 | -0.620618932628457 | -0.957622662047345 | 0.000889076415690 | 0.008633040932453 |
... | ... | ... | ... | ... | ... |
130 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000139735759219 | 0.000261626056302 |
131 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000137356944377 | 0.000255560371803 |
132 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000135038361076 | 0.000249703215311 |
133 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000132777992920 | 0.000244045136664 |
6 | -0.443941901199160 | -0.459628709385692 | -0.748750159238484 | ||
7 | -0.520100870902215 | -0.542645306108413 | -0.853977607768101 | ||
8 | -0.556536885257237 | -0.583659098382858 | -0.905284259156612 | 0.058729274542161 | -0.026306068852380 |
9 | -0.573518924659181 | -0.603467164338756 | -0.930955353820444 | 0.080344362906829 | -0.013649730593885 |
10 | -0.581223006893546 | -0.612863160691578 | -0.943971107021642 | 0.128671768726403 | -0.007465151139187 |
11 | -0.584595134560611 | -0.617236330531766 | -0.950616341128393 | 0.241624760699613 | -0.004174918547764 |
12 | -0.585992440180766 | -0.619223030479677 | -0.954022066928013 | 0.610528974899303 | -0.002190468901347 |
13 | -0.586518745810595 | -0.620095336445413 | -0.955771266316709 | ![]() |
-0.000850216542803 |
14 | -0.586679919269898 | -0.620459027321466 | -0.956670676554674 | 0.000121840268831 | |
15 | -0.586700746506923 | -0.620597972499906 | -0.957133341118791 | 0.000852932852074 | |
16 | -0.586676484989993 | -0.620642321779810 | -0.957371324937994 | 0.413586106013437 | 0.001415295083768 |
17 | -0.586644418330550 | -0.620649931914695 | -0.957493684559891 | 0.190305901218847 | 0.001859614403370 |
18 | -0.586617015202294 | -0.620645368625226 | -0.957556550137796 | 0.109832951493248 | 0.002225549011656 |
19 | -0.586596873075783 | -0.620638314972537 | -0.957588818205496 | 0.071664256512660 | 0.002544802357486 |
20 | -0.586583173943061 | -0.620632083909781 | -0.957605362078047 | 0.050499711583506 | 0.002842453889702 |
21 | -0.586574289856734 | -0.620627443753603 | -0.957613833086740 | 0.037509951787786 | 0.003138215394092 |
22 | -0.586568711100965 | -0.620624267704828 | -0.957618164219074 | 0.028949923195199 | 0.003447850331833 |
23 | -0.586565288541358 | -0.620622200775421 | -0.957620375131279 | 0.023005809098210 | 0.003784631126657 |
24 | -0.586563225447445 | -0.620620900146398 | -0.957621501752285 | 0.018709540891981 | 0.004160716244030 |
25 | -0.586561998839785 | -0.620620101158850 | -0.957622074742812 | 0.015504530869589 | 0.004588418968925 |
26 | -0.586561277586088 | -0.620619619108263 | -0.957622365545888 | 0.013051435125914 | 0.005081420311428 |
27 | -0.586560857315490 | -0.620619332326706 | -0.957622512790829 | 0.011133367899055 | 0.005656036360843 |
28 | -0.586560614274431 | -0.620619163618962 | -0.957622587155352 | 0.009606268344908 | 0.006332699555547 |
29 | -0.586560474623738 | -0.620619065278693 | -0.957622624605403 | 0.008371347061941 | 0.007137875304418 |
30 | -0.586560394821339 | -0.620619008392593 | -0.957622643405331 | 0.007359019822839 | 0.008106737165843 |
31 | -0.586560349435936 | -0.620618975698324 | -0.957622652809200 | 0.006519140961599 | 0.009287105370122 |
32 | -0.586560323731824 | -0.620618957011726 | -0.957622657494091 | 0.005814851668130 | 0.010745485981220 |
33 | -0.586560309227800 | -0.620618946382443 | -0.957622659817296 | 0.005218585512918 | 0.012576672031927 |
34 | -0.586560301070400 | -0.620618940361643 | -0.957622660963252 | 0.004709405818352 | 0.014919580168340 |
35 | -0.586560296495915 | -0.620618936963855 | -0.957622661525026 | 0.004271189936352 | 0.017984458981884 |
36 | -0.586560293937405 | -0.620618935052642 | -0.957622661798422 | 0.003891366100283 | 0.022101890826913 |
37 | -0.586560292509839 | -0.620618933980766 | -0.957622661930319 | 0.003560019024118 | 0.027816130104866 |
38 | -0.586560291715030 | -0.620618933381211 | -0.957622661993280 | 0.003269246510245 | 0.036075439595772 |
39 | -0.586560291273389 | -0.620618933046650 | -0.957622662022941 | 0.003012689961575 | 0.048654717261140 |
40 | -0.586560291028433 | -0.620618932860367 | -0.957622662036681 | 0.002785187299874 | 0.069202823204726 |
41 | -0.586560290892795 | -0.620618932756850 | -0.957622662042905 | 0.002582513285418 | 0.106252407643879 |
42 | -0.586560290817804 | -0.620618932699431 | -0.957622662045639 | 0.002401183065501 | 0.183931206227178 |
43 | -0.586560290776403 | -0.620618932667634 | -0.957622662046787 | 0.002238302019191 | 0.395871858811726 |
44 | -0.586560290753577 | -0.620618932650054 | -0.957622662047233 | 0.002091449881007 | ![]() |
45 | -0.586560290741007 | -0.620618932640348 | -0.957622662047384 | 0.001958590511266 | |
46 | -0.586560290734093 | -0.620618932634996 | -0.957622662047418 | 0.001838001042586 | |
47 | -0.586560290730294 | -0.620618932632049 | -0.957622662047412 | 0.001728215799801 | 0.656599165556123 |
48 | -0.586560290728209 | -0.620618932630427 | -0.957622662047395 | 0.001627981581417 | 0.254309960732174 |
49 | -0.586560290727065 | -0.620618932629537 | -0.957622662047380 | 0.001536221750036 | 0.135036401821108 |
50 | -0.586560290726439 | -0.620618932629048 | -0.957622662047367 | 0.001452007205293 | 0.083733839431576 |
51 | -0.586560290726096 | -0.620618932628780 | -0.957622662047359 | 0.001374532773379 | 0.057000237694008 |
52 | -0.586560290725908 | -0.620618932628633 | -0.957622662047354 | 0.001303097888832 | 0.041307018277047 |
53 | -0.586560290725806 | -0.620618932628552 | -0.957622662047350 | 0.001237090699874 | 0.031310539999293 |
54 | -0.586560290725750 | -0.620618932628508 | -0.957622662047348 | 0.001175974921344 | 0.024551153473680 |
55 | -0.586560290725719 | -0.620618932628484 | -0.957622662047347 | 0.001119278905643 | 0.019767365058109 |
56 | -0.586560290725702 | -0.620618932628471 | -0.957622662047346 | 0.001066586514221 | 0.016257632487938 |
57 | -0.586560290725693 | -0.620618932628464 | -0.957622662047345 | 0.001017529458438 | 0.013606423077801 |
58 | -0.586560290725688 | -0.620618932628460 | -0.957622662047345 | 0.000971780845602 | 0.011554839714071 |
59 | -0.586560290725686 | -0.620618932628458 | -0.957622662047345 | 0.000929049718241 | 0.009934735217316 |
60 | -0.586560290725684 | -0.620618932628457 | -0.957622662047345 | 0.000889076415690 | 0.008633040932453 |
... | ... | ... | ... | ... | ... |
130 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000139735759219 | 0.000261626056302 |
131 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000137356944377 | 0.000255560371803 |
132 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000135038361076 | 0.000249703215311 |
133 | -0.5865602907256825 | -0.6206189326284553 | -0.9576226620473447 | 0.000132777992920 | 0.000244045136664 |
12 | -0.585992440180766 | -0.619223030479677 | -0.954022066928013 | ||
13 | -0.679680789153866 | -0.586518745810595 | -0.955771266316709 | 0.611294033635953 | -0.348337217906535 |
14 | -0.832943120892727 | -0.680405597366770 | -0.956670676554674 | 1.10453178002677 | 0.611827028590535 |
15 | -0.971700859248744 | -0.833858856488816 | -0.957133341118791 | 1.42682237020761 | 1.10614894485762 |
16 | -1.06895034261354 | -0.972586339650497 | -0.957371324937994 | 1.62539036711981 | 1.42786267946187 |
17 | -1.12878180546213 | -1.06974378003823 | -0.957493684559891 | 1.74041613015774 | 1.62585984766665 |
18 | -1.16315948204869 | -1.12950135417794 | -0.957556550137796 | 1.80447633168710 | 1.74057500572435 |
19 | -1.18221081344211 | -1.16383344210474 | -0.957588818205496 | 1.83935532699776 | 1.80451355982745 |
20 | -1.19256842684817 | -1.18285911677470 | -0.957605362078047 | 1.85810857221899 | 1.83936683519620 |
21 | -1.19814270467075 | -1.19320271670588 | -0.957613833086740 | 1.86812125596453 | 1.85813290902146 |
22 | -1.20112659974971 | -1.19876937982291 | -0.957618164219074 | 1.87344594955960 | 1.86816902366844 |
23 | -1.20271933055663 | -1.20174912257985 | -0.957620375131279 | 1.87627085023485 | 1.87351627383757 |
24 | -1.20356821162661 | -1.20333957727649 | -0.957621501752285 | 1.87776722660084 | 1.87635922088281 |
25 | -1.20402028100878 | -1.20418720532070 | -0.957622074742812 | 1.87855899443983 | 1.87786878623627 |
26 | -1.20426092790010 | -1.20463858299438 | -0.957622365545888 | 1.87897756998574 | 1.87866968187920 |
27 | -1.20438900121782 | -1.20487884751668 | -0.957622512790829 | 1.87919868763727 | 1.87909434452461 |
28 | -1.20445715437191 | -1.20500670938961 | -0.957622587155352 | 1.87931541764533 | 1.87931941124216 |
29 | -1.20449341925509 | -1.20507474565432 | -0.957622624605403 | 1.87937700435579 | 1.87943864752752 |
30 | -1.20451271548097 | -1.20511094596593 | -0.957622643405331 | 1.87940948249360 | 1.87950179504452 |
31 | -1.20452298265588 | -1.20513020655494 | -0.957622652809200 | 1.87942660604971 | 1.87953522802574 |
32 | -1.20452844558530 | -1.20514045408248 | -0.957622657494091 | 1.87943563599648 | 1.87955292480737 |
33 | -1.20453135227139 | -1.20514590619155 | -0.957622659817296 | 1.87944040281495 | 1.87956229105465 |
34 | -1.20453289884137 | -1.20514880692492 | -0.957622660963252 | 1.87944292580798 | 1.87956724882028 |
35 | -1.20453372172880 | -1.20515035022345 | -0.957622661525026 | 1.87944426871036 | 1.87956987448293 |
36 | -1.20453415956431 | -1.20515117131478 | -0.957622661798422 | 1.87944499146173 | 1.87957126690649 |
37 | -1.20453439252431 | -1.20515160816511 | -0.957622661930319 | 1.87944538858269 | 1.87957200742329 |
38 | -1.20453451647578 | -1.20515184058522 | -0.957622661993280 | 1.87944561486440 | 1.87957240346345 |
39 | -1.20453458242685 | -1.20515196424107 | -0.957622662022941 | 1.87944575159381 | 1.87957261754713 |
40 | -1.20453461751756 | -1.20515203003041 | -0.957622662036681 | 1.87944584142170 | 1.87957273555679 |
41 | -1.20453463618833 | -1.20515206503269 | -0.957622662042905 | 1.87944590670690 | 1.87957280285885 |
42 | -1.20453464612252 | -1.20515208365516 | -0.957622662045639 | 1.87944595915983 | 1.87957284340823 |
43 | -1.20453465140822 | -1.20515209356298 | -0.957622662046787 | 1.87944600490796 | 1.87957286984881 |
44 | -1.20453465422059 | -1.20515209883429 | -0.957622662047233 | 1.87944604715423 | 1.87957288885154 |
45 | -1.20453465571697 | -1.20515210163882 | -0.957622662047384 | 1.87944608757114 | 1.87957290393459 |
46 | -1.20453465651316 | -1.20515210313093 | -0.957622662047418 | 1.87944612703108 | 1.87957291695257 |
47 | -1.20453465693678 | -1.20515210392479 | -0.957622662047412 | 1.87944616598852 | 1.87957292888273 |
48 | -1.20453465716218 | -1.20515210434715 | -0.957622662047395 | 1.87944620467997 | 1.87957294023975 |
49 | -1.20453465728211 | -1.20515210457186 | -0.957622662047380 | 1.87944624322833 | 1.87957295129461 |
50 | -1.20453465734592 | -1.20515210469141 | -0.957622662047367 | 1.87944628169738 | 1.87957296218987 |
51 | -1.20453465737988 | -1.20515210475502 | -0.957622662047359 | 1.87944632012018 | 1.87957297300052 |
52 | -1.20453465739794 | -1.20515210478886 | -0.957622662047354 | 1.87944635851383 | 1.87957298376597 |
53 | -1.20453465740755 | -1.20515210480686 | -0.957622662047350 | 1.87944639688716 | 1.87957299450695 |
54 | -1.20453465741267 | -1.20515210481644 | -0.957622662047348 | 1.87944643524473 | 1.87957300523431 |
55 | -1.20453465741539 | -1.20515210482154 | -0.957622662047347 | 1.87944647358889 | 1.87957301595377 |
56 | -1.20453465741684 | -1.20515210482425 | -0.957622662047346 | 1.87944651192084 | 1.87957302666832 |
57 | -1.20453465741761 | -1.20515210482569 | -0.957622662047345 | 1.87944655024121 | 1.87957303737953 |
58 | -1.20453465741802 | -1.20515210482646 | -0.957622662047345 | 1.87944658855031 | 1.87957304808822 |
59 | -1.20453465741823 | -1.20515210482687 | -0.957622662047345 | 1.87944662684832 | 1.87957305879483 |
60 | -1.20453465741835 | -1.20515210482709 | -0.957622662047345 | 1.87944666513531 | 1.87957306949957 |
... | ... | ... | ... | ... | ... |
123 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944905537658 | 1.87957374064019 |
124 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944909297353 | 1.87957375124201 |
125 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944913055985 | 1.87957376184224 |
126 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944916813554 | 1.87957377244087 |
12 | -0.585992440180766 | -0.619223030479677 | -0.954022066928013 | ||
13 | -0.679680789153866 | -0.586518745810595 | -0.955771266316709 | 0.611294033635953 | -0.348337217906535 |
14 | -0.832943120892727 | -0.680405597366770 | -0.956670676554674 | 1.10453178002677 | 0.611827028590535 |
15 | -0.971700859248744 | -0.833858856488816 | -0.957133341118791 | 1.42682237020761 | 1.10614894485762 |
16 | -1.06895034261354 | -0.972586339650497 | -0.957371324937994 | 1.62539036711981 | 1.42786267946187 |
17 | -1.12878180546213 | -1.06974378003823 | -0.957493684559891 | 1.74041613015774 | 1.62585984766665 |
18 | -1.16315948204869 | -1.12950135417794 | -0.957556550137796 | 1.80447633168710 | 1.74057500572435 |
19 | -1.18221081344211 | -1.16383344210474 | -0.957588818205496 | 1.83935532699776 | 1.80451355982745 |
20 | -1.19256842684817 | -1.18285911677470 | -0.957605362078047 | 1.85810857221899 | 1.83936683519620 |
21 | -1.19814270467075 | -1.19320271670588 | -0.957613833086740 | 1.86812125596453 | 1.85813290902146 |
22 | -1.20112659974971 | -1.19876937982291 | -0.957618164219074 | 1.87344594955960 | 1.86816902366844 |
23 | -1.20271933055663 | -1.20174912257985 | -0.957620375131279 | 1.87627085023485 | 1.87351627383757 |
24 | -1.20356821162661 | -1.20333957727649 | -0.957621501752285 | 1.87776722660084 | 1.87635922088281 |
25 | -1.20402028100878 | -1.20418720532070 | -0.957622074742812 | 1.87855899443983 | 1.87786878623627 |
26 | -1.20426092790010 | -1.20463858299438 | -0.957622365545888 | 1.87897756998574 | 1.87866968187920 |
27 | -1.20438900121782 | -1.20487884751668 | -0.957622512790829 | 1.87919868763727 | 1.87909434452461 |
28 | -1.20445715437191 | -1.20500670938961 | -0.957622587155352 | 1.87931541764533 | 1.87931941124216 |
29 | -1.20449341925509 | -1.20507474565432 | -0.957622624605403 | 1.87937700435579 | 1.87943864752752 |
30 | -1.20451271548097 | -1.20511094596593 | -0.957622643405331 | 1.87940948249360 | 1.87950179504452 |
31 | -1.20452298265588 | -1.20513020655494 | -0.957622652809200 | 1.87942660604971 | 1.87953522802574 |
32 | -1.20452844558530 | -1.20514045408248 | -0.957622657494091 | 1.87943563599648 | 1.87955292480737 |
33 | -1.20453135227139 | -1.20514590619155 | -0.957622659817296 | 1.87944040281495 | 1.87956229105465 |
34 | -1.20453289884137 | -1.20514880692492 | -0.957622660963252 | 1.87944292580798 | 1.87956724882028 |
35 | -1.20453372172880 | -1.20515035022345 | -0.957622661525026 | 1.87944426871036 | 1.87956987448293 |
36 | -1.20453415956431 | -1.20515117131478 | -0.957622661798422 | 1.87944499146173 | 1.87957126690649 |
37 | -1.20453439252431 | -1.20515160816511 | -0.957622661930319 | 1.87944538858269 | 1.87957200742329 |
38 | -1.20453451647578 | -1.20515184058522 | -0.957622661993280 | 1.87944561486440 | 1.87957240346345 |
39 | -1.20453458242685 | -1.20515196424107 | -0.957622662022941 | 1.87944575159381 | 1.87957261754713 |
40 | -1.20453461751756 | -1.20515203003041 | -0.957622662036681 | 1.87944584142170 | 1.87957273555679 |
41 | -1.20453463618833 | -1.20515206503269 | -0.957622662042905 | 1.87944590670690 | 1.87957280285885 |
42 | -1.20453464612252 | -1.20515208365516 | -0.957622662045639 | 1.87944595915983 | 1.87957284340823 |
43 | -1.20453465140822 | -1.20515209356298 | -0.957622662046787 | 1.87944600490796 | 1.87957286984881 |
44 | -1.20453465422059 | -1.20515209883429 | -0.957622662047233 | 1.87944604715423 | 1.87957288885154 |
45 | -1.20453465571697 | -1.20515210163882 | -0.957622662047384 | 1.87944608757114 | 1.87957290393459 |
46 | -1.20453465651316 | -1.20515210313093 | -0.957622662047418 | 1.87944612703108 | 1.87957291695257 |
47 | -1.20453465693678 | -1.20515210392479 | -0.957622662047412 | 1.87944616598852 | 1.87957292888273 |
48 | -1.20453465716218 | -1.20515210434715 | -0.957622662047395 | 1.87944620467997 | 1.87957294023975 |
49 | -1.20453465728211 | -1.20515210457186 | -0.957622662047380 | 1.87944624322833 | 1.87957295129461 |
50 | -1.20453465734592 | -1.20515210469141 | -0.957622662047367 | 1.87944628169738 | 1.87957296218987 |
51 | -1.20453465737988 | -1.20515210475502 | -0.957622662047359 | 1.87944632012018 | 1.87957297300052 |
52 | -1.20453465739794 | -1.20515210478886 | -0.957622662047354 | 1.87944635851383 | 1.87957298376597 |
53 | -1.20453465740755 | -1.20515210480686 | -0.957622662047350 | 1.87944639688716 | 1.87957299450695 |
54 | -1.20453465741267 | -1.20515210481644 | -0.957622662047348 | 1.87944643524473 | 1.87957300523431 |
55 | -1.20453465741539 | -1.20515210482154 | -0.957622662047347 | 1.87944647358889 | 1.87957301595377 |
56 | -1.20453465741684 | -1.20515210482425 | -0.957622662047346 | 1.87944651192084 | 1.87957302666832 |
57 | -1.20453465741761 | -1.20515210482569 | -0.957622662047345 | 1.87944655024121 | 1.87957303737953 |
58 | -1.20453465741802 | -1.20515210482646 | -0.957622662047345 | 1.87944658855031 | 1.87957304808822 |
59 | -1.20453465741823 | -1.20515210482687 | -0.957622662047345 | 1.87944662684832 | 1.87957305879483 |
60 | -1.20453465741835 | -1.20515210482709 | -0.957622662047345 | 1.87944666513531 | 1.87957306949957 |
... | ... | ... | ... | ... | ... |
123 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944905537658 | 1.87957374064019 |
124 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944909297353 | 1.87957375124201 |
125 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944913055985 | 1.87957376184224 |
126 | -1.20453465741848 | -1.20515210482733 | -0.9576226620473447 | 1.87944916813554 | 1.87957377244087 |
[1] |
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 |
[2] |
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 |
[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] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 |
[8] |
Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249 |
[9] |
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 |
[10] |
César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617 |
[11] |
Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357 |
[12] |
Ana Cristina Mereu, Marco Antonio Teixeira. Reversibility and branching of periodic orbits. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1177-1199. doi: 10.3934/dcds.2013.33.1177 |
[13] |
Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505 |
[14] |
Katrin Gelfert, Christian Wolf. On the distribution of periodic orbits. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 949-966. doi: 10.3934/dcds.2010.26.949 |
[15] |
Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451 |
[16] |
Motserrat Corbera, Jaume Llibre, Claudia Valls. Periodic orbits of perturbed non-axially symmetric potentials in 1:1:1 and 1:1:2 resonances. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2299-2337. doi: 10.3934/dcdsb.2018101 |
[17] |
Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics, 2021, 17: 305-317. doi: 10.3934/jmd.2021010 |
[18] |
Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331 |
[19] |
Peter Albers, Jean Gutt, Doris Hein. Periodic Reeb orbits on prequantization bundles. Journal of Modern Dynamics, 2018, 12: 123-150. doi: 10.3934/jmd.2018005 |
[20] |
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 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]