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A numerical renormalization method for quasi–conservative periodic attractors

  • * Corresponding author: Laura Tedeschini-Lalli

    * Corresponding author: Laura Tedeschini-Lalli

The research has been supported by CNR-Gruppo Nazionale di Fisica Matematica

Abstract Full Text(HTML) Figure(2) / Table(2) Related Papers Cited by
  • 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 [3]) and in other ranges of the period for the dynamical plane $ (x, y) $. For the first time we explain here how specific numerical instabilities depend on geometry displacements in dynamical plane $ (x, y) $. We describe how to take advantage of such displacement in the sequence, and of the high period, by moving forward from one branch to its image under dynamics. This, for high period, allows entering the hyperbolicity neighborhood of a saddle, where the dynamics is conjugate to a hyperbolic linear map.

    The subtle interplay of branches and of the hyperbolic neighborhood can hopefully help visualize the renormalization approach theoretically discussed in [7] for highly dissipative systems.

    Mathematics Subject Classification: Primary: 37J25, 37J46; Secondary: 37E20.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  Sequence of periodic orbits. Each periodic orbit of period $ q_j $ is shown on plane $ a = a_{sn(q_j)} $. We call "Branch" the sequence of blue dots $ Q_j^{(0)} $. In Table 1 all points lie on Branch

    Figure 2.  Hopping among branches. From segment $ (Q_{8}^{(0)}, Q_{9}^{(0)}, Q_{10}^{(0)}) $ we pass to segment $ (Q_{9}^{(1)}, Q_{10}^{(1)}, Q_{11}^{(1)}) $

    Table 1.  Values of $x_k, y_k, a_k$ for $q_k$-periodic points, at the corresponding value $a = a_{sn(q_k)}$ of their saddle-node bifurcation, together with the differences among the interpolating values of the sequences $\{x_k\}$ ($g_k(x) = \frac{x_k-x_{k-1}}{x_{k+1}-x_k}$) and $\{a_k\}$ ($g_k(a) = \frac{a_k-a_{k-1}}{a_{k+1}-a_k}$). Notice obstacles in their convergence

    $ q_k $ $ x_k $ $ y_k $ $ a_k $ $ g_{k+1}(x)-g_k(x) $ $ g_{k+1}(a)-g_k(a) $
    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
     | Show Table
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    Table 2.  Values of $ x_k, y_k, a_k $ for $ q_k $-periodic points at the corresponding value $ a = a_{sn(q_k)} $ of their saddle-node bifurcation, together with the interpolating values of the sequences $ \{x_k\} $ and $ \{a_k\} $. Note the monotone convergence of the interpolating values of $ x_k $ and $ y_k $. Sequence of periodic orbits is the same as in Table 1, but representative points in each orbit are picked by our "hopping method"

    $ q_k $ $ x_k $ $ y_k $ $ a_k $ $ \frac{x_k-x_{k-1}}{x_{k+1}-x_k} $ $ \frac{y_k-y_{k-1}}{y_{k+1}-y_k} $
    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
     | Show Table
    DownLoad: CSV
  • [1] R. C. CallejaA. CellettiC. 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.
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