American Institute of Mathematical Sciences

December  2020, 7(2): 461-468. doi: 10.3934/jcd.2020018

A numerical renormalization method for quasi–conservative periodic attractors

 Dipartimento di Architettura, Università degli Studi Roma Tre, Roma, Italy

* Corresponding author: Laura Tedeschini-Lalli

Received  October 2019 Published  July 2020

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

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.

Citation: Corrado Falcolini, Laura Tedeschini-Lalli. A numerical renormalization method for quasi–conservative periodic attractors. Journal of Computational Dynamics, 2020, 7 (2) : 461-468. doi: 10.3934/jcd.2020018
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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] C. Falcolini and L. Tedeschini-Lalli, Quasi-conservative Hénon: Coexisting sequences of coexisting sinks organized by their rotation number, in progress. Google Scholar [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.   Google Scholar [6] J. M. Greene, Method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201.  doi: 10.2172/6191337.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] C. Falcolini and L. Tedeschini-Lalli, Quasi-conservative Hénon: Coexisting sequences of coexisting sinks organized by their rotation number, in progress. Google Scholar [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.   Google Scholar [6] J. M. Greene, Method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201.  doi: 10.2172/6191337.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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
Hopping among branches. From segment $(Q_{8}^{(0)}, Q_{9}^{(0)}, Q_{10}^{(0)})$ we pass to segment $(Q_{9}^{(1)}, Q_{10}^{(1)}, Q_{11}^{(1)})$
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
 $q_k$ $x_k$ $y_k$ $a_k$ $g_{k+1}(x)-g_k(x)$ $g_{k+1}(a)-g_k(a)$ 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
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
 $q_k$ $x_k$ $y_k$ $a_k$ $\frac{x_k-x_{k-1}}{x_{k+1}-x_k}$ $\frac{y_k-y_{k-1}}{y_{k+1}-y_k}$ 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
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