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. 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.  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

show all references

References:
[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.  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

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
$ 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
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
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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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