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Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case

This research was partially supported by GNFM-INdAM.
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  • Hénon map is a well-studied classical example of area-contracting maps, modelling dissipative dynamics. The rich phenomena of coexistence of stable islands and their separatrices is typical of area-preserving maps, modelling conservative dynamics. In this paper we use the Hénon map to ascertain that coexistence of sinks is greater and greater approaching the conservative case, and that part of it can be organized following a renormalization argument. Using a numerical continuation that we devised, and called "dribbling method" [5], one can follow bifurcation paths from the area-preserving case into the dissipative one, organizing families of coexisting attractive periodic orbits with diverging period. When the dissipation parameter goes to zero, we will give numerical evidence of the increasing coexistence of such periodic orbits, in the coordinate and parameter space values. Vanishing dissipation and diverging period constitute a double limit that we study as such, giving evidence of a singularity in the limit. The families we study all appear as homoclinic bifurcation, and the fixed point causing the homoclinic onset also structures the renormalization scheme. One of the goals of this paper is to improve the results obtained by looking to higher periods, and to approach dissipation down to an area-contraction factor of $1- 10^{-8}$. Using the same dribbling method, as further promising application, we also deal with the dissipative Standard map.

    Mathematics Subject Classification: Primary: 37J25, 37J45; Secondary: 37F25.

    Citation:

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  • Figure 1.  "Dribbling" a possible singularity at $|b| = 1$, in 4-dimensional space $(x,y,b,a)$ to be able to then approach it from elsewhere. Starting at step 1 on a period-7 saddle in the conservative case, follow branches of continuation curves in $(x,y,a,b)$, by switching independent variable at each turning point 2, 3, 4. At steps 3 and 4 one can follow a bifurcation curve with known continuation methods. Upper and lower curves are respectively period-doubling and saddle-node bifurcation curves

    Figure 2.  The two conservative cases to initialize a dribbling method. Left: for $b = -1$ and $a = 0.244$ big dots represent a period 14 saddle. Right: for $b = 1$ and $a = -0.719$ big dots represent a period 7 saddle

    Figure 3.  Adjustments in renormalization scheme of the saddle node bifurcation value. If Newton's method initialized using linear prediction seems not to converge, adjust initial guess only in one variable $y$, and try again. This figure illustrates dependence of the final error in the calculation of $a_{sn}$ from $y-$adjusted guess

    Figure 4.  Blow-up of same error function of Fig. 3 but with $y-$range $2\cdot 10^{-4}$. The minimal error can now be detected using an automatic algorithm in the suitable interval where the error function is observed to be smooth

    Figure 5.  Stability ranges in the $a$ parameter for $b = 1-10^{-5}$ and periodic orbits of period $p$ between 6 and 50. For $a\equiv{\bar a}_{max}(\varepsilon) = -0.93294$ there are a maximum of 10 coexisting attractors of period between $p = 11$ and $p_1 = 20$

    Figure 6.  Width of the stability ranges in the $a$ parameter for $b = 1-10^{-5}$ and periodic orbits of period $p$ between 6 and 60. The rate of fast exponential convergence starts around $p = p_{2}\equiv 25$

    Figure 7.  Stability ranges in the $a-a_{min}$ parameter for $b = 1-10^{-5}$ and periodic orbits of period $p$ between 34 and 70. The values of $a_{min}$ is the minimum value of $a_{sn}^{(p)}$ (reached for $p = p_{3} = 38$). Observe that $p = 36$ is the largest period such that $a_{sn}^{(p)}\leq a_{sn}^{(36)}$ for any $p\geq 36$

    Figure 8.  Width of the stability ranges in the $a$ parameter and coexistence in $p$ of periodic attractors with $\varepsilon = 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$. Right: $b = 1-\varepsilon$

    Figure 9.  Rate of convergence in $p$ of the stability range $a_{pd}(\varepsilon) - a_{sn}(\varepsilon)$ with $\varepsilon = 10^{-5}, 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$ Right: $b = 1-\varepsilon$. The negative value of the slope $m_\varepsilon$ in logarithmic scale increases as $\varepsilon$ decreases.

    Figure 10.  Stability ranges in the $a-a_{min}$ parameter with $\varepsilon = 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$. Right: $b = 1-\varepsilon$

    Figure 11.  Coexistence of periodic attractors (isolated points), all other points are orbits of a discretized segment: on the unstable manifold (and inside a $10^{-8}$ neighborhood) of the fixed point (rightmost big dot). $b = -1+\varepsilon$. Up: ${\bar a}_{max}(10^{-5}) = 0.0769$. Down: for ${\bar a}_{max}(10^{-6}) = 0.0549$ the scale is five times smaller

    Figure 12.  Stability range in $(a,\nu)$, $b = 0.5$, for two periodic orbits with rotation number $\omega = 3/5$ and $\omega = 8/13$. The border curves corresponds to values of $a_{sn}(\nu)$ for which the residue is $0$. The middle curve corresponds, given $b$ and $\nu$, to values of $a$ for which the residue is maximum

    Table 1.  Values of $p_1(\varepsilon)<p_2(\varepsilon)<p_3(\varepsilon)$ for $b = -1+\varepsilon$ and decreasing $\varepsilon$

    k $\varepsilon = 10^{-k}$ $b=-1+\varepsilon$ $p_1(\varepsilon)$ $p_2(\varepsilon)$ $p_3(\varepsilon)$
    5 $10^{-5}$ -0.99999 68 86 144
    6 $10^{-6}$ -0.999999 96 114 188
    7 $10^{-7}$ -0.9999999 128 146 236
    8 $10^{-8}$ -0.99999999 164 182 288
     | Show Table
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    Table 2.  Values of $p_1(\varepsilon)<p_2(\varepsilon)<p_3(\varepsilon)$ for $b = 1-\varepsilon$ and decreasing $\varepsilon$

    k $\varepsilon = 10^{-k}$ $b=1-\varepsilon$ $p_1(\varepsilon)$ $p_2(\varepsilon)$ $p_3(\varepsilon)$
    5 $10^{-5}$ 0.99999 20 25 38
    6 $10^{-6}$ 0.999999 25 30 46
    7 $10^{-7}$ 0.9999999 31 36 55
    8 $10^{-8}$ 0.99999999 38 43 65
     | Show Table
    DownLoad: CSV

    Table 3.  Coexistence of attractors for $b = -1+\varepsilon$ with $\varepsilon$ ranging from $10^{-3}$ to $10^{-8}$

    $\varepsilon$ $b=-1+\varepsilon$ $N_{max}(\varepsilon)$ ${\bar a}_{max}(\varepsilon)$ $Periods$
    $10^{-3}$ -0.999 4 0.151600 20 - 30
    $10^{-4}$ -0.9999 9 0.098750 28 - 46
    $10^{-5}$ -0.99999 16 0.076900 30 - 74
    $10^{-6}$ -0.999999 27 0.054900 42 - 98
    $10^{-7}$ -0.9999999 39 0.045000 46 - 134
    $10^{-8}$ -0.99999999 54 0.035794 54 - 166
     | Show Table
    DownLoad: CSV

    Table 4.  Coexistence of attractors for $b = 1-\varepsilon$ with $\varepsilon$ ranging from $10^{-5}$ to $10^{-8}$

    $\varepsilon$ $b=1-\varepsilon$ $N_{max}(\varepsilon)$ ${\bar a}_{max}(\varepsilon)$ $Periods$
    $10^{-5}$ 0.99999 10 -0.93294 11 - 20
    $10^{-6}$ 0.999999 14 -0.95400 12 - 25
    $10^{-7}$ 0.9999999 18 -0.96490 12 - 29
    $10^{-8}$ 0.99999999 23 -0.97420 13 - 35
     | Show Table
    DownLoad: CSV

    Table 5.  $b = -1+10^{-6}$, $\bar a = 0.0549$. Orbits in Fig. 11 with period from 42 to 94 are attractors with complex eigenvalues

    $p$ $x_{\text{in}}$ $y_{\text{in}} $ first eigenvalue accuracy
    42 -0.02471212033 -0.004353351966 0.999666+0.025007 i 980
    44 0.003893968151 0.01400370243 0.998106+0.061164 i 981
    46 0.03287883220 0.03630238615 0.996571+0.082467 i 981
    48 0.06028145342 0.05973990689 0.994655+0.103021 i 980
    50 0.08552343060 0.08300646446 0.992221+0.124292 i 980
    52 0.1082722485 0.1051821971 0.989124+0.146904 i 979
    54 0.1283734421 0.1256447760 0.985195+0.171279 i 979
    56 0.1458237800 0.1440274072 0.980218+0.197778 i 979
    58 0.1607392037 0.1601751773 0.973924+0.226748 i 979
    60 0.1733182271 0.1740957680 0.965971+0.258534 i 979
    62 0.1838069083 0.1859098749 0.955930+0.293489 i 979
    64 0.1924697936 0.1958070451 0.943258+0.331965 i 978
    66 0.1995685673 0.2040103236 0.927268+0.374309 i 978
    68 0.2053481559 0.2107505750 0.907097+0.420841 i 978
    70 0.2100289888 0.2162495981 0.881651+0.471827 i 978
    72 0.2138038071 0.2207103044 0.849555+0.527432 i 978
    74 0.2168375173 0.2243120570 0.809070+0.587650 i 979
    76 0.2192688855 0.2272094757 0.758004+0.652192 i 979
    78 0.2212131915 0.2295333706 0.693593+0.720313 i 979
    80 0.2227652522 0.2313928385 0.612348+0.790538 i 979
    82 0.2240024479 0.2328778729 0.509870+0.860204 i 978
    84 0.2249875438 0.2340620792 0.380610+0.924690 i 977
    86 0.2257712060 0.2350052607 0.217565+0.976002 i 977
    88 0.2263941791 0.2357557547 0.011907+0.999885 i 976
    90 0.2268891314 0.2363524720 -0.247504+0.968841 i 975
    92 0.2272821942 0.2368266348 -0.574717+0.818296 i 975
    94 0.2275942304 0.2372032327 -0.987457+0.157593 i 974
    96 0.2278418727 0.2375022270 -0.379187 974
    98 0.2280383657 0.2377395373 -0.244788 973
     | Show Table
    DownLoad: CSV
  • [1] K. Banerjee, On the widths of the Arnol'd tongues, Ergodic Theory and Dynamical Systems, 34 (2014), 1451-1463.  doi: 10.1017/etds.2013.11.
    [2] R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative Standard map, Chaos, 20 (2010), 013121, 9pp. doi: 10.1063/1.3335408.
    [3] R. CallejaA. CellettiC. Falcolini and R. de la Llave, An extension of Greene's criterion for conformally symplectic systems and a partial justification, SIAM Journal on Mathematical Analysis, 46 (2014), 2350-2384.  doi: 10.1137/130929369.
    [4] C. Falcolini and R. de la Llave, A rigorous partial justification of Greene's criterion, Journal of Statistical Physics, 67 (1992), 609-643.  doi: 10.1007/BF01049722.
    [5] C. Falcolini and L. Tedeschini-Lalli, Hénon map: simple sinks gaining coexistence as $b \to 1$, International Journal of Bifurcation and Chaos, 23 (2013), 1330030, 13 pp. doi: 10.1142/S0218127413300309.
    [6] C. Falcolini and L. Tedeschini-Lalli, Backbones in the parameter plane of the Hénon map, Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2016), 013104, 9pp. doi: 10.1063/1.4939862.
    [7] S. V. GonchenkoYu. A. Kuznetsov and H. G. E. Meijer, Generalized Hénon map and bifurcation of homoclinic tangencies, SIAM Journal on Applied Dynamical Systems, 4 (2005), 407-436.  doi: 10.1137/04060487X.
    [8] S.V. GonchenkoI. Ovsyannikov and D. Turaev, On the effect of invisibility of stable periodic orbits at homoclinic bifurcations, Physica D Nonlinear Phenomena, 241 (2012), 1115-1122.  doi: 10.1016/j.physd.2012.03.002.
    [9] M. Hénon, A two dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.
    [10] P. Holmes and D. Whitley, Bifurcations of one- and two-dimensional maps, Phil. Trans. Roy. Soc. Lond. Ser. A, 311 (1984), 43–102. Erratum: Bifurcations of one- and two-dimensional maps, Phil. Trans. Roy. Soc. Lond., 312 (1984), 601–602. doi: 10.1098/rsta.1984.0020.
    [11] C. A. Jousseph, A. Abdulack, C. Manchein and M. W. Beims, Hierarchical collapse of regular islands via dissipation, J. Phys. A: Math. Theor., 51 (2018), 105101, 17pp.
    [12] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2010.
    [13] N. MiguelC. Simó and A. Vieiro, From the Hénon conservative map to the Chirikov standard map for large parameter values, Regul. Chaot. Dyn., 18 (2013), 469-489.  doi: 10.1134/S1560354713050018.
    [14] G. Schmidt and B. W. Wang, Dissipative standard map, Phys. Rev. A, 32 (1985), 2994-2999.  doi: 10.1103/PhysRevA.32.2994.
    [15] W. Wenzel, O. Biham and C. Jayaprakash, Periodic orbits in the dissipative standard map, Phys. Rev. A, 43 (1991), 6550. doi: 10.1103/PhysRevA.43.6550.
    [16] J. A. Yorke and L. Tedeschini-Lalli, How often do simple dynamical systems have infinitely many coexisting sinks, Comm. Math. Phys., 106 (1986), 635-657.  doi: 10.1007/BF01463400.
    [17] J. A. Yorke and K. T. Alligood, Period doubling cascades of attractors: A prerequisite for horseshoes, Comm. Math. Phys., 101 (1985), 305-321.  doi: 10.1007/BF01216092.
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