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December  2018, 38(12): 6105-6122. doi: 10.3934/dcds.2018263

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

Corrado Falcolini and  Laura Tedeschini-Lalli

Dipartimento di Architettura, Università Roma Tre, Via della Madonna dei Monti 40, I-00184 Rome, Italy

Received  October 2017 Revised  June 2018 Published  September 2018

Fund Project: This research was partially supported by GNFM-INdAM.

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.

Keywords: Periodic orbits, bifurcation curves, Hénon map, standard map, renormalization schemes.
Mathematics Subject Classification: Primary: 37J25, 37J45; Secondary: 37F25.
Citation: Corrado Falcolini, Laura Tedeschini-Lalli. Diverging period and vanishing dissipation: Families of periodic sinks in the quasi-conservative case. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6105-6122. doi: 10.3934/dcds.2018263
References:
[1]

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[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.  Google Scholar

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R. 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 Journal on Mathematical Analysis, 46 (2014), 2350-2384.  doi: 10.1137/130929369.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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[8]

S.V. Gonchenko, I. 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.  Google Scholar

[9]

M. Hénon, A two dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2010. Google Scholar

[13]

N. Miguel, C. 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.  Google Scholar

[14]

G. Schmidt and B. W. Wang, Dissipative standard map, Phys. Rev. A, 32 (1985), 2994-2999.  doi: 10.1103/PhysRevA.32.2994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

R. 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 Journal on Mathematical Analysis, 46 (2014), 2350-2384.  doi: 10.1137/130929369.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. V. Gonchenko, Yu. 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.  Google Scholar

[8]

S.V. Gonchenko, I. 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.  Google Scholar

[9]

M. Hénon, A two dimensional mapping with a strange attractor, Commun. Math. Phys., 50 (1976), 69-77.  doi: 10.1007/BF01608556.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2010. Google Scholar

[13]

N. Miguel, C. 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.  Google Scholar

[14]

G. Schmidt and B. W. Wang, Dissipative standard map, Phys. Rev. A, 32 (1985), 2994-2999.  doi: 10.1103/PhysRevA.32.2994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

$|b| = 1$, 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 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
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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
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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
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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 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
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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 $p_1 = 20$
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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 $p = p_{2}\equiv 25$
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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 $p\geq 36$
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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: $b = 1-\varepsilon$
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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.
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Figure 10.  Stability ranges in the $a-a_{min}$ parameter with $\varepsilon = 10^{-6}, 10^{-7}$ and $10^{-8}$. Left: $b = -1+\varepsilon$. Right: $b = 1-\varepsilon$
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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
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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
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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
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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
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
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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
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
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$\varepsilon$ $b=-1+\varepsilon$ $N_{max}(\varepsilon)$ ${\bar a}_{max}(\varepsilon)$ $Periods$
$10^{-3}$ -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
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
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$\varepsilon$ $b=1-\varepsilon$ $N_{max}(\varepsilon)$ ${\bar a}_{max}(\varepsilon)$ $Periods$
$10^{-5}$ 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
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
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$p$ $x_{\text{in}}$ $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
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