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Computer-assisted estimates for Birkhoff normal forms

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  • Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Hénon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.

    Mathematics Subject Classification: Primary: 68V05; Secondary: 37J40, 37N05, 70F07, 70H08.


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  • Figure 1.  On the left, plot of the optimal normalization step $ r_{\rm opt} $ as a function of the ball radius $ {\varrho}\, $; on the right, graph of the evaluation of our lower bound about the escape time $ T $ as a function of $ 1/\sqrt{{\varrho}}\, $. Both the plots refer to results obtained by applying computer-assisted estimates to the Hénon-Heiles model with frequencies $ \omega_1 = 1 $ and $ \omega_2 = - (\sqrt 5 -1)/2 $

    Figure 2.  Plots of the evaluation of our lower bound of the escape time $ T $ (in semi-log scale). On the left, the graph is a function of $ {\varrho}_0\, $, on the right, of $ {{\varrho}^*_2}\, $. The horizontal line corresponds to $ T_{ \rm e. l. t.} = 5\times 10^8 $. See the text for more details

    Figure 3.  Growth of the norms (in semi-log scale) of the generating functions for the non-resonant Birkhoff normal form (continuous line) and the resonant one (dashed line). From top to down and from left to right, the boxes refer to the cases of the systems having Sun-Jupiter, Sun-Uranus, Sun-Mars and Saturn-Janus as primary bodies, respectively

    Table 1.  In this table we report the results obtained for the Hénon-Heiles model with frequencies $\omega_1 = 1$ and $\omega_2 = -(\sqrt 5 -1)/2$

    $\rho_0$ $\rho$ $r_{\rm opt}$ $a_r$ $ \log_{10}{| \mathcal{R}^{(r_{\rm opt})}|_\rho}$ $\log_{10}|\dot I_j|_\rho$ $\log_{10}T$
    9.96e-04 1.00e-03 232 1.00e+03 -1.82e+02 -1.80e+02 1.72e+02
    1.24e-03 1.25e-03 230 8.02e+02 -1.59e+02 -1.57e+02 1.49e+02
    1.55e-03 1.56e-03 164 6.40e+02 -1.42e+02 -1.39e+02 1.32e+02
    1.94e-03 1.95e-03 144 5.13e+02 -1.28e+02 -1.26e+02 1.18e+02
    2.42e-03 2.44e-03 110 4.10e+02 -1.16e+02 -1.14e+02 1.07e+02
    3.02e-03 3.05e-03 102 3.28e+02 -1.06e+02 -1.04e+02 9.73e+01
    3.78e-03 3.81e-03 100 2.63e+02 -9.63e+01 -9.43e+01 8.77e+01
    4.72e-03 4.77e-03 100 2.11e+02 -8.63e+01 -8.43e+01 7.79e+01
    5.90e-03 5.96e-03 100 1.69e+02 -7.63e+01 -7.43e+01 6.82e+01
    7.38e-03 7.45e-03 100 1.35e+02 -6.63e+01 -6.43e+01 5.84e+01
    9.22e-03 9.31e-03 100 1.08e+02 -5.64e+01 -5.43e+01 4.86e+01
    1.15e-02 1.16e-02 74 8.63e+01 -4.78e+01 -4.59e+01 4.05e+01
    1.43e-02 1.46e-02 58 7.07e+01 -4.18e+01 -4.00e+01 3.48e+01
    1.79e-02 1.82e-02 52 5.66e+01 -3.67e+01 -3.49e+01 3.00e+01
    2.23e-02 2.27e-02 52 4.49e+01 -3.13e+01 -2.96e+01 2.49e+01
    2.79e-02 2.84e-02 48 3.57e+01 -2.67e+01 -2.50e+01 2.05e+01
    3.46e-02 3.55e-02 38 2.84e+01 -2.27e+01 -2.11e+01 1.68e+01
    4.30e-02 4.44e-02 30 2.32e+01 -1.97e+01 -1.82e+01 1.43e+01
    5.36e-02 5.55e-02 26 1.86e+01 -1.71e+01 -1.56e+01 1.19e+01
    6.70e-02 6.94e-02 26 1.49e+01 -1.42e+01 -1.28e+01 9.30e+00
    8.37e-02 8.67e-02 26 1.15e+01 -1.14e+01 -9.94e+00 6.65e+00
     | Show Table
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    Table 2.  Comparison for the estimates on the stability time between the non-resonant and resonant Birkhoff normal forms. The Jupiter case ($\mu\simeq 0.000954$) with $T_{ \rm e. l. t.} \simeq 5\times 10^8$

    $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
    2.49e-04 2.59e-04 6.36e+08 2.05e-04 1.83e-04 2.07e-04 5.93e+08
    2.47e-04 2.57e-04 1.01e+09 2.02e-04 1.80e-04 2.04e-04 7.23e+08
     | Show Table
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    Table 3.  As in Table 2 for the Uranus case ($\mu\simeq 4.36\times 10^{-5}$) with $T_{ \rm e. l. t.} \simeq 6\times 10^7$

    $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
    8.30e-05 8.80e-05 6.03e+07 9.23e-04 7.57e-04 9.24e-04 7.18e+07
    8.13e-05 8.63e-05 1.44e+08 9.04e-04 7.44e-04 9.05e-04 1.27e+08
     | Show Table
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    Table 4.  As in Table 2 for the Mars case ($\mu\simeq 3.21\times 10^{-7}$) with $T_{ \rm e. l. t.} \simeq 3 \times 10^9$

    $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
    7.36e-06 7.84e-06 3.09e+09 1.28e-04 1.08e-04 1.28e-04 3.87e+09
    7.22e-06 7.69e-06 6.15e+09 1.27e-04 1.07e-04 1.27e-04 5.86e+09
     | Show Table
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    Table 5.  As in Table 2 for the Janus case ($\mu\simeq 3.36\times 10^{-9}$) with $T_{ \rm e. l. t.} \simeq 3 \times 10^{12}$

    $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
    6.00e-07 6.37e-07 3.10e+12 1.18e-05 1.10e-05 1.18e-05 3.50e+12
    5.89e-07 6.24e-07 5.40e+12 1.15e-05 1.08e-05 1.15e-05 6.83e+12
     | Show Table
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    Table 6.  Comparisons between the values of the radii $\rho_0^2$ and $({\rho^*_2})^2$ which refer to the stability domains for the non-resonant Birkhoff normal form and the resonant one, respectively. The results are reported as a function of different values of the mass ratio $\mu$, the name of the smaller primary in the corresponding CPRTBP model is reported in the first column

    $\mu$ $\rho_0^2\ \, {\rm (non-res.)}$ $({\rho^*_2})^2\ \, {\rm (reson.)}$ $({\rho^*_2}/\rho_0)^2$
    Jupiter $9.54 \times 10^{-4}$ $2.49\times10^{-4}$ $1.83\times10^{-4}$ 0.73
    Uranus $4.36 \times 10^{-5}$ $8.30\times 10^{-5}$ $7.57\times 10^{-4}$ 9.12
    Mars $3.21\times 10^{-7}$ $7.36\times 10^{-6}$ $1.08\times 10^{-4}$ 14.67
    Janus $3.36\times 10^{-9}$ $6.00\times 10^{-7}$ $1.10\times 10^{-5}$ 18.33
     | Show Table
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