# American Institute of Mathematical Sciences

December  2020, 7(2): 425-460. doi: 10.3934/jcd.2020017

## Computer-assisted estimates for Birkhoff normal forms

 Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", via della Ricerca Scientifica 1, 00133— Rome, Italy

Received  November 2019 Published  July 2020

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.

Citation: Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017
##### References:

show all references

##### References:
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 $\omega_2 = - (\sqrt 5 -1)/2$
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
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
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
 $\rho_0$ $\rho$ $r_{\rm opt}$ $a_r$ $\log_{10}{| \mathcal{R}^{(r_{\rm opt})}|_\rho}$ $\log_{10}|\dot I_j|_\rho$ $\log_{10}T$ 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
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
 $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$ 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
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
 $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$ 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
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
 $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$ 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
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
 $\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$ 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
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
 $\mu$ $\rho_0^2\ \, {\rm (non-res.)}$ $({\rho^*_2})^2\ \, {\rm (reson.)}$ $({\rho^*_2}/\rho_0)^2$ 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
 [1] Maxime Breden, Jean-Philippe Lessard. Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2825-2858. doi: 10.3934/dcdsb.2018164 [2] István Balázs, Jan Bouwe van den Berg, Julien Courtois, János Dudás, Jean-Philippe Lessard, Anett Vörös-Kiss, JF Williams, Xi Yuan Yin. Computer-assisted proofs for radially symmetric solutions of PDEs. Journal of Computational Dynamics, 2018, 5 (1&2) : 61-80. doi: 10.3934/jcd.2018003 [3] Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 [4] Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045 [5] Maciej J. Capiński, Emmanuel Fleurantin, J. D. Mireles James. Computer assisted proofs of two-dimensional attracting invariant tori for ODEs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6681-6707. doi: 10.3934/dcds.2020162 [6] A. Aschwanden, A. Schulze-Halberg, D. Stoffer. Stable periodic solutions for delay equations with positive feedback - a computer-assisted proof. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 721-736. doi: 10.3934/dcds.2006.14.721 [7] Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569 [8] Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67 [9] Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533 [10] Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95 [11] Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703 [12] Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295 [13] Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529 [14] Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 [15] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [16] Shalela Mohd--Mahali, Song Wang, Xia Lou, Sungging Pintowantoro. Numerical methods for estimating effective diffusion coefficients of three-dimensional drug delivery systems. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 377-393. doi: 10.3934/naco.2012.2.377 [17] P. Balseiro, M. de León, Juan Carlos Marrero, D. Martín de Diego. The ubiquity of the symplectic Hamiltonian equations in mechanics. Journal of Geometric Mechanics, 2009, 1 (1) : 1-34. doi: 10.3934/jgm.2009.1.1 [18] Michael Kastner, Jacques-Alexandre Sepulchre. Effective Hamiltonian for traveling discrete breathers in the FPU chain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 719-734. doi: 10.3934/dcdsb.2005.5.719 [19] Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. Effective Hamiltonian dynamics via the Maupertuis principle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1395-1410. doi: 10.3934/dcdss.2020078 [20] Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004

Impact Factor: