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:
[1]

K. Appel and W. Haken, Every planar map is four colorable. I. Discharging, Illinois J. Math., 21 (1977), 429-490.  doi: 10.1215/ijm/1256049011.  Google Scholar

[2]

K. Appel and W. Haken, Every Planar Map Is Four Colorable, Contemporary Mathematics, 98, American Mathematical Society, Providence, RI, 1989. doi: 10.1090/conm/098.  Google Scholar

[3]

K. AppelW. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math., 21 (1977), 491-567.  doi: 10.1215/ijm/1256049012.  Google Scholar

[4]

V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surv., 18 (1963). doi: 10.1070/RM1963v018n05ABEH004130.  Google Scholar

[5]

I. BalázsJ. B. van den BergJ. CourtoisJ. Dudás and J.-P. Lessard, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5 (2018), 61-80.  doi: 10.3934/jcd.2018003.  Google Scholar

[6]

G. D. Birkhoff, Dynamical Systems, American Mathematical Society Colloquium Publications, IX, American Mathematical Society, Providence, RI, 1966.  Google Scholar

[7]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007). doi: 10.1090/memo/0878.  Google Scholar

[8]

A. Celletti and A. Giorgilli, On the stability of the Lagrangian points in the spatial restricted problem of three bodies, Celestial Mech. Dynam. Astronom., 50 (1991), 31-38.  doi: 10.1007/BF00048985.  Google Scholar

[9]

A. CellettiA. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems, Nonlinearity, 13 (2000), 397-412.  doi: 10.1088/0951-7715/13/2/304.  Google Scholar

[10]

T. M. Cherry, On integrals developable about a singular point of a Hamiltonian system of differential equations, Proc. Camb. Phil. Soc., 22 (1924), 325-349.  doi: 10.1017/S0305004100014249.  Google Scholar

[11]

T. M. Cherry, On integrals developable about a singular point of a Hamiltonian system of differential equations Part II, Proc. Camb. Phil. Soc., 22 (1925), 510-533.  doi: 10.1017/S0305004100003224.  Google Scholar

[12]

G. Contopoulos, A Review of the "Third" Integral, Math. Engrg., 2 (2020), 472-511.  doi: 10.3934/mine.2020022.  Google Scholar

[13]

C. EfthymiopoulosA. Giorgilli and G. Contopoulos, Nonconvergence of formal integrals. II. Improved estimates for the optimal order of truncation, J. Phys. A, 37 (2004), 10831-10858.  doi: 10.1088/0305-4470/37/45/008.  Google Scholar

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C. Efthymiopoulos and Z. Sándor, Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion, Mon. Not. Roy. Astron. Soc., 364 (2005), 253-271.  doi: 10.1111/j.1365-2966.2005.09572.x.  Google Scholar

[15]

J.-Ll. FiguerasA. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3.  Google Scholar

[16]

F. Gabern and À. Jorba, A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 143-182.  doi: 10.3934/dcdsb.2001.1.143.  Google Scholar

[17]

F. GabernÀ. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids, Nonlinearity, 18 (2005), 1705-1734.  doi: 10.1088/0951-7715/18/4/017.  Google Scholar

[18]

A. Giorgilli, Exponential stability of Hamiltonian systems, in Dynamical Systems. Part I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, 87-198.  Google Scholar

[19]

A. GiorgilliA. DelshamsE. FontichL. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, 77 (1989), 167-198.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[20]

A. Giorgilli and U. Locatelli, Canonical perturbation theory for nearly integrable systems, in Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems, Proc. NATO Adv. Study Institute, 227, Cortina, Italy, 2003, 1-41. doi: 10.1007/978-1-4020-4706-0_1.  Google Scholar

[21]

A. GiorgilliU. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celestial Mech. Dynam. Astronom., 104 (2009), 159-173.  doi: 10.1007/s10569-009-9192-7.  Google Scholar

[22]

A. GiorgilliU. Locatelli and M. Sansottera, Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; Effective stability in the light of Kolmogorov and Nekhoroshev theories, Regul. Chaotic Dyn., 22 (2017), 54-77.  doi: 10.1134/S156035471701004X.  Google Scholar

[23]

A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory, preprint, arXiv: 1303.7398. Google Scholar

[24]

A. Giorgilli and Ch. Skokos, On the stability of the Trojan asteroids, Astron. Astroph., 317 (1997), 254-261.   Google Scholar

[25]

W. Gröbner and H. Knapp, Contributions to the method of Lie series, Bibliographisches Institut, Mannheim, 1967.  Google Scholar

[26]

F. G. Gustavson, Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point, Astron. J., 71 (1966), 670-686.  doi: 10.1086/110172.  Google Scholar

[27]

M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, 3 (1966), 49-66.   Google Scholar

[28]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79.  doi: 10.1086/109234.  Google Scholar

[29]

T. Johnson and W. Tucker, Automated computation of robust normal forms of planar analytic vector fields, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 769-782.  doi: 10.3934/dcdsb.2009.12.769.  Google Scholar

[30]

À. Jorba and J. Masdemont, Dynamics in the center manifold of the collinear points of the restricted three body problem, Phys. D, 132 (1999), 189-213.  doi: 10.1016/S0167-2789(99)00042-1.  Google Scholar

[31]

H. KochA. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604.  doi: 10.1137/S0036144595284180.  Google Scholar

[32]

A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Phys., 93, Springer, Berlin-New York, 1979, 51-56. doi: 10.1007/BFb0021737.  Google Scholar

[33]

C. LhotkaC. Efthymiopoulos and R. Dvorak, Nekhoroshev stability at L4 or L5 in the elliptic-restricted three-body problem -- Application to the Trojan asteroids, Mon. Not. R. Astron. Soc., 384 (2008), 1165-1177.   Google Scholar

[34]

J. E. Littlewood, On the equilateral configuration in the restricted problem of three bodies, Proc. London Math. Soc. (3), 9 (1959), 343–372. doi: 10.1112/plms/s3-9.3.343.  Google Scholar

[35]

J. E. Littlewood, The Lagrange configuration in celestial mechanics, Proc. London Math. Soc. (3), 9 (1959), 525–543. doi: 10.1112/plms/s3-10.1.640-t.  Google Scholar

[36]

R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5 (1992), 867-888.  doi: 10.1088/0951-7715/5/4/002.  Google Scholar

[37]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.   Google Scholar

[38]

N. N. Nekhorošev, An exponential estimates of the stability time of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5-66,287. doi: 10.1070/RM1977v032n06ABEH003859.  Google Scholar

[39]

N. N. Nekhorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50.   Google Scholar

[40]

R. I. Páez and U. Locatelli, Trojans dynamics well approximated by a new Hamiltonian normal form, Mon. Not. Roy. Astron. Soc., 453 (2015), 2177-2188.  doi: 10.1093/mnras/stv1792.  Google Scholar

[41]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, 1892. Google Scholar

[42]

M. SansotteraA. Giorgilli and T. Carletti, High-order control for symplectic maps, Phys. D, 316 (2016), 1-15.  doi: 10.1016/j.physd.2015.10.012.  Google Scholar

[43]

M. SansotteraC. Lhotka and A. Lemaître, Effective stability around the Cassini state in the spin-orbit problem, Celestial Mech. Dynam. Astronom., 119 (2014), 75-89.  doi: 10.1007/s10569-014-9547-6.  Google Scholar

[44]

M. SansotteraU. Locatelli and A. Giorgilli, On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system, Math. Comput. Simulation, 88 (2013), 1-14.  doi: 10.1016/j.matcom.2010.11.018.  Google Scholar

[45]

A. SchenkelJ. Wehr and P. Wittwer, Computer-assisted proofs for fixed point problems in Sobolev spaces, Math. Phys. Electron. J., 6 (2000), 50-117.  doi: 10.1142/9789812777874_0009.  Google Scholar

[46]

Ch. Skokos and A. Dokoumetzidis, Effective stability of the Trojan asteroids, Astron. Astroph., 367 (2001), 729-736.  doi: 10.1051/0004-6361:20000456.  Google Scholar

[47]

V. Szebehely, Theory of Orbits, Academic Press, New York, 1967. doi: 10.1016/B978-0-12-395732-0.X5001-6.  Google Scholar

[48]

E. T. Whittaker, On the adelphic integral of the differential equations of dynamics, Proc. Roy Soc. Edinburgh, Sect. A, 37 (1918), 95-116. doi: 10.1017/S037016460002352X.  Google Scholar

show all references

References:
[1]

K. Appel and W. Haken, Every planar map is four colorable. I. Discharging, Illinois J. Math., 21 (1977), 429-490.  doi: 10.1215/ijm/1256049011.  Google Scholar

[2]

K. Appel and W. Haken, Every Planar Map Is Four Colorable, Contemporary Mathematics, 98, American Mathematical Society, Providence, RI, 1989. doi: 10.1090/conm/098.  Google Scholar

[3]

K. AppelW. Haken and J. Koch, Every planar map is four colorable. Part II. Reducibility, Illinois J. Math., 21 (1977), 491-567.  doi: 10.1215/ijm/1256049012.  Google Scholar

[4]

V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Russ. Math. Surv., 18 (1963). doi: 10.1070/RM1963v018n05ABEH004130.  Google Scholar

[5]

I. BalázsJ. B. van den BergJ. CourtoisJ. Dudás and J.-P. Lessard, Computer-assisted proofs for radially symmetric solutions of PDEs, J. Comput. Dyn., 5 (2018), 61-80.  doi: 10.3934/jcd.2018003.  Google Scholar

[6]

G. D. Birkhoff, Dynamical Systems, American Mathematical Society Colloquium Publications, IX, American Mathematical Society, Providence, RI, 1966.  Google Scholar

[7]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Mem. Amer. Math. Soc., 187 (2007). doi: 10.1090/memo/0878.  Google Scholar

[8]

A. Celletti and A. Giorgilli, On the stability of the Lagrangian points in the spatial restricted problem of three bodies, Celestial Mech. Dynam. Astronom., 50 (1991), 31-38.  doi: 10.1007/BF00048985.  Google Scholar

[9]

A. CellettiA. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems, Nonlinearity, 13 (2000), 397-412.  doi: 10.1088/0951-7715/13/2/304.  Google Scholar

[10]

T. M. Cherry, On integrals developable about a singular point of a Hamiltonian system of differential equations, Proc. Camb. Phil. Soc., 22 (1924), 325-349.  doi: 10.1017/S0305004100014249.  Google Scholar

[11]

T. M. Cherry, On integrals developable about a singular point of a Hamiltonian system of differential equations Part II, Proc. Camb. Phil. Soc., 22 (1925), 510-533.  doi: 10.1017/S0305004100003224.  Google Scholar

[12]

G. Contopoulos, A Review of the "Third" Integral, Math. Engrg., 2 (2020), 472-511.  doi: 10.3934/mine.2020022.  Google Scholar

[13]

C. EfthymiopoulosA. Giorgilli and G. Contopoulos, Nonconvergence of formal integrals. II. Improved estimates for the optimal order of truncation, J. Phys. A, 37 (2004), 10831-10858.  doi: 10.1088/0305-4470/37/45/008.  Google Scholar

[14]

C. Efthymiopoulos and Z. Sándor, Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion, Mon. Not. Roy. Astron. Soc., 364 (2005), 253-271.  doi: 10.1111/j.1365-2966.2005.09572.x.  Google Scholar

[15]

J.-Ll. FiguerasA. Haro and A. Luque, Rigorous computer-assisted application of KAM theory: A modern approach, Found. Comput. Math., 17 (2017), 1123-1193.  doi: 10.1007/s10208-016-9339-3.  Google Scholar

[16]

F. Gabern and À. Jorba, A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system, Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 143-182.  doi: 10.3934/dcdsb.2001.1.143.  Google Scholar

[17]

F. GabernÀ. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids, Nonlinearity, 18 (2005), 1705-1734.  doi: 10.1088/0951-7715/18/4/017.  Google Scholar

[18]

A. Giorgilli, Exponential stability of Hamiltonian systems, in Dynamical Systems. Part I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, 87-198.  Google Scholar

[19]

A. GiorgilliA. DelshamsE. FontichL. Galgani and C. Simó, Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem, J. Differential Equations, 77 (1989), 167-198.  doi: 10.1016/0022-0396(89)90161-7.  Google Scholar

[20]

A. Giorgilli and U. Locatelli, Canonical perturbation theory for nearly integrable systems, in Chaotic Worlds: From Order to Disorder in Gravitational N-Body Dynamical Systems, Proc. NATO Adv. Study Institute, 227, Cortina, Italy, 2003, 1-41. doi: 10.1007/978-1-4020-4706-0_1.  Google Scholar

[21]

A. GiorgilliU. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celestial Mech. Dynam. Astronom., 104 (2009), 159-173.  doi: 10.1007/s10569-009-9192-7.  Google Scholar

[22]

A. GiorgilliU. Locatelli and M. Sansottera, Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; Effective stability in the light of Kolmogorov and Nekhoroshev theories, Regul. Chaotic Dyn., 22 (2017), 54-77.  doi: 10.1134/S156035471701004X.  Google Scholar

[23]

A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory, preprint, arXiv: 1303.7398. Google Scholar

[24]

A. Giorgilli and Ch. Skokos, On the stability of the Trojan asteroids, Astron. Astroph., 317 (1997), 254-261.   Google Scholar

[25]

W. Gröbner and H. Knapp, Contributions to the method of Lie series, Bibliographisches Institut, Mannheim, 1967.  Google Scholar

[26]

F. G. Gustavson, Oil constructing formal integrals of a Hamiltonian system near ail equilibrium point, Astron. J., 71 (1966), 670-686.  doi: 10.1086/110172.  Google Scholar

[27]

M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, 3 (1966), 49-66.   Google Scholar

[28]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astronom. J., 69 (1964), 73-79.  doi: 10.1086/109234.  Google Scholar

[29]

T. Johnson and W. Tucker, Automated computation of robust normal forms of planar analytic vector fields, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 769-782.  doi: 10.3934/dcdsb.2009.12.769.  Google Scholar

[30]

À. Jorba and J. Masdemont, Dynamics in the center manifold of the collinear points of the restricted three body problem, Phys. D, 132 (1999), 189-213.  doi: 10.1016/S0167-2789(99)00042-1.  Google Scholar

[31]

H. KochA. Schenkel and P. Wittwer, Computer-assisted proofs in analysis and programming in logic: A case study, SIAM Rev., 38 (1996), 565-604.  doi: 10.1137/S0036144595284180.  Google Scholar

[32]

A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Lecture Notes in Phys., 93, Springer, Berlin-New York, 1979, 51-56. doi: 10.1007/BFb0021737.  Google Scholar

[33]

C. LhotkaC. Efthymiopoulos and R. Dvorak, Nekhoroshev stability at L4 or L5 in the elliptic-restricted three-body problem -- Application to the Trojan asteroids, Mon. Not. R. Astron. Soc., 384 (2008), 1165-1177.   Google Scholar

[34]

J. E. Littlewood, On the equilateral configuration in the restricted problem of three bodies, Proc. London Math. Soc. (3), 9 (1959), 343–372. doi: 10.1112/plms/s3-9.3.343.  Google Scholar

[35]

J. E. Littlewood, The Lagrange configuration in celestial mechanics, Proc. London Math. Soc. (3), 9 (1959), 525–543. doi: 10.1112/plms/s3-10.1.640-t.  Google Scholar

[36]

R. S. MacKay and J. Stark, Locally most robust circles and boundary circles for area-preserving maps, Nonlinearity, 5 (1992), 867-888.  doi: 10.1088/0951-7715/5/4/002.  Google Scholar

[37]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20.   Google Scholar

[38]

N. N. Nekhorošev, An exponential estimates of the stability time of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5-66,287. doi: 10.1070/RM1977v032n06ABEH003859.  Google Scholar

[39]

N. N. Nekhorošev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. II, Trudy Sem. Petrovsk., 5 (1979), 5-50.   Google Scholar

[40]

R. I. Páez and U. Locatelli, Trojans dynamics well approximated by a new Hamiltonian normal form, Mon. Not. Roy. Astron. Soc., 453 (2015), 2177-2188.  doi: 10.1093/mnras/stv1792.  Google Scholar

[41]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Gauthier-Villars, Paris, 1892. Google Scholar

[42]

M. SansotteraA. Giorgilli and T. Carletti, High-order control for symplectic maps, Phys. D, 316 (2016), 1-15.  doi: 10.1016/j.physd.2015.10.012.  Google Scholar

[43]

M. SansotteraC. Lhotka and A. Lemaître, Effective stability around the Cassini state in the spin-orbit problem, Celestial Mech. Dynam. Astronom., 119 (2014), 75-89.  doi: 10.1007/s10569-014-9547-6.  Google Scholar

[44]

M. SansotteraU. Locatelli and A. Giorgilli, On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system, Math. Comput. Simulation, 88 (2013), 1-14.  doi: 10.1016/j.matcom.2010.11.018.  Google Scholar

[45]

A. SchenkelJ. Wehr and P. Wittwer, Computer-assisted proofs for fixed point problems in Sobolev spaces, Math. Phys. Electron. J., 6 (2000), 50-117.  doi: 10.1142/9789812777874_0009.  Google Scholar

[46]

Ch. Skokos and A. Dokoumetzidis, Effective stability of the Trojan asteroids, Astron. Astroph., 367 (2001), 729-736.  doi: 10.1051/0004-6361:20000456.  Google Scholar

[47]

V. Szebehely, Theory of Orbits, Academic Press, New York, 1967. doi: 10.1016/B978-0-12-395732-0.X5001-6.  Google Scholar

[48]

E. T. Whittaker, On the adelphic integral of the differential equations of dynamics, Proc. Roy Soc. Edinburgh, Sect. A, 37 (1918), 95-116. doi: 10.1017/S037016460002352X.  Google Scholar

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
$\rho_0$ $\rho$ $r_{\rm opt}$ $a_r$ $ \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
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
$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
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
$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
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
$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
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
$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
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
$\mu$ $\rho_0^2\ \, {\rm (non-res.)}$ $({\rho^*_2})^2\ \, {\rm (reson.)}$ $({\rho^*_2}/\rho_0)^2$
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
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