American Institute of Mathematical Sciences

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

Computer-assisted estimates for Birkhoff normal forms

Chiara Caracciolo and  Ugo Locatelli

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.

Keywords: Computer-assisted proofs, normal form methods, effective stability, Hamiltonian systems, Celestial Mechanics.
Mathematics Subject Classification: Primary: 68V05; Secondary: 37J40, 37N05, 70F07, 70H08.
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. Appel, W. 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. Balázs, J. B. van den Berg, J. Courtois, J. 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. Celletti, A. 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. Efthymiopoulos, A. 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. Figueras, A. 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. Gabern, À. 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. Giorgilli, A. Delshams, E. Fontich, L. 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. Giorgilli, U. 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. Giorgilli, U. 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. Koch, A. 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. Lhotka, C. 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. Sansottera, A. 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. Sansottera, C. 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. Sansottera, U. 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. Schenkel, J. 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. Appel, W. 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. Balázs, J. B. van den Berg, J. Courtois, J. 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. Celletti, A. 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. Efthymiopoulos, A. 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. Figueras, A. 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. Gabern, À. 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. Giorgilli, A. Delshams, E. Fontich, L. 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. Giorgilli, U. 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. Giorgilli, U. 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. Koch, A. 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. Lhotka, C. 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. Sansottera, A. 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. Sansottera, C. 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. Sansottera, U. 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. Schenkel, J. 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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$\rho_0$ $\rho$ $r_{\rm opt}$ $a_r$ $ \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
Table Options

Download as excel

$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
Table Options

Download as excel

$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
Table Options

Download as excel

$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
Table Options

Download as excel

$\rho_0^2$ $\rho^2$ $T$ $\rho_0^2$ $({\rho^*_2})^2$ $\rho^2$ $T$
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
Table Options

Download as excel

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

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  doi: 10.3934/dcds.2020162
[5]

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
[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]

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
[17]

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
[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: 

Tools

Metrics

  • PDF downloads (42)
  • HTML views (51)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences