doi: 10.3934/dcdss.2020062

A Galilean dance 1:2:4 resonant periodic motions and their librations of Jupiter and his Galilean moons

1. 

Bernoulli Instituut, Rijksuniversiteit Groningen, Postbus 407, NL 9700 AK Groningen, The Netherlands

2. 

Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, NL 3508 TA Utrecht, The Netherlands

Received  November 2017 Revised  March 2018 Published  April 2019

The four Galilean moons of Jupiter were discovered by Galileo in the early 17th century, and their motion was first seen as a miniature solar system. Around 1800 Laplace discovered that the Galilean motion is subjected to an orbital $ 1{:}2{:}4 $-resonance of the inner three moons Io, Europa and Ganymedes. In the early 20th century De Sitter gave a mathematical explanation for this in a Newtonian framework. In fact, he found a family of stable periodic solutions by using the seminal work of Poincaré, which at the time was quite new. In this paper we review and summarize recent results of Broer, Hanßmann and Zhao on the motion of the entire Galilean system, so including the fourth moon Callisto. To this purpose we use a version of parametrised Kolmogorov-Arnol'd-Moser theory where a family of multi-periodic isotropic invariant three-dimensional tori is found that combines the periodic motions of De Sitter and Callisto. The $ 3 $-tori are normally elliptic and excite a family of invariant Lagrangean $ 8 $-tori that project down to librational motions. Both the $ 3 $- and the $ 8 $-tori occur for an almost full Hausdorff measure set in the product of corresponding dimension in phase space and a parameter space, where the external parameters are given by the masses of the moons.

Citation: Henk W. Broer, Heinz Hanssmann. A Galilean dance 1:2:4 resonant periodic motions and their librations of Jupiter and his Galilean moons. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020062
References:
[1]

V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Mathematical Surveys, 18: 6 (1963), 91-192.  Google Scholar

[2]

V. I. Arnol'd, Instability of dyanmical systems with several degrees of freedom, Soviet Mathematics, 5 (1964), 581-585.   Google Scholar

[3]

V. I. Arnol'd, On matrices depending on parameters, Russian Uspehi Mathematical Surveys, 26: 2 (1971), 101-114.  Google Scholar

[4]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar

[5]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1983.  Google Scholar

[6]

V. I. Arnol'd, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, 2006.  Google Scholar

[7]

H. W. Broer, Normal forms in perturbation theory, In R. Meyers, editor, Encyclopædia of Complexity & System Science, 2009, 6310-6329. Google Scholar

[8]

H. W. Broer and H. Hanßmann, On Jupiter and his Galilean satellites: Librations of De Sitter's periodic motions, Indagationes Mathematicæ, 27 (2016), 1305-1336. doi: 10.1016/j.indag.2016.09.002.  Google Scholar

[9]

H. W. BroerH. Hanßmann and J. Hoo, The quasi-periodic Hamiltonian Hopf bifurcation, Nonlinearity, 20 (2007), 417-460.  doi: 10.1088/0951-7715/20/2/009.  Google Scholar

[10]

H. W. BroerH. HanßmannÀ JorbaJ. Villanueva and F. O. O. Wagener, Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach, Nonlinearity, 16 (2003), 1751-1791.  doi: 10.1088/0951-7715/16/5/312.  Google Scholar

[11]

H. W. Broer, H. Hanßmann and F. O. O. Wagener, Quasi-Periodic Bifurcation Theory, the Geometry of KAM, Springer, in preparation. Google Scholar

[12]

H. W. BroerJ. Hoo and V. Naudot, Normal stability of quasi-periodic tori, Journal of Differential Equations, 232 (2007), 355-418.  doi: 10.1016/j.jde.2006.08.022.  Google Scholar

[13]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, LNM 1645, Springer, 1996.  Google Scholar

[14]

H. W. Broer, G. B. Huitema and F. Takens, Unfoldings of quasi-periodic tori, Memoirs of the American Mathematical Society, 83 # 421 (1990), 1-82.  Google Scholar

[15]

H. W. Broer and M. B. Sevryuk, KAM theory: Quasi-periodicity in dynamical systems, Handbook of Dynamical Systems, 3 (2010), 249-344.  doi: 10.1016/S1874-575X(10)00314-0.  Google Scholar

[16]

H. W. Broer and G. Vegter, Bifurcational aspects of parametric resonance, Dynamics Reported, New Series, 1 (1992), 1-53.   Google Scholar

[17]

H. W. Broer and G. Vegter, Generic Hopf-Neimark-Sacker bifurcations in feed forward systems, Nonlinearity, 21 (2008), 1547-1578.  doi: 10.1088/0951-7715/21/7/010.  Google Scholar

[18]

H. W. Broer and L. Zhao, De Sitter's theory of Galilean satellites, Celestial Mechanics and Dynamical Astronomy, 127 (2017), 95-119.  doi: 10.1007/s10569-016-9718-8.  Google Scholar

[19]

A. Celletti, F. Paita and G. Pucacco, The dynamics of the de Sitter resonance, Celestial Mechanics and Dynamical Astronomy, 130, Art. 15 (2018), 1-15. doi: 10.1007/s10569-017-9815-3.  Google Scholar

[20]

J. Féjoz, Démonstration du théorème d'Arnold sur la stabilité du système planétaire (d'après Herman), Ergodic Theory and Dynamical Systems, 24 (2004), 1521-1582. Revised version, available at https://www.ceremade.dauphine.fr/~fejoz/Articles/Fejoz_2004_Arnold.pdf  Google Scholar

[21]

G. Galileo, Sidereus Nuncius, 1610. Translated as Sidereus Nuncius, or, The Sidereal Messenger, University of Chicago Press, Chicago, IL, 1989. doi: 10.7208/chicago/9780226279046.001.0001.  Google Scholar

[22]

D. M. Galin, Deformations of linear Hamiltonian systems, Translations of the American Mathematical Society, 118 (1982), 1-12.   Google Scholar

[23]

J. Guichelaar, De Sitter. Een alternatief voor Einsteins heelalmodel, Natuurwetenschappen & Techniek, Van Veen Magazines, 2009. Google Scholar

[24]

Y. HanY. Li and Y. Yi, Invariant tori in Hamiltonian systems with high order proper degeneracy, Annales Henri Poincaré, 10 (2010), 1419-1436.  doi: 10.1007/s00023-010-0026-7.  Google Scholar

[25]

H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems - Results and Examples, LNM 1893, Springer, 2007.  Google Scholar

[26]

G. B. Huitema, Unfoldings of Quasi-Periodic Tori, Thesis, University of Groningen, 1988. Google Scholar

[27]

À. Jorba and J. Villanueva, On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems, Nonlinearity, 10 (1997), 783-822. doi: 10.1088/0951-7715/10/4/001.  Google Scholar

[28] P. C. van der Kruit, De Inrichting van de Hemel: Een Biografie van de Astronoom Jacobus C. Kapteyn, Amsterdam University Press, 2016.   Google Scholar
[29]

V. LaineyL. Duriez and A. Vienne, New accurate ephemerides for the Galilean satellites of Jupiter I. Numerical integration of elaborated equations of motion, Astronomy & Astrophysics, 420 (2004), 1171-1183.   Google Scholar

[30]

P. S. de Laplace, Traité de Mécanique Céleste, Œuvres Complètes, tome, 4 (1799), 1-501. Google Scholar

[31]

R. Malhotra, Tidal origin of the Laplace Resonance and the Resurfacing of Ganymede, Icarus, 94 (1991), 399-412.  doi: 10.1016/0019-1035(91)90237-N.  Google Scholar

[32]

J. K. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  doi: 10.1007/BF01399536.  Google Scholar

[33]

J. K. Moser and E. J. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes, 12, AMS 2005. doi: 10.1090/cln/012.  Google Scholar

[34]

H. Poincaré, Thèse, Œuvres I, pp. LIX-CXXIX, Gauthier VIllars 1928, 1879. Google Scholar

[35]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, tome Ⅱ., Librairie Scientifique et Technique Albert Blanchard, Paris, 1987, 1892-1899.  Google Scholar

[36]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications Pure Applied Mathematics, 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[37]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, 2nd ed, Applied Mathematical Sciences 59, Springer, 2007.  Google Scholar

[38]

M. B. Sevryuk, Partial preservation of frequencies and Floquet exponents in KAM theory, Proceedings Steklov Institute Mathematics, 259 (2007), 167-195.  doi: 10.1134/S0081543807040128.  Google Scholar

[39]

M. B. Sevryuk, KAM tori: Persistence and smoothness, Nonlinearity, 21 (2008), T177-T185. doi: 10.1088/0951-7715/21/10/T01.  Google Scholar

[40]

W. de Sitter, On the periodic solutions of a particular case of the problem of four bodies, KNAW Proceedings, 11 (1909), 682-698.   Google Scholar

[41]

W. de Sitter, Outlines of a new mathematical theory of Jupiter's satellites, Annalen van de Sterrewacht te Leiden, 12 (1925), 1-55.   Google Scholar

[42]

W. de Sitter, Jupiter's galilean satellites (George Darwin Lecture), Monthly Notices of the Royal Astronomical Society, 91 (1931), 706-738.   Google Scholar

[43]

F. Tisserand, Traité de Mécanique Céleste, volume 1., Gauthier-Villars, 1889. Google Scholar

[44] R. S. Westfall, Never at Rest: A Biography of Isaac Newton, Cambridge University Press, Cambridge, 1980.   Google Scholar
[45]

L. Zhao, Quasi-periodic solutions of the spatial lunar three-body problem, Celestial Mechanics and Dynamical Astronomy, 119 (2014), 91-118.  doi: 10.1007/s10569-014-9549-4.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Mathematical Surveys, 18: 6 (1963), 91-192.  Google Scholar

[2]

V. I. Arnol'd, Instability of dyanmical systems with several degrees of freedom, Soviet Mathematics, 5 (1964), 581-585.   Google Scholar

[3]

V. I. Arnol'd, On matrices depending on parameters, Russian Uspehi Mathematical Surveys, 26: 2 (1971), 101-114.  Google Scholar

[4]

V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, 60. Springer-Verlag, New York-Heidelberg, 1978.  Google Scholar

[5]

V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer, 1983.  Google Scholar

[6]

V. I. Arnol'd, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, 2006.  Google Scholar

[7]

H. W. Broer, Normal forms in perturbation theory, In R. Meyers, editor, Encyclopædia of Complexity & System Science, 2009, 6310-6329. Google Scholar

[8]

H. W. Broer and H. Hanßmann, On Jupiter and his Galilean satellites: Librations of De Sitter's periodic motions, Indagationes Mathematicæ, 27 (2016), 1305-1336. doi: 10.1016/j.indag.2016.09.002.  Google Scholar

[9]

H. W. BroerH. Hanßmann and J. Hoo, The quasi-periodic Hamiltonian Hopf bifurcation, Nonlinearity, 20 (2007), 417-460.  doi: 10.1088/0951-7715/20/2/009.  Google Scholar

[10]

H. W. BroerH. HanßmannÀ JorbaJ. Villanueva and F. O. O. Wagener, Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach, Nonlinearity, 16 (2003), 1751-1791.  doi: 10.1088/0951-7715/16/5/312.  Google Scholar

[11]

H. W. Broer, H. Hanßmann and F. O. O. Wagener, Quasi-Periodic Bifurcation Theory, the Geometry of KAM, Springer, in preparation. Google Scholar

[12]

H. W. BroerJ. Hoo and V. Naudot, Normal stability of quasi-periodic tori, Journal of Differential Equations, 232 (2007), 355-418.  doi: 10.1016/j.jde.2006.08.022.  Google Scholar

[13]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, LNM 1645, Springer, 1996.  Google Scholar

[14]

H. W. Broer, G. B. Huitema and F. Takens, Unfoldings of quasi-periodic tori, Memoirs of the American Mathematical Society, 83 # 421 (1990), 1-82.  Google Scholar

[15]

H. W. Broer and M. B. Sevryuk, KAM theory: Quasi-periodicity in dynamical systems, Handbook of Dynamical Systems, 3 (2010), 249-344.  doi: 10.1016/S1874-575X(10)00314-0.  Google Scholar

[16]

H. W. Broer and G. Vegter, Bifurcational aspects of parametric resonance, Dynamics Reported, New Series, 1 (1992), 1-53.   Google Scholar

[17]

H. W. Broer and G. Vegter, Generic Hopf-Neimark-Sacker bifurcations in feed forward systems, Nonlinearity, 21 (2008), 1547-1578.  doi: 10.1088/0951-7715/21/7/010.  Google Scholar

[18]

H. W. Broer and L. Zhao, De Sitter's theory of Galilean satellites, Celestial Mechanics and Dynamical Astronomy, 127 (2017), 95-119.  doi: 10.1007/s10569-016-9718-8.  Google Scholar

[19]

A. Celletti, F. Paita and G. Pucacco, The dynamics of the de Sitter resonance, Celestial Mechanics and Dynamical Astronomy, 130, Art. 15 (2018), 1-15. doi: 10.1007/s10569-017-9815-3.  Google Scholar

[20]

J. Féjoz, Démonstration du théorème d'Arnold sur la stabilité du système planétaire (d'après Herman), Ergodic Theory and Dynamical Systems, 24 (2004), 1521-1582. Revised version, available at https://www.ceremade.dauphine.fr/~fejoz/Articles/Fejoz_2004_Arnold.pdf  Google Scholar

[21]

G. Galileo, Sidereus Nuncius, 1610. Translated as Sidereus Nuncius, or, The Sidereal Messenger, University of Chicago Press, Chicago, IL, 1989. doi: 10.7208/chicago/9780226279046.001.0001.  Google Scholar

[22]

D. M. Galin, Deformations of linear Hamiltonian systems, Translations of the American Mathematical Society, 118 (1982), 1-12.   Google Scholar

[23]

J. Guichelaar, De Sitter. Een alternatief voor Einsteins heelalmodel, Natuurwetenschappen & Techniek, Van Veen Magazines, 2009. Google Scholar

[24]

Y. HanY. Li and Y. Yi, Invariant tori in Hamiltonian systems with high order proper degeneracy, Annales Henri Poincaré, 10 (2010), 1419-1436.  doi: 10.1007/s00023-010-0026-7.  Google Scholar

[25]

H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems - Results and Examples, LNM 1893, Springer, 2007.  Google Scholar

[26]

G. B. Huitema, Unfoldings of Quasi-Periodic Tori, Thesis, University of Groningen, 1988. Google Scholar

[27]

À. Jorba and J. Villanueva, On the normal behaviour of partially elliptic lower-dimensional tori of Hamiltonian systems, Nonlinearity, 10 (1997), 783-822. doi: 10.1088/0951-7715/10/4/001.  Google Scholar

[28] P. C. van der Kruit, De Inrichting van de Hemel: Een Biografie van de Astronoom Jacobus C. Kapteyn, Amsterdam University Press, 2016.   Google Scholar
[29]

V. LaineyL. Duriez and A. Vienne, New accurate ephemerides for the Galilean satellites of Jupiter I. Numerical integration of elaborated equations of motion, Astronomy & Astrophysics, 420 (2004), 1171-1183.   Google Scholar

[30]

P. S. de Laplace, Traité de Mécanique Céleste, Œuvres Complètes, tome, 4 (1799), 1-501. Google Scholar

[31]

R. Malhotra, Tidal origin of the Laplace Resonance and the Resurfacing of Ganymede, Icarus, 94 (1991), 399-412.  doi: 10.1016/0019-1035(91)90237-N.  Google Scholar

[32]

J. K. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  doi: 10.1007/BF01399536.  Google Scholar

[33]

J. K. Moser and E. J. Zehnder, Notes on Dynamical Systems, Courant Lecture Notes, 12, AMS 2005. doi: 10.1090/cln/012.  Google Scholar

[34]

H. Poincaré, Thèse, Œuvres I, pp. LIX-CXXIX, Gauthier VIllars 1928, 1879. Google Scholar

[35]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, tome Ⅱ., Librairie Scientifique et Technique Albert Blanchard, Paris, 1987, 1892-1899.  Google Scholar

[36]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications Pure Applied Mathematics, 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.  Google Scholar

[37]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, 2nd ed, Applied Mathematical Sciences 59, Springer, 2007.  Google Scholar

[38]

M. B. Sevryuk, Partial preservation of frequencies and Floquet exponents in KAM theory, Proceedings Steklov Institute Mathematics, 259 (2007), 167-195.  doi: 10.1134/S0081543807040128.  Google Scholar

[39]

M. B. Sevryuk, KAM tori: Persistence and smoothness, Nonlinearity, 21 (2008), T177-T185. doi: 10.1088/0951-7715/21/10/T01.  Google Scholar

[40]

W. de Sitter, On the periodic solutions of a particular case of the problem of four bodies, KNAW Proceedings, 11 (1909), 682-698.   Google Scholar

[41]

W. de Sitter, Outlines of a new mathematical theory of Jupiter's satellites, Annalen van de Sterrewacht te Leiden, 12 (1925), 1-55.   Google Scholar

[42]

W. de Sitter, Jupiter's galilean satellites (George Darwin Lecture), Monthly Notices of the Royal Astronomical Society, 91 (1931), 706-738.   Google Scholar

[43]

F. Tisserand, Traité de Mécanique Céleste, volume 1., Gauthier-Villars, 1889. Google Scholar

[44] R. S. Westfall, Never at Rest: A Biography of Isaac Newton, Cambridge University Press, Cambridge, 1980.   Google Scholar
[45]

L. Zhao, Quasi-periodic solutions of the spatial lunar three-body problem, Celestial Mechanics and Dynamical Astronomy, 119 (2014), 91-118.  doi: 10.1007/s10569-014-9549-4.  Google Scholar

Figure 1.  Left: Galileo Galilei 1564-1642. Right: The Galilean moons of Jupiter
Figure 2.  Left: Pierre Simon de Laplace 1749-1827. Right: Willem de Sitter 1872-1934
Figure 3.  Delaunay angles: 'mean anomaly' $\ell$ and 'argument' $g$ of the pericenter of a Keplerian motion
Figure 4.  Poincaré's Ansatz: look near the 16 possible collinearities of Jupiter-Io-Europa-Ganymedes. It turns out that the one in the last column in the first row from above corresponds to the stable situation, where all moons are in their perijoves and the ellipse of Europa is $\pi$-rotated with respect to those of Io and Ganymedes
Figure 5.  Sketch of the set $(\mathbb{R}^2)_{\tau,\gamma}$
Figure 6.  Sketch of $\Gamma^\gamma_{\tau,\gamma} \subseteq \Gamma^\gamma \subseteq \Gamma$
[1]

V. Afraimovich, J. Schmeling, Edgardo Ugalde, Jesús Urías. Spectra of dimensions for Poincaré recurrences. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 901-914. doi: 10.3934/dcds.2000.6.901

[2]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835

[3]

Juan Wang, Xiaodan Zhang, Yun Zhao. Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2315-2332. doi: 10.3934/dcds.2014.34.2315

[4]

Godofredo Iommi, Bartłomiej Skorulski. Multifractal analysis for the exponential family. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 857-869. doi: 10.3934/dcds.2006.16.857

[5]

V. Afraimovich, Jean-René Chazottes, Benoît Saussol. Pointwise dimensions for Poincaré recurrences associated with maps and special flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 263-280. doi: 10.3934/dcds.2003.9.263

[6]

Julien Barral, Yan-Hui Qu. On the higher-dimensional multifractal analysis. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 1977-1995. doi: 10.3934/dcds.2012.32.1977

[7]

Mario Roy, Mariusz Urbański. Multifractal analysis for conformal graph directed Markov systems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 627-650. doi: 10.3934/dcds.2009.25.627

[8]

Zhihui Yuan. Multifractal analysis of random weak Gibbs measures. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5367-5405. doi: 10.3934/dcds.2017234

[9]

Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345

[10]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[11]

Valentin Afraimovich, Jean-Rene Chazottes and Benoit Saussol. Local dimensions for Poincare recurrences. Electronic Research Announcements, 2000, 6: 64-74.

[12]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[13]

Jean-Pierre Francoise, Claude Piquet. Global recurrences of multi-time scaled systems. Conference Publications, 2011, 2011 (Special) : 430-436. doi: 10.3934/proc.2011.2011.430

[14]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[15]

Lars Olsen. First return times: multifractal spectra and divergence points. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 635-656. doi: 10.3934/dcds.2004.10.635

[16]

Imen Bhouri, Houssem Tlili. On the multifractal formalism for Bernoulli products of invertible matrices. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1129-1145. doi: 10.3934/dcds.2009.24.1129

[17]

Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473

[18]

Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049

[19]

João Lopes Dias. Brjuno condition and renormalization for Poincaré flows. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 641-656. doi: 10.3934/dcds.2006.15.641

[20]

Antonio Giorgilli, Stefano Marmi. Convergence radius in the Poincaré-Siegel problem. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 601-621. doi: 10.3934/dcdss.2010.3.601

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (34)
  • HTML views (432)
  • Cited by (0)

Other articles
by authors

[Back to Top]