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Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem
Henri Poincaré's neglected ideas
Mathematisch Instituut, University of Utrecht, PO Box 80.010, 3508 TA Utrecht, The Netherlands |
The purpose of this article is to discuss two basic ideas of Henri Poincaré in the theory of dynamical systems. The first one, the recurrence theorem, got at first a lot of attention but most scientists lost interest when finding out that long timescales were involved. We will show that recurrence can be a tool to find complex dynamics in resonance zones of Hamiltonian systems; this is related to the phenomenon of quasi-trapping. To demonstrate the use of recurrence phenomena we will explore the $ 2:2:3 $ Hamiltonian resonance near stable equilibrium. This will involve interaction of low and higher order resonance. A second useful idea is concerned with the characteristic exponents of periodic solutions of dynamical systems. If a periodic solution of a Hamiltonian system has more than two zero characteristic exponents, this points at the existence of an integral of motion besides the energy. We will apply this idea to examples of two and three degrees-of-freedom (dof), the Hénon-Heiles (or Braun's) family and the $ 1:2:2 $ resonance.
References:
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, 2006. |
[2] |
G. D. Birkhoff,
Proof of Poincaré's geometric theorem, Trans. AMS, 14 (1913), 14-22.
doi: 10.2307/1988766. |
[3] |
T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painlevé property, Phys. Review A, General Physics, 3rd series, 25 (1982), 1257–1264.
doi: 10.1103/PhysRevA.25.1257. |
[4] |
H. W. Broer and F. Takens, Dynamical Systems and Chaos, Applied Math. Sciences 172, Springer, 2011.
doi: 10.1007/978-1-4419-6870-8. |
[5] |
H. W. Broer, B. Hasselblatt and F. Takens (eds.), Handbook of Dynamical Systems, Elsevier/North-Holland, Amsterdam, 2010. |
[6] |
O. Christov,
Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom, Celest. Mech. Dyn. Astr., 112 (2012), 149-167.
doi: 10.1007/s10569-011-9389-4. |
[7] |
R. C. Churchill, G. Pecelli and D. L. Rod,
Stability transitions for periodic orbits in Hamiltonian systems, Arch. Rat. Mech. Anal., 73 (1980), 313-347.
doi: 10.1007/BF00247673. |
[8] |
S. Yu. Dobrokhotov and M. A. Poteryakhin,
Normal forms near two-dimensional resonance tori for the multidimensional anharmonic oscillator, Math. Notes, 76 (2004), 653-664.
doi: 10.1023/B:MATN.0000049664.77226.52. |
[9] |
F. G. Gustavson, On constructing formal integrals of a Hamiltonian system near an equilibrium point, Astron. J., 71 (1966), 670-686. Google Scholar |
[10] |
R. Haberman,
Slow passage through the nonhyperbolic homoclinic orbit associated with a subcritical pitchfork bifurcation for Hamiltonian systems and the change of action, SIAM J. Appl. Math., 62 (2006), 488-513.
doi: 10.1137/S0036139900373836. |
[11] |
P. Holmes, J. Marsden and J. Scheurle, Exponentially small splitting of separatrices and degenerate bifurcations, Contemp. Math., 81 (1988), 213-143. Google Scholar |
[12] |
J. Kevorkian,
Perturbation techniques for oscillatory systems with slowly varying coefficients, SIAM Rev., 29 (1987), 391-461.
doi: 10.1137/1029076. |
[13] |
A. Neishtadt and Tan Su,
An asymptotic description of passage through a resonance in quasilinear Hamiltonian systems, SIAM J. Appl. Dyn. Systems., 12 (2013), 1436-1473.
doi: 10.1137/120898061. |
[14] |
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Célèste, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987. |
[15] |
H. Poincaré, Sur un theorème de géometrie, Rend. Circolo Mat. Palermo, 33 (1912), 375-407. Google Scholar |
[16] |
J. A. Sanders,
Are higher order resonances really interesting?, Celestial Mechanics, 16 (1978), 421-440.
doi: 10.1007/BF01229286. |
[17] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, 2007. |
[18] |
V. S. Steckline, Zermelo, Boltzmann and the recurrence paradox, Am. J. Physics, 51 (1983), 894-897. Google Scholar |
[19] |
E. Van der Aa and F. Verhulst,
Asymptotic integrability and periodic solutions of a Hamiltonian system in $1:2:2$ resonance, SIAM J. Math. Anal., 15 (1984), 890-911.
doi: 10.1137/0515067. |
[20] |
F. Verhulst, Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies, Phil. Trans. Royal Soc. A, 290 (1979), 435-465. Google Scholar |
[21] |
F. Verhulst, Henri Poincaré, Impatient Genius, Springer, New York, 2012.
doi: 10.1007/978-1-4614-2407-9. |
[22] |
F. Verhulst, Near-integrability and recurrence in FPU-cells, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650230, 23 pp.
doi: 10.1142/S0218127416502308. |
[23] |
G. M. Zaslavsky, The Physics of Chaos in Hamiltonian Systems, Imperial College Press (2nd extended ed.), 2007. Google Scholar |
show all references
References:
[1] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer, 2006. |
[2] |
G. D. Birkhoff,
Proof of Poincaré's geometric theorem, Trans. AMS, 14 (1913), 14-22.
doi: 10.2307/1988766. |
[3] |
T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painlevé property, Phys. Review A, General Physics, 3rd series, 25 (1982), 1257–1264.
doi: 10.1103/PhysRevA.25.1257. |
[4] |
H. W. Broer and F. Takens, Dynamical Systems and Chaos, Applied Math. Sciences 172, Springer, 2011.
doi: 10.1007/978-1-4419-6870-8. |
[5] |
H. W. Broer, B. Hasselblatt and F. Takens (eds.), Handbook of Dynamical Systems, Elsevier/North-Holland, Amsterdam, 2010. |
[6] |
O. Christov,
Non-integrability of first order resonances in Hamiltonian systems in three degrees of freedom, Celest. Mech. Dyn. Astr., 112 (2012), 149-167.
doi: 10.1007/s10569-011-9389-4. |
[7] |
R. C. Churchill, G. Pecelli and D. L. Rod,
Stability transitions for periodic orbits in Hamiltonian systems, Arch. Rat. Mech. Anal., 73 (1980), 313-347.
doi: 10.1007/BF00247673. |
[8] |
S. Yu. Dobrokhotov and M. A. Poteryakhin,
Normal forms near two-dimensional resonance tori for the multidimensional anharmonic oscillator, Math. Notes, 76 (2004), 653-664.
doi: 10.1023/B:MATN.0000049664.77226.52. |
[9] |
F. G. Gustavson, On constructing formal integrals of a Hamiltonian system near an equilibrium point, Astron. J., 71 (1966), 670-686. Google Scholar |
[10] |
R. Haberman,
Slow passage through the nonhyperbolic homoclinic orbit associated with a subcritical pitchfork bifurcation for Hamiltonian systems and the change of action, SIAM J. Appl. Math., 62 (2006), 488-513.
doi: 10.1137/S0036139900373836. |
[11] |
P. Holmes, J. Marsden and J. Scheurle, Exponentially small splitting of separatrices and degenerate bifurcations, Contemp. Math., 81 (1988), 213-143. Google Scholar |
[12] |
J. Kevorkian,
Perturbation techniques for oscillatory systems with slowly varying coefficients, SIAM Rev., 29 (1987), 391-461.
doi: 10.1137/1029076. |
[13] |
A. Neishtadt and Tan Su,
An asymptotic description of passage through a resonance in quasilinear Hamiltonian systems, SIAM J. Appl. Dyn. Systems., 12 (2013), 1436-1473.
doi: 10.1137/120898061. |
[14] |
H. Poincaré, Les Méthodes Nouvelles de la Mécanique Célèste, Librairie Scientifique et Technique Albert Blanchard, Paris, 1987. |
[15] |
H. Poincaré, Sur un theorème de géometrie, Rend. Circolo Mat. Palermo, 33 (1912), 375-407. Google Scholar |
[16] |
J. A. Sanders,
Are higher order resonances really interesting?, Celestial Mechanics, 16 (1978), 421-440.
doi: 10.1007/BF01229286. |
[17] |
J. A. Sanders, F. Verhulst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Springer, 2007. |
[18] |
V. S. Steckline, Zermelo, Boltzmann and the recurrence paradox, Am. J. Physics, 51 (1983), 894-897. Google Scholar |
[19] |
E. Van der Aa and F. Verhulst,
Asymptotic integrability and periodic solutions of a Hamiltonian system in $1:2:2$ resonance, SIAM J. Math. Anal., 15 (1984), 890-911.
doi: 10.1137/0515067. |
[20] |
F. Verhulst, Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies, Phil. Trans. Royal Soc. A, 290 (1979), 435-465. Google Scholar |
[21] |
F. Verhulst, Henri Poincaré, Impatient Genius, Springer, New York, 2012.
doi: 10.1007/978-1-4614-2407-9. |
[22] |
F. Verhulst, Near-integrability and recurrence in FPU-cells, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650230, 23 pp.
doi: 10.1142/S0218127416502308. |
[23] |
G. M. Zaslavsky, The Physics of Chaos in Hamiltonian Systems, Imperial College Press (2nd extended ed.), 2007. Google Scholar |









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