PaperNo Access

Mathematisch Instituut, P. O. Box 80.010, 508 TA Utrecht, Netherlands

Corresponding author.

Search for more papers by this author

and

Taoufik Bakri

https://orcid.org/0009-0009-8225-1033

TNO Sustainable Urban Mobility and Safety, P. O. Box 96800, 2509 JE The Hague, The Netherlands

E-mail Address: taoufik.bakri@tno.nl

Search for more papers by this author

https://doi.org/10.1142/S0218127424501682Cited by:0 (Source: Crossref)

Abstract

We review a number of methods to prove nonintegrability of Hamiltonian systems and focus on 3 Degrees-of-Freedom (DoF) systems listing the known results for the prominent resonances. Associated with the Hamiltonian systems are the averaged-normal forms that provide us with geometric insight, approximations of orbits and measures of chaos. Symmetries do change the qualitative and quantitative pictures; we illustrate this for the 1:2:1 resonance with discrete symmetry in the 1st and 3rd DoF. In this case, the averaged-normal form is still nonintegrable, but it becomes integrable when adding discrete symmetry in all DoF. Apart from the short-periodic solutions obtained by averaging, we find many periodic solutions. There is numerical evidence of the presence of Šilnikov bifurcation which clarifies the presence of nonintegrability phenomena qualitatively and quantitatively.

Keywords:

References

Arnold, V. I. [1978] Mathematical Methods of Classical Mechanics (Springer). Crossref, Google Scholar
Arnold, V. I. [1983] Geometrical Methods in the Theory of Ordinary Differential Equations (Springer). Crossref, Google Scholar
Bakri, T. & Verhulst, F. [2022] “Time-reversal, tori families and canards in the Sprott A and NE9 systems,” Chaos 32, 083119. Crossref, Google Scholar
Broer, H. W. & Sevryuk, M. B. [2010] “KAM theory: Quasi-periodicity in dynamical systems,” in Handbook of Dynamical Systems, eds. Broer, H. W., Hasselblatt, B. & Takens, F., Vol. 3 (Elsevier), pp. 249–344. Google Scholar
Christov, O. [2012] “Non-integrability of first order resonances in Hamiltonian systems with three degrees of freedom,” Celest. Mech. Dyn. Astron. 112, 149–167. Crossref, Google Scholar
Christov, O. [2020] “On the integrability of the 1:2:2 resonance,” Nonlin. Dyn. 102, 2295–2309. Crossref, Google Scholar
Christov, O. [2023] private communication. Google Scholar
Devaney, R. L. [1976] “Homoclinic orbits in Hamiltonian systems,” J. Differ. Eqs. 21, 431–438. Crossref, Web of Science, Google Scholar
Duistermaat, J. J. [1984] “Non-integrability of the 1:1:2 resonance,” Ergod. Theory Dyn. Syst. 4, 553–568. Crossref, Google Scholar
Duval, A. & Loday-Richaud, M. [1992] “Kovacic’s algorithm and its application to some families of special functions,” Appl. Algebra Eng. Commun. Comput. 3, 211–246. Crossref, Google Scholar
Fermi, E., Pasta, J. & Ulam, S. [1955] “Los Alamos Report LA-1940,” in Collected Papers of Enrico Fermi, ed. Fermi, E., Vol. 2 (University of Chicago Press, Chicago), pp. 977–988. Google Scholar
Gustavson, F. G. [1966] “On constructing formal integrals of a Hamiltonian system near an equilibrium point,” Astron. J. 71, 670. Crossref, Web of Science, Google Scholar
Holmes, P., Marsden, J. & Scheurle, J. [1988] “Exponentially small splitting of separatrices with applications to KAM theory and degenerate bifurcations,” Contemp. Math. 81, 213–244. Crossref, Google Scholar
Hoveijn, I. & Verhulst, F. [1990] “Chaos in the 1:2:3 Hamiltonian normal form,” Physica D 44, 397–406. Crossref, Web of Science, Google Scholar
Kozlov, V. V. [1996] Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergebnisse der Mathematik Bd 31 (Springer). Crossref, Google Scholar
Laskar, J. [1993] “Frequency analysis of a dynamical system,” Celest. Mech. Dyn. Astron. 56, 191–196. Crossref, Web of Science, Google Scholar
Levi, M. [2010] “Some applications of Moser’s twist theorem,” in Handbook of Dynamical Systems, eds. Broer, H. W., Hasselblatt, B. & Takens, F., Vol. 3 (Elsevier), pp. 225–247. Google Scholar
Martinet, L., Magnenat, P. & Verhulst, F. [1981] “On the number of isolating integrals in resonant systems with 3 degrees of motion,” Celest. Mech. Dyn. Astron. 29, 93–99. Crossref, Google Scholar
Morales-Ruiz, J. [2000] “Kovalevskaya, Liapounov, Painlevé, Ziglin and the differential Galois theory,” Regul. Chaotic Dyn. 5, 251–271. Crossref, Web of Science, Google Scholar
Morales-Ruiz, J. & Ramis, J. P. [2010] “Integrability of dynamical systems through differential Galois theory: Practical guide,” Contemp. Math. 509, 143–220. Crossref, Google Scholar
Poincaré, H. [1892, 1893, 1899] Les Méthodes Nouvelles de la Mécanique Célèste, 3 Vols (Gauthier-Villars, Paris). Crossref, Google Scholar
Rink, B. W. & Verhulst, F. [2000] “Near-integrability of periodic FPU-chains,” Physica A 285, 467–482. Crossref, Web of Science, Google Scholar
Sanders, J. A., Verhulst, F. & Murdock, J. [2007] Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 2nd edition, Vol. 59 (Springer). Google Scholar
van der Aa, E. & Verhulst, F. [1984] “Asymptotic integrability and periodic solutions of a system in 1:2:2 resonance,” SIAM J. Appl. Math. Anal. 15, 890–911. Crossref, Google Scholar
Verhulst, F. [1979] “Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies,” Philos. Trans. R. Soc. Lond. A 290, 435–465. Crossref, Web of Science, Google Scholar
Verhulst, F. [2016] “Near-integrability and recurrence in FPU cell-chains,” Int. J. Bifurcation and Chaos 26, 1650230. Link, Google Scholar
Verhulst, F. [2023] A Toolbox of Averaging Theorems, Ordinary and Partial Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 12 (Springer). Crossref, Google Scholar
Weinstein, A. [1973] “Normal modes for nonlinear Hamiltonian systems,” Invent. Math. 20, 47–57. Crossref, Web of Science, Google Scholar