Dynamics of Axially Symmetric Perturbed Hamiltonians in 1:1:1 Resonance

We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.

[1]  R. Cushman,et al.  A Universal Reduction Procedure for Hamiltonian Group Actions , 1991 .

[2]  Kenneth R. Meyer,et al.  Symmetries and Integrals in Mechanics , 1973 .

[3]  D. Farrelly,et al.  Normalization and the detection of integrability: The generalized Van Der Waals potential , 1995 .

[4]  H. Broer,et al.  The quasi-periodic Hamiltonian Hopf bifurcation , 2007 .

[5]  Jan Cornelis van der Meer,et al.  The Hamiltonian Hopf Bifurcation , 1985 .

[6]  O. Cornea.  Lusternik-Schnirelmann Category , 2003 .

[7]  A. Deprit.  The Lissajous transformation I. Basics , 1991 .

[8]  Konstantinos Efstathiou,et al.  Analysis of Rotation-Vibration Relative Equilibria on the Example of a Tetrahedral Four Atom Molecule , 2004, SIAM J. Appl. Dyn. Syst..

[9]  J. Marsden,et al.  Reduction of symplectic manifolds with symmetry , 1974 .

[10]  George Huitema,et al.  Quasi-Periodic Motions in Families of Dynamical Systems: Order amidst Chaos , 2002 .

[11]  T. Zeeuw.  Dynamical Models for Axisymmetric and Triaxial Galaxies , 1987 .

[12]  Jesús F. Palacián,et al.  Periodic Solutions and KAM Tori in a Triaxial Potential , 2017, SIAM J. Appl. Dyn. Syst..

[13]  Els van der Aa.  First-order resonances in three-degrees-of-freedom systems , 1983 .

[14]  Kenneth R. Meyer,et al.  Periodic Solutions in Hamiltonian Systems, Averaging, and the Lunar Problem , 2008, SIAM J. Appl. Dyn. Syst..

[15]  M. Kummer,et al.  On averaging, reduction, and symmetry in hamiltonian systems , 1983 .

[16]  Richard Cushman,et al.  Global Aspects of Classical Integrable Systems , 2004 .

[17]  George Haller,et al.  Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems , 1996 .

[18]  Bernd Sturmfels,et al.  Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation) , 2008 .

[19]  Kenneth R. Meyer,et al.  Periodic orbits near ℒ4 for mass ratios near the critical mass ratio of routh , 1971 .

[20]  R. Cushman,et al.  Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator , 1999 .

[21]  V. Lanchares,et al.  Perturbed ion traps: A generalization of the three-dimensional Henon-Heiles problem. , 2002, Chaos.

[22]  Y. Yi,et al.  Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy , 2010 .

[23]  A. Elipe,et al.  Analytical investigation of the orbital structure close to the 1:1:1 resonance in spheroidal galaxies , 2005 .

[24]  André Deprit,et al.  Canonical transformations depending on a small parameter , 1969 .

[25]  C. Valls,et al.  PERIODIC SOLUTIONS OF A GALACTIC POTENTIAL , 2014 .

[26]  Konstantinos Efstathiou,et al.  Metamorphoses of Hamiltonian Systems with Symmetries , 2005 .

[27]  K. Efstathiou,et al.  Perturbations of the 1:1:1 resonance with tetrahedral symmetry: a three degree of freedom analogue of the two degree of freedom Henon-Heiles Hamiltonian , 2004 .

[28]  Kenneth R. Meyer,et al.  Introduction to Hamiltonian Dynamical Systems and the N-Body Problem , 1991 .

[29]  N. Caranicolas.  1:1:1 resonant periodic orbits in 3-dimensional galactic-type Hamiltonians , 1994 .

[30]  M. Stiavelli,et al.  Structure and dynamics of elliptical galaxies , 1993 .

[31]  D. Merritt,et al.  Stellar orbits in a triaxial galaxy. I - Orbits in the plane of rotation , 1983 .

[32]  V. Arnold,et al.  Mathematical aspects of classical and celestial mechanics , 1997 .

[33]  S. Ferrer,et al.  Hamiltonian fourfold 1:1 resonance with two rotational symmetries , 2007 .

[34]  M. Kummer.  On resonant classical Hamiltonians with frequencies , 1990 .

[35]  Kenneth R. Meyer,et al.  Geometric Averaging of Hamiltonian Systems: Periodic Solutions, Stability, and KAM Tori , 2011, SIAM Journal on Applied Dynamical Systems.

[36]  Qualitative features of Hamiltonian systems through averaging and reduction , 2007 .

[37]  H. Hanßmann.  Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples , 2006 .

[38]  Jesús F. Palacián,et al.  Hamiltonian Oscillators in 1—1—1 Resonance: Normalization and Integrability , 2000, J. Nonlinear Sci..

[39]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[40]  H. Hanßmann,et al.  Algebraic Methods for Determining Hamiltonian Hopf Bifurcations in Three-Degree-of-Freedom Systems , 2005 .

[41]  E. Zotos,et al.  Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits , 2012, 1206.5394.

[42]  G. Haller,et al.  Chaos near resonance , 1999 .

[43]  Luis Alberto Ibort Latre.  The geometry of dynamics , 1996 .

[44]  J. Llibre,et al.  Periodic orbits for the generalized Yang–Mills Hamiltonian system in dimension 6 , 2014 .

[45]  Alberto Ibort,et al.  Geometry from Dynamics, Classical and Quantum , 2014 .

[46]  K. Meyer,et al.  Canonical forms for symplectic and Hamiltonian matrices , 1974 .

[47]  Yanheng Ding,et al.  Periodic Solutions of Hamiltonian Systems , 2000, SIAM J. Math. Anal..

[48]  Normal forms of 3 degree of freedom Hamiltonian systems at equilibrium in the resonant case , 2007 .

[49]  Bruce R. Miller,et al.  Normalization in the Face of Integrability , 1988 .

[50]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[51]  H. Schomerus.  Periodic orbits near bifurcations of codimension two: Classical mechanics, semiclassics and Stokes transitions. , 1997, chao-dyn/9711013.

[52]  Periodic orbits of a perturbed 3-dimensional isotropic oscillator with axial symmetry , 2016 .

[53]  Heinz Hanßmann,et al.  On the Hamiltonian Hopf Bifurcations in the 3D Hénon–Heiles Family , 2001 .

[54]  George Huitema,et al.  Quasi-periodic motions in families of dynamical systems , 1996 .

[55]  Jorge V. José,et al.  Chaos in classical and quantum mechanics , 1990 .

[56]  The Hénon and Heiles Problem in Three Dimensions. , 1998 .

[57]  A. Weinstein.  Bifurcations and Hamilton's principle , 1978 .

[58]  K. Meyer,et al.  Invariant tori in the lunar problem , 2014 .

[59]  Alan Weinstein,et al.  Normal modes for nonlinear hamiltonian systems , 1973 .

[60]  A. Weinstein.  Symplectic v‐manifolds, periodic orbits of hamiltonian systems, and the volume of certain riemannian manifolds , 1977 .

[61]  New 1:1:1 periodic solutions in $$3$$3-dimensional galactic-type Hamiltonian systems , 2014 .

[62]  Jesús F. Palacián,et al.  On perturbed oscillators in 1-1-1 resonance: the case of axially symmetric cubic potentials , 2002 .

[64]  J. Moser,et al.  Regularization of kepler's problem and the averaging method on a manifold , 1970 .

[65]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[66]  K. Meyer,et al.  Singular reduction of resonant Hamiltonians , 2018 .

[67]  J. Llibre,et al.  New families of periodic orbits for a galactic potential , 2016 .

[68]  R. Cushman,et al.  Linear Hamiltonian Hopf bifurcation for point–group–invariant perturbations of the 1:1:1 resonance , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[69]  Darryl D. Holm,et al.  Hamiltonian chaos in nonlinear optical polarization dynamics , 1990 .

[70]  J. C. van der Meer,et al.  Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry , 2011, J. Nonlinear Sci..