On perturbed oscillators in 1-1-1 resonance: The case of axially symmetric cubic potentials
type: Item
instance of: scholarly article
title: On perturbed oscillators in 1-1-1 resonance: The case of axially symmetric cubic potentials
zbMATH Open document ID: 1037.34031
DOI: 10.1016/S0393-0440(01)00041-9
author: S. Ferrer
author: Heinz Hanßmann
author: Patricia Yanguas
author: Jesús F. Palacián
published in: Journal of Geometry and Physics
publication date: 2002-06-23T00:00:00Z
review text: From authors' abstract: Axially symmetric perturbations of the isotropic harmonic oscillator in three dimensions are studied. A normal form transformation introduces a second symmetry, after truncation. The reduction of the two symmetries leads to a one-degree-of-freedom system. To this end, we use a special set of action-angle variables, as well as conveniently chosen generators of the ring of invariant functions. Also, we analyse the 2-parameter family of (arbitrary) axially symmetric cubic potentials. This family has rich dynamics, displaying all local bifurcations of co-dimension one. With the exception of six ratios of the parameter values, the dynamic behavior close to the origin turns out to be completely determined by the normal form of order 1.
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 70H08
Mathematics Subject Classification ID: 70H15
zbMATH DE Number: 1758073
zbMATH Keywords: perturbed oscillators
zbMATH Keywords: isotropic harmonic oscillator
zbMATH Keywords: genuine resonance
zbMATH Keywords: axial symmetry
zbMATH Keywords: cubic potentials
reviewed by: Antonio Cañada
describes a project that uses: Mathematica
MaRDI profile type: Publication
cites work: Q4174811
cites work: SMALL DENOMINATORS AND PROBLEMS OF STABILITY OF MOTION IN CLASSICAL AND CELESTIAL MECHANICS
cites work: A proof of the isoenergetic KAM-theorem from the ``ordinary'' one
cites work: Quasi-periodic motions in families of dynamical systems. Order amidst chaos
cites work: Q4012889
cites work: Q4023642
cites work: Geometry of the bifurcations of the normalized reduced Henon–Heiles family
cites work: Q2785808
cites work: Singular reduction of axially symmetric perturbations of the isotropic harmonic oscillator
cites work: Canonical transformations depending on a small parameter
cites work: The elimination of the parallax in satellite theory
cites work: The Lissajous transformation. I: Basics
cites work: The Lissajous transformation. II: Normalization
cites work: Non-integrability of the 1:1:2–resonance
cites work: The monodromy in the Hamiltonian Hopf bifurcation
cites work: On perturbed 3D elliptic oscillators: a case of critical inclination in galactic dynamics
cites work: The Hénon and Heiles Problem in Three Dimensions.
cites work: Q4514114
cites work: Hamiltonian oscillators in \(1\)-\(1\)-\(1\) resonance: Normalization and integrability
cites work: Three-Dimensional Isotropic Harmonic Oscillator and SU3
cites work: Q5539876
cites work: Chaos in classical and quantum mechanics
cites work: Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems
cites work: Q4863263
cites work: On the Hamiltonian Hopf bifurcations in the 3D Hénon-Heiles family
cites work: A degenerate bifurcation in the Hénon-Heiles family
cites work: A generalisation of the Runge-Lenz constant of classical motion in a central potential
cites work: On the Problem of Degeneracy in Quantum Mechanics
cites work: On resonant classical Hamiltonians with n frequencies
cites work: Q3721476
cites work: KAM theorem for generic analytic perturbations of the Euler system
cites work: The Hamiltonian Hopf bifurcation
cites work: Generic Bifurcation of Periodic Points
cites work: Q3223187
cites work: The Lissajous transformation. III: Parametric bifurcations
cites work: Regularization of kepler's problem and the averaging method on a manifold
cites work: Q4771891
cites work: A note on focus-focus singularities
cites work: VISUALIZING RELATIVE EQUILIBRIA AND BIFURCATIONS BY PAINTING HAMILTONIANS ON PERSONAL COMPUTERS
cites work: A Formal Third Integral of Motion in a Nearly Spherical Stellar System
cites work: Averaging methods in nonlinear dynamical systems
cites work: SINGULARITIES OF FUNCTIONS ON BOUNDARIES, CORNERS, ETC.
cites work: Discrete symmetric dynamical systems at the main resonances with application to axi-symmetric galaxies
cites work: Q4718428
cites work: Motion in the core of a triaxial potential
description: scientific article; zbMATH DE number 1758073