# Symmetry and Resonance in Hamiltonian Systems

@article{Verhulst2001SymmetryAR, title={Symmetry and Resonance in Hamiltonian Systems}, author={Ferdinand Verhulst and J. M. Tuwankotta}, journal={SIAM J. Appl. Math.}, year={2001}, volume={61}, pages={1369-1385} }

In this paper we study resonances in two degrees of freedom, autonomous, Hamiltonian systems. Due to the presence of a symmetry condition on one of the degrees of freedom, we show that some of the resonances vanish as lower order resonances. After giving a sharp estimate of the resonance domain, we investigate this order change of resonance in a rather general potential problem with discrete symmetry and consider as an example the Henon--Heiles family of Hamiltonians. We also study a classical…

## 40 Citations

### Higher-Order Resonances in Dynamical Systems

- Mathematics
- 2002

This thesis is a collection of studies on higher-order resonances in an important class of dynamical systems called coupled oscillators systems. After giving an overview of the mathematical…

### Higher Order Resonance in Two Degree of Freedom Hamiltonian System

- Mathematics
- 2001

This paper reviews higher order resonance in two degrees of freedom Hamilto-
nian systems. We consider a positive semi-definite Hamiltonian around the origin.
Using normal form theory, we give an…

### Bifurcation Sequences in the Symmetric 1: 1 Hamiltonian Resonance

- PhysicsInt. J. Bifurc. Chaos
- 2016

We present a general review of the bifurcation sequences of periodic orbits in general position of a family of resonant Hamiltonian normal forms with nearly equal unperturbed frequencies, invariant…

### Hamiltonian systems with widely separated frequencies

- Mathematics
- 2001

In this paper we study two degrees of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that the linearized system has purely imaginary eigenvalues:…

### Hamiltonian systems with widely separated frequencies

- Mathematics
- 2003

In this paper we study two degrees of freedom Hamiltonian systems and applications to nonlinear wave equations. Near the origin, we assume that the linearized system has purely imaginary eigenvalues:…

### Interaction of Lower and Higher Order Hamiltonian Resonances

- PhysicsInt. J. Bifurc. Chaos
- 2018

Applications are given to the three dof [Formula: see text] resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

### Continuation of periodic orbits in symmetric Hamiltonian and conservative systems

- Mathematics, Physics
- 2014

We present and review results on the continuation and bifurcation of periodic solutions in conservative, reversible and Hamiltonian systems in the presence of symmetries. In particular we show how…

### Evolution to Mirror-Symmetry in Rotating Systems

- PhysicsSymmetry
- 2021

The problem is inspired by the dynamics of axisymmetric, rotating galaxies that evolve slowly to mirror symmetry with respect to the galactic plane, the model formulation is quite general and a remarkable feature is the vanishing and emergence of normal modes, stability changes and strong changes of the velocity distribution in phase-space.

### The symmetric 1:2 resonance

- Mathematics, Physics
- 2013

Abstract.This paper illustrates the application of the Lie transform normal-form theory to the construction of the 1:2 resonant normal form corresponding to a wide class of natural Hamiltonian…

### Evolution to symmetry

- Physics
- 2021

A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the…

## References

SHOWING 1-10 OF 44 REFERENCES

### Approximations of higher order resonances with an application to Contopoulos' model problem

- Mathematics
- 1979

Higher order resonances in two degrees of freedom Hamiltonian systems are studied by using Birkhoff normalization. The normal forms can be used as a starting point to develop a theory of asymptotic…

### Discrete symmetric dynamical systems at the main resonances with application to axi-symmetric galaxies

- MathematicsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1979

A study of two-degrees-of-freedom systems with a potential which is discrete-symmetric (even in one of the position variables) is carried out for the resonance cases 1:2, 1:1, 2:1 and 1:3. To produce…

### Resonances in a spring-pendulum: algorithms for equivariant singularity theory

- Mathematics
- 1998

A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is…

### Symmetry and integrability in Hamiltonian normal forms

- Mathematics
- 1997

In studies in the natural sciences assumptions of symmetries abound spherical symmetry cylindrical symmetry reection symmetry etc This is partly because such an assumption simplies the mathematical…

### Symmetries, topology and resonances in Hamiltonian mechanics

- Mathematics
- 1996

I Hamiltonian Mechanics.- 1 The Hamilton Equations.- 2 Euler-Poincare Equations on Lie Algebras.- 3 The Motion of a Rigid Body.- 4 Pendulum Oscillations.- 5 Some Problems of Celestial Mechanics.- 6…

### On the Stability of Periodic Orbits for Nonlinear Oscillator Systems in Regions Exhibiting Stochastic Behavior

- Physics
- 1972

A computer has been used to determine the stability character of periodic orbits for the Hamiltonian oscillator system H=12(p12+p22+q12+q22)+q12q2−13q23. Using procedures developed by Greene [J.…

### An empirical study of the stability of periodic motion in the forced spring-pendulum

- PhysicsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1993

The elastic pendulum is a two-degree-of-freedom, nonlinear device in which the primary mass slides up and down the pendulum arm subject to the restoring force of a linear spring. In this study,…

### Semiclassical calculation of bound states in multidimensional systems with Fermi resonance

- Physics
- 1979

A method is devised to calculate eigenvalues semiclassically for an anharmonic system whose two unperturbed modes are 2:1 degenerate. For some special states the periodic energy exchange between…