Interaction of Lower and Higher Order Hamiltonian Resonances

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. 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.

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RECONSIDERING TRIGONOMETRIC INTEGRATORS

The ANZIAM Journal ◽

10.1017/s1446181109000042 ◽

2009 ◽

Vol 50 (3) ◽

pp. 320-332 ◽

Cited By ~ 1

Author(s):

DION R. J. O’NEALE ◽

ROBERT I. MCLACHLAN

Keyword(s):

Hamiltonian System ◽

Differential Equations ◽

Nonlinear Stability ◽

Higher Order ◽

Statistical Properties ◽

Highly Oscillatory ◽

Time Averages ◽

Highly Oscillatory Differential Equations ◽

Low Order

AbstractIn this paper we look at the performance of trigonometric integrators applied to highly oscillatory differential equations. It is widely known that some of the trigonometric integrators suffer from low-order resonances for particular step sizes. We show here that, in general, trigonometric integrators also suffer from higher-order resonances which can lead to loss of nonlinear stability. We illustrate this with the Fermi–Pasta–Ulam problem, a highly oscillatory Hamiltonian system. We also show that in some cases trigonometric integrators preserve invariant or adiabatic quantities but at the wrong values. We use statistical properties such as time averages to further evaluate the performance of the trigonometric methods and compare the performance with that of the mid-point rule.

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Non- linear stability of triangular librations points in circular restricted three body under radiating and oblate primaries in presence of resonance

International Journal of Advanced Astronomy ◽

10.14419/ijaa.v3i2.4772 ◽

2015 ◽

Vol 3 (2) ◽

pp. 58

Author(s):

Ashutosh Narayan ◽

Nutan Singh

Keyword(s):

Radiation Pressure ◽

Fourth Order ◽

Nonlinear Stability ◽

Binary Systems ◽

Normal Forms ◽

Boundary Region ◽

Third Order ◽

Order Resonance ◽

The Stability ◽

Three Body

The nonlinear stability of the triangular librations points is studied in the presence resonance considering both the primaries as radiating and oblate. The study is carried out for various values of radiation pressure and oblateness parameter in general and binary systems in particular. It is found that the normal forms of the Hamiltonian contains both the resonance cases; ω1= 2ω2 and ω1= 3ω2. The case ω1= ω2 corresponds to the boundary region of the stability for the system.It is investigated that for the motion is unstable for third order resonance but stable for fourth order resonance.

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the effects of resonance with Jupiter in the system of short-period comets

International Astronomical Union Colloquium ◽

10.1017/s0252921100033959 ◽

1974 ◽

Vol 22 ◽

pp. 193-203

Author(s):

L̆ubor Kresák

Keyword(s):

Structural Effects ◽

Order Resonance ◽

Mean Motion ◽

Short Period ◽

The Mean ◽

Compound Effect ◽

Low Order

AbstractStructural effects of the resonance with the mean motion of Jupiter on the system of short-period comets are discussed. The distribution of mean motions, determined from sets of consecutive perihelion passages of all known periodic comets, reveals a number of gaps associated with low-order resonance; most pronounced are those corresponding to the simplest commensurabilities of 5/2, 2/1, 5/3, 3/2, 1/1 and 1/2. The formation of the gaps is explained by a compound effect of five possible types of behaviour of the comets set into an approximate resonance, ranging from quick passages through the gap to temporary librations avoiding closer approaches to Jupiter. In addition to the comets of almost asteroidal appearance, librating with small amplitudes around the lower resonance ratios (Marsden, 1970b), there is an interesting group of faint diffuse comets librating in characteristic periods of about 200 years, with large amplitudes of about±8% in μ and almost±180° in σ, around the 2/1 resonance gap. This transient type of motion appears to be nearly as frequent as a circulating motion with period of revolution of less than one half that of Jupiter. The temporary members of this group are characteristic not only by their appearance but also by rather peculiar discovery conditions.

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