Stability of equilibria and fixed points of conservative systems

Cabral, Hildeberto E.; Meyer, Kenneth R.

doi:10.1088/0951-7715/12/5/309

Cited by 41 publications

(43 citation statements)

References 11 publications

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“…The fixed point is stable if |a 32 | > |a 05 | [24]. At the point in Table 1 where ω = 1/6, we find a 3,2 = i31 √ 3/72 while the resonant term a 0,5 = i13 √ 3/72 is much smaller; thus this point is stable [24].…”

mentioning

confidence: 69%

“…The map is unstable when τ 0 = 0, providing a 0,n−1 = 0 [15,24]. For the cases in Table 1 with τ 0 = 0 we find: when ω = 1/4 a 0,3 = −i4/3, and when ω = 1/5, 2/5 we find a 0,…”

mentioning

confidence: 79%

“…When ω = 1/3 the twist is singular and the fixed point is unstable, except at the point (l, k) = (−1, 0) where the map has the spatial symmetry S, (9). The additional symmetry implies that the resonant term vanishes, and since πτ 0 = 1/2 = 0 at this point, the fixed point is stable [23,24].…”

mentioning

confidence: 99%

“…2. For this case, the normal form (26) applies, and the fixed point is unstable if |a 2,1 | < |a 0,3 | [15], and stable if the inequality is reversed according to KAM theory [24]. We find that a 2,1 = ±3i(1 − l)/8 and a 0,3 = ±i(1 + 3l)/8, so that the fixed point of C − is unstable for l > 1/3 and stable when 0 ≤ l < 1/3.…”

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confidence: 99%

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Generalized Hénon maps: the cubic diffeomorphisms of the plane

Dullin

Meiss

2000

Physica D: Nonlinear Phenomena

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In general a polynomial automorphism of the plane can be written as a composition of generalized Hénon maps. These maps exhibit some of the familiar properties of the quadratic Hénon map, including a bounded set of bounded orbits and an anti-integrable limit. We investigate in particular the cubic, area-preserving case, which reduces to two, two-parameter families of maps. The bifurcations of low period orbits of these maps are discussed in detail.

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confidence: 69%

“…The map is unstable when τ 0 = 0, providing a 0,n−1 = 0 [15,24]. For the cases in Table 1 with τ 0 = 0 we find: when ω = 1/4 a 0,3 = −i4/3, and when ω = 1/5, 2/5 we find a 0,…”

mentioning

confidence: 79%

mentioning

confidence: 99%

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confidence: 99%

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Generalized Hénon maps: the cubic diffeomorphisms of the plane

Dullin

Meiss

2000

Physica D: Nonlinear Phenomena

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“…Markeev [17,18], Sokolsky [24,25,26]), where specific resonances are treated. Recently, Cabral and Meyer [6] revisited the problem and gave a general result that applies for both non-resonant and resonant cases.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Stability in Resonant Cases: A Geometrical Approach

et al. 2001

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Summary. In systems with two degrees of freedom Arnold's theorem is used for studying non linear stability of the origin when the quadratic part of the Hamiltonian is a non definite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases we determine the stability from the analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables.

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Stability of the Rydberg atom in the crossed magnetic and electric fields

Guirao

Vera

2011

Int J of Quantum Chemistry

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ABSTRACT:The stability of equilibrium positions of the Rydberg atom exposed to the uniform crossed electric and magnetic fields is analyzed. The dynamics of the system is described by an autonomous Hamiltonian depending on parameters a and f . By the normalization of the quadratic part of the Hamiltonian expansion in the neighborhood of the equilibrium position it is proved that for any f < 0 and 1 2, the equilibrium solution of the equations of motion is stable in Liapunov sense, while for f > 0 and a < 1/2, there is a domain of instability in the plain of parameters Ofa bounded by the curve d 3 = 0. In the domain of linear stability, it is proved that there are two curves in the plane Ofa, where the resonance conditions of third (ω 1 = 2ω 2 ) and fourth (ω 1 = 3ω 2 ) order are fulfilled. Moreover, by the normalization of the third-and fourth-order terms in the Hamiltonian expansion it is proved that in the case of the third-order resonance, the equilibrium position is unstable for all f > 0 different from f = 0.111572 and f = 0.281144, for which the stability takes place. In the case of the fourth-order resonance, there are two intervals of parameters for which the equilibrium position is unstable.

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Stability of equilibria and fixed points of conservative systems

Cited by 41 publications

References 11 publications

Generalized Hénon maps: the cubic diffeomorphisms of the plane

Generalized Hénon maps: the cubic diffeomorphisms of the plane

Nonlinear Stability in Resonant Cases: A Geometrical Approach

Stability of the Rydberg atom in the crossed magnetic and electric fields

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