Asymptotic analysis of hamiltonian systems

Verhulst, Ferdinand

doi:10.1007/bfb0062366

Cited by 7 publications

(5 citation statements)

References 66 publications

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“…This modification in the case of polynomial transformations is often called Birkhoff-or Birkhoff-Gustavson normalisation. The reader may consult Arnold (1976) appendix 7, Sanders and Verhulst (1985), Arnold (1988) and the survey by Verhulst (1983).…”

Section: Birkhoff-normalisationmentioning

confidence: 99%

Nonlinear Differential Equations and Dynamical Systems

1996

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Section: Birkhoff-normalisationmentioning

confidence: 99%

Nonlinear Differential Equations and Dynamical Systems

1996

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“…Therefore, this research is in the Hamiltonian tradition of e.g. Meyer [29], Sanders [39], Van der Meer [46] and Verhulst [49]. It is our aim to describe versal, viz.…”

Section: Setting Of the Problem Outlinementioning

confidence: 99%

“…44, 1993 A normally elliptic Hamiltonian bifurcation 391 may explain why certain 'integrable' characteristics in the unfoldings are so persistent. For a related comment we refer to Verhulst [49].(ii) Our approach differs from e.g. Van der Meer [46], also compare Duistermaat [19].…”

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confidence: 99%

A normally elliptic Hamiltonian bifurcation

Broer

Chow

Kim

et al. 1993

Z. angew. Math. Phys.

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The unfolding theory, to be developed below, thenThe conjugate pair of imaginary eigenvalues gives rise to a formal rotational symmetry in all unfoldings, i.e. a rotational symmetry in their Taylor series. Here the series is considered in dependence on both the phase space variables and the parameters. This is an application of Normal Form Theory, where the terms of the formal power series are changed by canonical coordinate transformations in an inductive process. The symmetry, thus obtained, enables a formal reduction to 1 degree of freedom, around an equilibrium point with a double eigenvalue 0. For similar approaches see some of the above references, also compare e.g. Arnold [6], Takens [43,44], Broer [10,11], Golubitsky and Stewart [23] and Broer and Vegter [16]. This Normal Form Theory will be presented in Section 3.In Section 2, we begin studying the 1 degree of freedom 'backbone'-problem in its own right. Since in the plane the integral curves of the systems are the level sets of the Hamiltonian functions, here the problem reduces to Singularity Theory, e.g. In Section 4 the connection is made between the planar reduction of the symmetric system and the 'backbone'-system of Section 2. It turns out that the formal integral, obtained by normalization, is a distinguished parameter in the sense of Golubitsky and Schaeffer [51] and Schecter [41], also compare Wassermann [52]. Now Singularity Theory yields new normal forms, that are polynomial of degree 3, at least in the phase-space variables. We note that here the normalizing transformations no longer need to be canonical. Technical details from Singularity Theory have been collected in an appendix (Section 7). After this we suspend, or dereduce, to the original 4-dimensional setting, so obtaining an integrable, i.e. rotationally symmetric, approximation of the original unfolding.Thus we obtain a perturbation problem, similar to e.g. Broer [10,12] or Braaksma and Broer [8]. The perturbation term is of arbitrarily high order, both in the phase space variables and the parameters. This problem is briefly addressed in Section 5.Remarks. (i) Although in the original family only generic restrictions are imposed on the lower order terms, by Normal Form Theory and Singularity Theory, the corresponding unfolding is reduced to an arbitrarily flat perturbation of a normal form, completely determined by a 1 degree of freedom system, which is polynomial of degree 3. This polynomial character Vol. 44, 1993 A normally elliptic Hamiltonian bifurcation 391 may explain why certain 'integrable' characteristics in the unfoldings are so persistent. For a related comment we refer to Verhulst [49].(ii) Our approach differs from e.g. Van der Meer [46], also compare Duistermaat [19]. In [46,19] the dynamics is reduced by the energy-momentum map as well as the Moser-Weinstein method. This gives information related to specific periodic solutions, that is also important for the dynamics as a whole. Instead, we just factor out a formal rotational symmetry, and so seem to get a more ...

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“…The simplest case with w^O for which multiple resonance occurs is in 3 degrees of freedom, and then this condition means that multiplying the a>j with a common factor (which can be arranged by a change in the time scale) one can take &», • e J. for j = 1, 2, 3. For a survey of the asymptotic analysis of such systems for combinations with small w,, see [7]. 1:1:2-resonance…”

Section: =Imentioning

confidence: 99%

“…at (e, 77) of the complex curve 9 = constant is given bỹ (e, V ) ^(e, TJ) • log Ŵ inding around 0 with r\ arbitrarily many times, one sees that (3.19) can only be finitely valued if f<.,,)/f(.,,) = ^,,>/^,,>,(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) …”

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Non-integrability of the 1:1:2–resonance

Duistermaat

1984

Ergod. Th. Dynam. Sys.

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A Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist A A formal power series , functionally independent, in involution, and such that the Taylor expansion of F is a formal power series in the .Take n = 3, , F(k) homogeneous of degree k, F(2) > 0 and the eigenfrequencies in ratio 1:1:2. If F(3) avoids a certain hypersurface of ‘symmetric’ third order terms, then the F system is not formally integrable. If F(3) is symmetric but F(4) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.

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Asymptotic analysis of hamiltonian systems

Cited by 7 publications

References 66 publications

Nonlinear Differential Equations and Dynamical Systems

Nonlinear Differential Equations and Dynamical Systems

A normally elliptic Hamiltonian bifurcation

Non-integrability of the 1:1:2–resonance

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