New Aspects on the Geometry and Dynamics of Quadratic Hamiltonian Systems on (𝔰𝔬(3))*

Daniasa, Cora; Gîrban, Anania; Tudoran, Răzvan M.

doi:10.1142/s0219887811005889

Cited by 8 publications

(10 citation statements)

References 15 publications

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“…Quadratic Hamilton-Poisson systems on Lie-Poisson spaces have recently been considered by several authors [30,15,7,8]. Specifically on se(1, 1) * − , spectral stability (and in some cases Lyapunov stability) as well as numerical integration was investigated in [6].…”

Section: Resultsmentioning

confidence: 99%

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Quadratic Hamilton–Poisson systems on $\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}$: The homogeneous case

Barrett

Biggs

Remsing

2014

Int. J. Geom. Methods Mod. Phys.

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In this paper we consider quadratic Hamilton-Poisson systems on the semi-Euclidean Lie-Poisson space se(1, 1) * − . The homogeneous positive semidefinite systems are classified; there are exactly six equivalence classes. In each case, the stability nature of the equilibrium states is determined. Explicit expressions for the integral curves are found. A characterization of the equivalence classes, in terms of the equilibria, is identified. Finally, the relation of this work to optimal control is briefly discussed.

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Section: Resultsmentioning

confidence: 99%

“…Thus far, equivalence of systems has received relatively little attention. However, in the last few years, some promising directions have been identified [29,15,10,11].…”

Section: Resultsmentioning

confidence: 99%

Quadratic Hamilton–Poisson systems on $\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}$: The homogeneous case

Barrett

Biggs

Remsing

2014

Int. J. Geom. Methods Mod. Phys.

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“…Note that a large number of concrete dynamical systems from various sciences admits three-dimensional Hamiltonian realizations of this type, e.g., Euler's equations from the free rigid body dynamics ( [5], [2]), the Lotka-Volterra system( [12]), the Rikitake system( [11]), the Rössler system ( [13]), the Rabinovich system ( [14]), etc.…”

Section: D Hamiltonian Systems Versus 3d Completely Integrable Systemsmentioning

confidence: 99%

“…Moreover, if there are many periodic orbits located on the same common level set of the Hamiltonian and the Casimir, then the same perturbation can be used in order to asymptotically stabilize all of them in the same time. The method can be applied for a large number of concrete dynamical systems coming from various sciences, which admit three-dimensional Hamiltonian realizations, e.g., Euler's equations of free rigid body dynamics ( [5], [2]), the Rikitake system( [11]), the Rössler system ( [13]), the Rabinovich system ( [14]), etc.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stabilization with phase of periodic orbits of three-dimensional Hamiltonian systems

Tudoran

2017

Journal of Geometry and Physics

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We provide a geometric method to stabilize asymptotically with phase an arbitrary fixed periodic orbit of a locally generic three-dimensional Hamiltonian dynamical system. The main advantage of this method is that one needs not know a parameterization of the orbit to be stabilized, but only the values of the Hamiltonian and a fixed Casimir (of the Poisson configuration manifold) at that orbit. The stabilization procedure is illustrated in the case of the Rikitake model of geomagnetic reversal.

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“…Moreover, any such element is a regular value of (F 1 , F 2 ), as well as its components for the corresponding maps, F 1 and respectively F 2 . For more details see, e.g., [3].…”

Section: )mentioning

confidence: 99%

On the control of stability of periodic orbits of completely integrable systems

Tudoran¹

2015

Journal of Geometric Mechanics

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We provide a constructive method designed in order to control the stability of a given periodic orbit of a general completely integrable system. The method consists of a specific type of perturbation, such that the resulting perturbed system becomes a codimension-one dissipative dynamical system which also admits that orbit as a periodic orbit, but whose stability can be a-priori prescribed. The main results are illustrated in the case of a three dimensional dissipative perturbation of the harmonic oscillator, and respectively Euler's equations form the free rigid body dynamics.

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New Aspects on the Geometry and Dynamics of Quadratic Hamiltonian Systems on (𝔰𝔬(3))*

Abstract: In this paper we analyze the quadratic and homogeneous Hamiltonian systems on (so(3)) * from the Poisson dynamics and geometry point of view.

Cited by 8 publications

References 15 publications

Quadratic Hamilton–Poisson systems on $\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}$: The homogeneous case

Quadratic Hamilton–Poisson systems on $\mathfrak{s}\mathfrak{e}(1, 1)^{*}_{-}$: The homogeneous case

Asymptotic stabilization with phase of periodic orbits of three-dimensional Hamiltonian systems

On the control of stability of periodic orbits of completely integrable systems

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