On the completely integrable case of the Rössler system

Tudoran, Răzvan M.; Gîrban, Anania

doi:10.1063/1.4708621

Cited by 8 publications

(8 citation statements)

References 9 publications

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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%

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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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Section: D Hamiltonian Systems Versus 3d Completely Integrable Systemsmentioning

confidence: 99%

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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“…was introduced in [16] as a prototype of 3D system of ODEs exhibiting the phenomenon of continuous chaos. When a = b = c = 0, the system (10) is known to be integrable in the Liouville sense [22,23,24,25], and gives rise toẋ…”

Section: Integrable Rössler Systemsmentioning

confidence: 99%

Integrable deformations of Rössler and Lorenz systems from Poisson–Lie groups

Ballesteros

Blasco

Musso

2016

Journal of Differential Equations

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A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable deformations can be obtained through the construction of the appropriate Poisson-Lie groups that deform the initial symmetry. The approach is applied in order to construct integrable deformations of both uncoupled and coupled versions of certain integrable types of Rössler and Lorenz systems. It is worth stressing that such deformations are of non-polynomial type since they are obtained through an exponentiation process that gives rise to the Poisson-Lie group from its infinitesimal Lie bialgebra structure. The full deformation procedure is essentially algorithmic and can be computerized to a large extent.

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“…This unique integrable Rössler differential system was first proved in [18]. Recently system (10) was studied from the Poisson dynamics point of view, see [17].…”

Section: J Llibre C Valls and X Zhangmentioning

confidence: 99%

The Completely Integrable Differential Systems are Essentially Linear Differential Systems

2015

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Abstract. Letẋ = f (x) be a C k autonomous differential system with k ∈ N ∪ {∞, ω} defined in an open subset Ω of R n . Assume that the systemẋ = f (x) is C r completely integrable, i.e. there exist n − 1 functionally independent first integrals of class C r with 2 ≤ r ≤ k. If the divergence of systemẋ = f (x) is non-identically zero, then any Jacobian multiplier is functionally independent of the n − 1 first integrals. Moreover the systemẋ = f (x) is C r−1 orbitally equivalent to the linear differential systemẏ = y in a full Lebesgue measure subset of Ω. For Darboux and polynomial integrable polynomial differential systems we characterize their type of Jacobian multipliers.

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On the completely integrable case of the Rössler system

Abstract: In this paper, we present some relevant dynamical properties of the Rössler system from the Poisson dynamics point of view.

Cited by 8 publications

References 9 publications

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

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

Integrable deformations of Rössler and Lorenz systems from Poisson–Lie groups

The Completely Integrable Differential Systems are Essentially Linear Differential Systems

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