Periodic orbits in analytically perturbed Poisson systems

García, Isaac A.; Hernández‐Bermejo, Benito

doi:10.1016/j.physd.2014.02.008

Cited by 4 publications

(6 citation statements)

References 20 publications

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“…Notice that this system is only well defined for r > 0. Moreover, in this region, since for sufficiently small ε we have θ < 0 in an arbitrarily large ball centered at the origin, we can rewrite the differential system (6) in such ball into the form…”

Section: The Lagrange Standard Form Of Averaging Theorymentioning

confidence: 99%

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

García

Hernández‐Bermejo

2020

JNMP

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A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the bifurcation phenomena of periodic orbits as a result of these perturbations in the period annulus associated to the unperturbed harmonic oscillator. This is accomplished via the averaging theory up to an arbitrary order in the perturbation parameter ε. In that theory we shall also use both branching theory and singularity theory of smooth maps to analyze the bifurcation phenomena at points where the implicit function theorem is not applicable. When the perturbation is given by a polynomial family, the associated Melnikov functions are polynomial and tools of computational algebra based on Gröbner basis are employed in order to reduce the generators of some polynomial ideals needed to analyze the bifurcation problem. When the most general perturbation of the harmonic oscillator by a quadratic perturbation field is considered, the complete bifurcation diagram (except at a high codimension subset) in the parameter space is obtained. Examples are given.

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Section: The Lagrange Standard Form Of Averaging Theorymentioning

confidence: 99%

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

García

Hernández‐Bermejo

2020

JNMP

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“…Moreover, in the particular (but prominent) case of classical Hamiltonian systems, the use of NTTs is well-known for the integrability analysis [22]- [24], in stability theory [25], etc. Given that NTTs preserve topology in phase-space, very often the global reduction to the Darboux canonical form is possible by means of a diffeomorphism followed by an NTT [3], [26]- [28]. Note however that NTTs do not preserve in general the Poisson structure, namely η(x)J (x) may not be a structure matrix in spite that J (x) is.…”

Section: Examples Example 1 Two-dimensional Poisson Systemsmentioning

confidence: 99%

Congruence method for global Darboux reduction of finite-dimensional Poisson systems

Hernández‐Bermejo

2018

Journal of Mathematical Physics

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A new procedure for the global construction of the Casimir invariants and Darboux canonical form for finite-dimensional Poisson systems is developed. This approach is based on the concept of matrix congruence, and can be applied without the previous determination of the Casimir invariants (recall that their prior knowledge is unavoidable for the standard reduction methods, thus requiring either the integration of a system of PDEs or solving some equivalent problem). Well the opposite, in the new congruence method both the Darboux coordinates and the Casimir invariants arise simultaeously as the outcome of the reduction algorithm. In fact, the congruence algorithm proceeds only in terms of matrix-algebraic transformations and direct quadratures, thus avoiding the need of previously integrating a system of PDEs and therefore improving previously known approaches. Physical examples illustrating different aspects of the theory are provided.PACS codes: 45.20.Jj, 02.30.Hq.

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“…Let x * ∈ ∂Ω i ⊂ V −1 (0) be a point of the boundary of Ω i . Observe that the rank of the Poisson structure matrix (17) vanishes on x * . Accordingly, the Darboux canonical form is not defined on such boundary, since there is no constant-rank neighborhood of x * .…”

Section: Conservativeness and Poisson Structurementioning

confidence: 99%

“…The presence of finite-dimensional Poisson systems (see [29,32] and references therein for an overview of the classical theory) is ubiquitous in many branches of physics and applied mathematics. The specific format of Poisson systems has allowed the development of many tools for their analysis (for instance, see [15][16][17][19][20][21] and references therein for a sample). In addition, Poisson dynamical systems are significant due to several reasons.…”

Section: Introductionmentioning

confidence: 99%

Inverse Jacobi multiplier as a link between conservative systems and Poisson structures

García¹,

Hernández‐Bermejo²

2017

J. Phys. A: Math. Theor.

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Some aspects of the relationship between conservativeness of a dynamical system (namely the preservation of a finite measure) and the existence of a Poisson structure for that system are analyzed. From the local point of view, due to the Flow-Box Theorem we restrict ourselves to neighborhoods of singularities. In this sense, we characterize Poisson structures around the typical zero-Hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non-vanishing first jet by connecting with the classical Poincaré center problem. From the global point of view, we connect the property of being strictly conservative (the invariant measure must be positive) with the existence of a Poisson structure depending on the phase space dimension. Finally, weak conservativeness in dimension two is introduced by the extension of inverse Jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed. Examples including Lotka-Volterra systems, quadratic isochronous centers, and non-smooth oscillators are provided. 973702702.From the point of view of the relationship with Poisson systems and their diffeomorphic transformation properties, it will be useful for us to know how inverse integrating factors change under orbital equivalence of vector fields, see [3,8] for further details.Proposition 3. Let Φ be a diffeomorphism in Ω ⊂ R n with non-vanishing Jacobian determinant J Φ on Ω and let η : Ω → R be such that η ∈ C 1 (Ω) and η(x) = 0 everywhere in Ω. If V is an inverse Jacobi multiplier of the C 1 -vector field Y in Ω then η(V • Φ)/J Φ is an inverse Jacobi multiplier of the orbitally equivalent vector field η Φ * (Y).

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Periodic orbits in analytically perturbed Poisson systems

Cited by 4 publications

References 20 publications

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

Congruence method for global Darboux reduction of finite-dimensional Poisson systems

Inverse Jacobi multiplier as a link between conservative systems and Poisson structures

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