Global bifurcation of planar and spatial periodic solutions in the restricted n-body problem

García-Azpeitia, Carlos; Ize, Jorge

doi:10.1007/s10569-011-9354-2

Cited by 17 publications

(25 citation statements)

References 30 publications

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“…Therefore these solutions follow the planar π-periodic curve (x, y) twice; one time with the spatial coordinate z and a second time with −z. This fact was proved in [3]. Fig.…”

Section: Breaking Of Symmetriesmentioning

confidence: 68%

“…Several papers have been devoted to study the stability and bifurcation of periodic solutions for the restricted N -body problem in the Maxwell configuration. For example, a study of the existence and linear stability of equilibrium positions can be found in [1], an analysis of the bifurcation of planar and vertical families of periodic solutions in [3], and a numerical exploration in [6].…”

Section: Introductionmentioning

confidence: 99%

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Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

Calleja

Doedel

García-Azpeitia

2016

Eur. Phys. J. Spec. Top.

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Abstract. We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three equilibrium Zn-orbits from which families of Lyapunov orbits emerge. Numerical continuation of these families using a boundary value formulation is used to construct the bifurcation diagram for the case n = 7, also including some secondary and tertiary bifurcating families. We observe symmetrybreaking bifurcations in this system, as well as certain period-doubling bifurcations.

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“…Therefore these solutions follow the planar π-periodic curve (x, y) twice; one time with the spatial coordinate z and a second time with −z. This fact was proved in [3]. Fig.…”

Section: Breaking Of Symmetriesmentioning

confidence: 68%

Section: Introductionmentioning

confidence: 99%

Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

Calleja

Doedel

García-Azpeitia

2016

Eur. Phys. J. Spec. Top.

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“…This will enable us to apply these results to a wide class of problems, as the dNLS equations at the final section. Also, notice that the change of variables was done in complex coordinates, and these will allow us to prove bifurcation of periodic solutions in a series of forthcoming papers analogous to [5]: as a matter of fact, the natural approach to the study of periodic solutions is, in this context, the use of Fourier series. Thus, the change of variables, which we have introduced, will be helpful.…”

Section: Remarkmentioning

confidence: 99%

“…A nj e j(ikI+J)ζ (12) for k ∈ {1, ..., n/2, n}, and the signs σ(µ) are defined as before in (5). Furthermore, since, in this case, there is no collision points, then the bifurcation is inadmissible only when the parameter µ or the norm of the branch goes to infinity.…”

Section: Dnlsmentioning

confidence: 99%

Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators

García-Azpeitia

Ize

2011

Journal of Differential Equations

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Given a regular polygonal arrangement of identical objects, turning around a central object (masses, vortices or dNLS oscillators), this paper studies the global bifurcation of relative equilibria in function of a natural parameter (central mass, central circulation or amplitude of the oscillation). The symmetries of the problem are used in order to find the irreducible representations, the linearization and, with the help of a degree theory, the symmetries of the bifurcated solutions.

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“…Many methods applied to (1) were inspired by the Hamiltonian systems of type where V = R 2n and J = 0 −Id Id 0 is the symplectic matrix, for which the existence of 2πperiodic solutions was intensively studied (see for example [1,3,14,18,19,20,21,22,27,28,39,40,41,43,44,45,46,50]). Similar methods were also developed for the system (1) (see [2,10,30,31,32,33,34,58]). After K. Geba introduced the concept of the gradient equivariant degree (cf.…”

Section: Introductionmentioning

confidence: 99%

Multiple periodic solutions for Γ-symmetric Newtonian systems

Da̧bkowski

Krawcewicz

et al. 2017

Journal of Differential Equations

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The existence of periodic solutions in Γ-symmetric Newtonian systemsẍ = −∇f (x) can be effectively studied by means of the Γ × O(2)-equivariant gradient degree with values in the Euler ring U (Γ × O(2)). In this paper we show that in the case of Γ being a finite group, the Euler ring U (Γ × O(2)) and the related basic degrees are effectively computable using Euler ring homomorphisms, the Burnside ring A(Γ × O(2)), and the reduced Γ × O(2)-degree with no free parameters. We present several examples of Newtonian systems with various symmetries, for which we show existence of multiple periodic solutions. We also provide exact value of the equivariant topological invariant for those problems.1 for example a difference of the equivariant gradient degrees on large and small ball). Such cases are for example when the system (1) is asymptotically linear or satisfy a Nagumo-type growth condition. Then clearly the existence of non-trivial solutions (i.e. outside the small ball) can be concluded by the fact that ω = 0. However, one can be also interested to predict the existence of multiple 2π-periodic solutions with different types of symmetry. In such a case, the coefficients of ω corresponding to the so-called maximal orbit type can provide the crucial information in order to formulate such results. But the maximality of such obit types implies that it is a generator of the Burnside ring A(G), therefore it can actually be detected by the Γ × O(2)-equivariant degree with no free parameter, which can be much easier computed than the equivariant gradient degree. Similar arguments apply to the system (2), which we can consider as a bifurcation problem with a parameter λ. More precisely, in this case we are looking for critical values λ o of the parameter λ > 0, to which we can associate the Γ × O(2)-equivariant gradient bifurcation invariants ω(λ o ) ∈ U (Γ×O(2)) classifying the bifurcation of 2π-periodic solutions from the zero solution. The existence and multiplicity of such bifurcating branches of 2π-periodic solutions can be described from the information contained in the invariants ω(λ o ). Consequently, all the essential information needed to establish the existence and multiplicity results for the systems (1) and (2) can be extracted from the Γ × O(2)-equivariant degree (with no free parameter) of J which takes values in the Burnside ring A(G). It is clear that the Γ × O(2)-equivariant degree without free parameter can be easily computed (without getting entangled in complicated technical details), has similar properties and provides enough information for analyzing these problems.Nevertheless, let us emphasize that only the equivariant invariants ω ∈ U (G) (without truncation of its coefficients) provide a complete equivariant topological classification for the related solution sets to (1) or (2).To illustrate the usage and the computations of the associated with the systems (1) and (2) equivariant invariants, in section 7 we present several examples of symmetric Newtonian systems, for which the exact values of the a...

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Global bifurcation of planar and spatial periodic solutions in the restricted n-body problem

Cited by 17 publications

References 30 publications

Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators

Multiple periodic solutions for Γ-symmetric Newtonian systems

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