Periodic Solutions and KAM Tori in a Triaxial Potential

Palacián, Jesús F.; Vidal, Cláudio; Vidarte, Jhon; Yanguas, Patricia

doi:10.1137/16m1082925

Cited by 9 publications

(9 citation statements)

References 17 publications

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“…To improve the procedure, we use a type of symplectic variables

\left(L,Q,l,P\right)

as in Palacián et al, 18 where the authors construct these variables such that the unperturbed Hamiltonian in () in the new coordinates depends only on

L

. They make a particularization for the 1 : 1 : 1 resonance of the construction of local symplectic maps for resonant Hamiltonian systems with

n

degrees of freedom (see Carrasco et al 19 and Meyer et al 20 ).…”

Section: Preliminary and Statements Of The Main Resultsmentioning

confidence: 99%

Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

Rothen

Vidal

2022

Math Methods in App Sciences

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The aim of this work is to prove analytically the existence of symmetric periodic solutions of the family of Hamiltonian systems with Hamiltonian function Hfalse(q1,q2,p1,p2false)=12false(q12+p12false)+12false(q22+p22false)+a0.1emq14+b0.1emq12q22+c0.1emq24$$ H\left({q}_1,{q}_2,{p}_1,{p}_2\right)=\frac{1}{2}\left({q}_1^2+{p}_1^2\right)+\frac{1}{2}\left({q}_2^2+{p}_2^2\right)+a\kern0.1em {q}_1^4+b\kern0.1em {q}_1^2{q}_2^2+c\kern0.1em {q}_2^4 $$ with three real parameters a,b$$ a,b $$, and c$$ c $$. Moreover, we characterize the stability of these periodic solutions as a function of the parameters. Also, we find a first‐order analytical approach of these symmetric periodic solutions. We emphasize that these families of periodic solutions are different from those that exist in the literature.

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“…To improve the procedure, we use a type of symplectic variables

\left(L,Q,l,P\right)

as in Palacián et al, 18 where the authors construct these variables such that the unperturbed Hamiltonian in () in the new coordinates depends only on

L

. They make a particularization for the 1 : 1 : 1 resonance of the construction of local symplectic maps for resonant Hamiltonian systems with

n

degrees of freedom (see Carrasco et al 19 and Meyer et al 20 ).…”

Section: Preliminary and Statements Of The Main Resultsmentioning

confidence: 99%

Symmetric periodic solutions of symmetric Hamiltonians in 1 : 1 resonance

Rothen

Vidal

2022

Math Methods in App Sciences

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“…Note that these invariants are different from the ones of the 1:1:1 resonance (see [38]). The normalised Hamiltonian (3) can be expressed in terms of the invariants.…”

Section: Normalisation and Reductionmentioning

confidence: 91%

“…As a difference with respect to the perturbed Hamiltonian in 1:1:1 resonance treated in[38], here we have a greater number of bifurcation curves due to the restrictions given in (5) together with the sign of each nonzero energy level h.The upcoming results provide information on the linear stability of the periodic solutions established in Theorems 4.1 and 4.2.…”

mentioning

confidence: 90%

Periodic solutions, KAM tori and bifurcations in a cosmology-inspired potential

et al. 2019

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A family of perturbed Hamiltonians Hε = 1 2 (x 2 + X 2) − 1 2 (y 2 + Y 2) + 1 2 (z 2 + Z 2) + ε 2 [α(x 4 + y 4 + z 4) + β(x 2 y 2 + x 2 z 2 + y 2 z 2)] in 1:−1:1 resonance depending on two real parameters is considered. We show the existence and stability of periodic solutions using reduction and averaging. In fact, there are at most thirteen families for every energy level h < 0 and at most twenty six families for every h > 0. The different types of periodic solutions for every nonzero energy level, as well as their bifurcations, are characterised in terms of the parameters. The linear stability of each family of periodic solutions, together with the determination of KAM 3-tori encasing some of the linearly stable periodic solutions is proved. Critical Hamiltonian bifurcations on the reduced space are characterised. We find important differences with respect to the dynamics of the 1:1:1 resonance with the same perturbation as the one given here. We end up with an intuitive interpretation of the results from a cosmological viewpoint.

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“…det ∂ 2 y H 0 (y) = 0, then for the perturbed system H(x, y) = H 0 (y) + εP (x, y, ε), most of nonresonant invariant tori still survive ( [1,12,16]). For some recent developments and applications of KAM theory, refer to [8,9,10,18,21,22,23]. However, in the presence of resonance, the persistence problem will become very complicated.…”

Section: Introductionmentioning

confidence: 99%