On theC1non-integrability of differential systems via periodic orbits

Llibre, Jaume; Valls, Clàudìa

doi:10.1017/s0956792511000143

Cited by 10 publications

(6 citation statements)

References 29 publications

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“…The eigenvector tangent to the periodic orbit has associated an eigenvalue equal to 1. So the periodic orbit has at least one multiplier equal to 1, for more details see for instance Proposition 1 in [10].…”

Section: Basic Results On the Continuation Of Periodic Solutionsmentioning

confidence: 99%

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Periodic Solutions for the Generalized Anisotropic Lennard-Jones Hamiltonian

Llibre

2015

Qual. Theory Dyn. Syst.

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Abstract. We characterize the circular periodic solutions of the generalized Lennard-Jones Hamiltonian system with two particles in R n , and we analyze what of these periodic solutions can be continued to periodic solutions of the anisotropic generalized Lennard-Jones Hamiltonian system.We also characterize the periods of antiperiodic solutions of the generalized Lennard-Jones Hamiltonian system on R 2n , and prove the existences of 0 < τ * ≤ τ * * such that this system possesses no τ /2-antiperiodic solution for all τ ∈ (0, τ * ), at least one τ /2-antiperiodic solution when τ = τ * , precisely 2 n families of τ /2-antiperiodic circular solutions when τ = τ * * , and precisely 2 n+1 families of τ /2-antiperiodic circular solutions when τ > τ * * . Each of these circular solution families is of dimension n − 1 module the S 1 -action.

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Section: Basic Results On the Continuation Of Periodic Solutionsmentioning

confidence: 99%

“…, ∇F k (x) and f (x; ε) are linearly independent. Then 1 is a multiplier of the periodic orbit γ with multiplicity at least k + 1, see for instance Theorem 2 of [10].…”

Section: Basic Results On the Continuation Of Periodic Solutionsmentioning

confidence: 99%

Periodic Solutions for the Generalized Anisotropic Lennard-Jones Hamiltonian

Llibre

2015

Qual. Theory Dyn. Syst.

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“…The eigenvalues of the monodromy matrix associated to the periodic solution ϕ(t, x 0 ) are called the multipliers of the periodic orbit. In [9] the authors proved the following result which goes back to Poincaré (see [12]). Theorem 7.…”

Section: Proof Of Theoremmentioning

confidence: 97%

A note on the first integrals of the ABC system

Llibre

Valls

2012

Journal of Mathematical Physics

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Abstract. Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C ≥ 0, or A = 1 and 0 < B, C ≤ 1. In the first case it is known that the ABC system is completely integrable, here we provide its explicit first integrals. In the second case Ziglin in [14] proved that the ABC system with 0 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin's result showing that there are no C 1 first integrals under convenient assumptions.

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“…Theorem 5 is due to Poincaré [14] (see sect. 36), and see also [20]. It provides a tool for studying the non Liouville-Arnol'd integrability, independently of the class of differentiability of the second first integral.…”

Section: Periodic Orbits and The Liouville-arnol'd Integrabilitymentioning

confidence: 99%