Abstract
In this paper we study the periodic orbits of the Hamiltonian system with a fifth-order generalized Hénon–Heiles potential and its \({\mathcal{C}}^{1}\) non-integrability in the sense of Liouville–Arnold.
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M. Alvarez-Ramírez was partially supported by the grant Sistemas Hamiltonianos, Mecánica y Geometría from the PAPDI2021 CBI-UAMI.
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(Submitted by M. A.Malakhaltsev)
AVERAGING THEORY OF SECOND ORDER
AVERAGING THEORY OF SECOND ORDER
Based on the averaging theory of first, second and third order, whose key point is the Brouwer degree technique, Buică and Llibre [3] provided a way to find periodic solutions for a differential system whose vector field depends on a small parameter \(\varepsilon\). Next, we give a brief description of their results.
Theorem 5 (Second order averaging theory). Consider the differential system
where \(F_{1}\), \(F_{2}:\mathbb{R}\times D\to\mathbb{R}^{n}\), \(R:\mathbb{R}\times D\times(-\varepsilon_{f},\varepsilon_{f})\to\mathbb{R}^{n}\) are continuous functions, \(T\)-periodic in \(t\), and \(D\) is an open subset of \(\mathbb{R}^{n}\). Assume that
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\((i)\) \(F_{1}(t,\cdot)\in\mathcal{C}^{1}(D)\) for all \(t\in\mathbb{R}\) , \(F_{1}\) , \(F_{2}\) , \(R\) and \(D_{x}F_{1}\) are locally Lipschitz with respect to \(x\) , and \(R\) is differentiable with respect to \(\varepsilon\) . We define \(f_{1},\,f_{2}:D\to\mathbb{R}^{n}\) as
$$f_{1}(z)=\frac{1}{T}\int\limits_{0}^{T}F_{1}(s,z)ds,$$$$f_{2}(z)=\frac{1}{T}\int\limits_{0}^{T}\left[D_{z}F_{1}(s,z)\cdot\int\limits_{0}^{s}F_{1}(t,z)dt+F_{2}(s,z)\right]ds,$$and
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\((ii)\) for \(V\subset D\) an open and bounded set and for each \(\varepsilon\in(-\varepsilon_{f},\varepsilon_{f})\setminus\{0\}\), there exist \(a_{\varepsilon}\in V\) such that \(f_{1}(a_{\varepsilon})+\varepsilon f_{2}(a_{\varepsilon})=0\) and \(d_{B}(f_{1}+\varepsilon f_{2},V,0)\neq 0\).
Then, for \(|\varepsilon|>0\) sufficiently small, there exist a \(T\)–periodic solution \(\varphi(\cdot,\varepsilon)\) of system (A1) such that \(\varphi(0,\varepsilon)=a_{\varepsilon}\).
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Alvarez-Ramírez, M., Cornelio, J.L. & Medina, M. Periodic Orbits for a Fifth-Order Generalized Hénon–Heiles Hamiltonian System. Lobachevskii J Math 43, 1–9 (2022). https://doi.org/10.1134/S1995080222040023
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DOI: https://doi.org/10.1134/S1995080222040023