On the cyclicity of hyperbolic polycycles

Autor(es)

Buzzi, Claudio ; Gasull, Armengol ; Santana, Paulo

• Tipo de documento: Artigo Científico
• Fonte: Journal of Differential Equations; v. 429, p. 32-pg., 2025-02-25.

Resumo

Let X be a planar smooth vector field with a polycycle Gamma n with n sides and all its corners, that are at most n singularities, being hyperbolic saddles. In this paper we study the cyclicity of Gamma n in terms of the hyperbolicity ratios of these saddles, giving explicit conditions that ensure that it is at least k, for any k <= n. Our result extends old results and also provides a more accurate proof of the known ones because we rely on some recent powerful works that study in more detail the regularity with respect to initial conditions and parameters of the Dulac map of hyperbolic saddles for families of vector fields. We also prove that when Xis polynomial there is a polynomial perturbation (in general with degree much higher that the one of X) that attains each of the obtained lower bounds for the cyclicities. Finally, we also study some related inverse problems and provide concrete examples of applications in the polynomial world. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. (AU)

• Processo FAPESP: 19/10269-3 - Teorias ergódica e qualitativa dos sistemas dinâmicos II
• Beneficiário: Claudio Aguinaldo Buzzi
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 23/02959-5 - Sistemas Não Suaves e Perturbações Singulares
• Beneficiário: Paulo Ricardo da Silva
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 22/14353-1 - Estudo de policiclos com aplicações na teoria dos jogos
• Beneficiário: Paulo Henrique Reis Santana
• Modalidade de apoio: Bolsas no Exterior - Estágio de Pesquisa - Doutorado
• Processo FAPESP: 21/01799-9 - Estudo de campos de vetores com aplicações na teoria dos jogos
• Beneficiário: Paulo Henrique Reis Santana
• Modalidade de apoio: Bolsas no Brasil - Doutorado