A new simple proof for Lum-Chua's conjecture

Autor(es)

Carmona, Victoriano ^{[1, 2]} ; Fernandez-Sanchez, Fernando ^{[3, 4]} ; Novaes, Douglas D. ^[5]

Afiliação do(s) autor(es)

^[1] Univ Seville, Dept Matemat Aplicada 2, Escuela Politecn Super, Calle Virgen Africa 7, Seville 741011 - Spain

^[2] Univ Seville, IMUS, Escuela Politecn Super, Calle Virgen Africa 7, Seville 741011 - Spain

^[3] Univ Seville, Dept Matemat Aplicada 2, Escuela Tecn Super Ingn, Camino Descubrimientos S-N, Seville 41092 - Spain

^[4] Univ Seville, IMUS, Escuela Tecn Super Ingn, Camino Descubrimientos S-N, Seville 41092 - Spain

^[5] Univ Estadual Campinas, Dept Matemat, Rua Sergio Buarque de Holanda 651, Cidade Univ, BR-13083859 Campinas, SP - Brazil

• Tipo de documento: Artigo Científico
• Fonte: NONLINEAR ANALYSIS-HYBRID SYSTEMS; v. 40, MAY 2021.
• Citações Web of Science: 1

Resumo

The already proved Lum-Chua's conjecture says that a continuous planar piecewise linear differential system with two zones separated by a straight line has at most one limit cycle. In this paper, we provide a new proof by using a novel characterization for Poincare half-maps in planar linear systems. This proof is very short and straightforward, because this characterization avoids the inherent flaws of the usual methods to study piecewise linear systems (the appearance of large case-by-case analysis due to the unnecessary discrimination between the different spectra of the involved matrices, to deal with transcendental equations forced by the implicit occurrence of flight time,...). In addition, the application of the characterization allow us to prove that if a limit cycle exists, then it is hyperbolic and its stability is determined by a simple relationship between the parameters. To the best of our knowledge, the hyperbolicity of the limit cycle and this simple expression for its stability have not been pointed out before. (C) 2020 Elsevier Ltd. All rights reserved. (AU)

• Processo FAPESP: 18/16430-8 - Dinâmica global das equações diferenciais não suaves
• Beneficiário: Douglas Duarte Novaes
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• 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: 18/13481-0 - Geometria de sistemas de controle, sistemas dinâmicos e estocásticos
• Beneficiário: Marco Antônio Teixeira
• Modalidade de apoio: Auxílio à Pesquisa - Temático