Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Novaes, Douglas D. (Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. Departamento de Matemática)
Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)

Data:	2023
Resum:	The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations x ⃛ + | x | q x ¨ + k | x | q x x ˙ 2 = 0, which has its regularity varying with q, a positive integer. Indeed, for q = 1, it is discontinuous on the straight line x = 0, whereas for q a positive even integer it is polynomial, and for q > 1 a positive odd integer it is continuous but not differentiable on the straight line x = 0. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for q = 2 and k = 3. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for k = q + 1 and any positive integer q, has actually an invariant topological cylinder foliated by periodic solutions in the (x, x ˙, x ¨) -space. In order to set forth the bases of our approach, we start by considering q = 1, 2, 3, which are representatives of the different classes of regularity. For an arbitrary positive integer q, an algorithm is provided for checking the sufficient conditions for the existence of such an invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to q = 100.
Ajuts:	Agencia Estatal de Investigación PID2019-104658GB-I00 European Commission 777911
Drets:	Tots els drets reservats.
Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Generalized Chazy differential equation ; Discontinuous differential system ; Continuous differential system ; Analytic differential system ; Periodic solution
Publicat a:	Chaos, Vol. 33, Issue 7 (July 2023) , art. 73104, ISSN 1089-7682

DOI: 10.1063/5.0138309

Postprint 22 p, 1.3 MB

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