| Full text | |
| Author(s): |
Llibre, Jaume
;
Novaes, Douglas D.
;
Valls, Claudia
Total Authors: 3
|
| Document type: | Journal article |
| Source: | Chaos; v. 33, n. 7, p. 17-pg., 2023-07-01. |
| Abstract | |
The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations (x) triple over dot + vertical bar x vertical bar(q) (x) triple over dot + k|x|(q)/x (x)over dot(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)over dot, (x) triple over dot)-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. (AU) | |
| FAPESP's process: | 18/13481-0 - Geometry of control, dynamical and stochastic systems |
| Grantee: | Marco Antônio Teixeira |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 21/10606-0 - On limit cycles in piecewise linear vector fields with algebraic discontinuity variety |
| Grantee: | Douglas Duarte Novaes |
| Support Opportunities: | Scholarships abroad - Research |
| FAPESP's process: | 19/10269-3 - Ergodic and qualitative theories of dynamical systems II |
| Grantee: | Claudio Aguinaldo Buzzi |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 22/09633-5 - Averaging theory for studying invariant tori and periodic behavior in differential equations and inclusions |
| Grantee: | Douglas Duarte Novaes |
| Support Opportunities: | Regular Research Grants |