| Full text | |
| Author(s): |
Carvalho, Tiago
;
Antunes, Andre do Amaral
Total Authors: 2
|
| Document type: | Journal article |
| Source: | Journal of Differential Equations; v. 419, p. 25-pg., 2024-11-29. |
| Abstract | |
It is well known that many results obtained for piecewise smooth vector fields do not have an analogous for smooth vector fields and vice-versa. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between shift maps and piecewise smooth vector fields presenting homoclinic loops which are associated to symbols in such a way that the flow restricted to a homoclinic loop is codified with a symbol. The construction of the topological conjugation between the quoted piecewise smooth vector fields and the respective shift spaces needs several technicality which were solved considering a specific family of piecewise smooth vector fields (Theorem A) and then generalizing the result for an entire class of piecewise smooth vector fields (Theorem B). By means of the results obtained and the techniques employed, a new perspective on the study of piecewise smooth vector fields is brought to light and, through already established results for discrete dynamic systems, we will be able to obtain results regarding piecewise smooth vector fields. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies. (AU) | |
| 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: | 19/10450-0 - Piecewise smooth vector fields: Closing Lemmas, shifts and horseshoe dynamics. |
| Grantee: | Tiago de Carvalho |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 21/12395-6 - Piecewise smooth vector fields: theory and applications |
| Grantee: | Tiago de Carvalho |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 17/18255-6 - Closing lemmas and shifts for piecewise smooth vector fields |
| Grantee: | Andre do Amaral Antunes |
| Support Opportunities: | Scholarships in Brazil - Doctorate (Direct) |
| FAPESP's process: | 22/02819-6 - Intermittent vector fields: theoretical aspects and applications |
| Grantee: | Tiago de Carvalho |
| Support Opportunities: | Research Grants - Initial Project |
| FAPESP's process: | 17/00883-0 - Piecewise smooth vector fields with applications in biology |
| Grantee: | Tiago de Carvalho |
| Support Opportunities: | Regular Research Grants |