Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-24T05:51:14.981Z Has data issue: false hasContentIssue false
  • English
  • Français

Chaotic planar piecewise smooth vector fields with non-trivial minimal sets

Published online by Cambridge University Press:  05 August 2014

CLAUDIO A. BUZZI ,
TIAGO DE CARVALHO  and
RODRIGO D. EUZÉBIO
CLAUDIO A. BUZZI
Affiliation:
IBILCE-UNESP, CEP 15054-000, S. J. Rio Preto, São Paulo, Brazil email buzzi@ibilce.unesp.br, rodrigo.euzebio@sjrp.unesp.br
TIAGO DE CARVALHO
Affiliation:
FC-UNESP, CEP 17033-360, Bauru, São Paulo, Brazil email tcarvalho@fc.unesp.br
RODRIGO D. EUZÉBIO
Affiliation:
IBILCE-UNESP, CEP 15054-000, S. J. Rio Preto, São Paulo, Brazil email buzzi@ibilce.unesp.br, rodrigo.euzebio@sjrp.unesp.br
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. Some relations between minimality and orientable minimality are also investigated and the existence of new kinds of non-trivial minimal sets in chaotic systems is observed. The approach is geometrical and involves the ordinary techniques of non-smooth systems.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 2 , April 2016 , pp. 458 - 469
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

di Bernardo, M., Budd, C. J., Champneys, A. R. and Kowalczyk, P.. Piecewise-Smooth Dynamical Systems. Theory and Applications (Applied Mathematical Sciences, 163). Springer, London, 2008.Google Scholar
Buzzi, C. A., de Carvalho, T. and Euzebio, R. D.. On Poincare–Bendixson theorem and non-trivial minimal sets in planar nonsmooth vector fields. Preprint, 2013, http://arxiv.org/pdf/1307.6825v1.pdf.Google Scholar
Colombo, A. and Jeffrey, M. R.. Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows. SIAM J. Appl. Dyn. Syst. 102 (2011), 423–451.CrossRefGoogle Scholar
Devaney, R. L.. Introduction to Chaotic Dynamical Systems. Westview Press, Boulder, CO, 1989.Google Scholar
Filippov, A. F.. Differential equations with discontinuous righthand sides. Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers, Dordrecht, 1988.Google Scholar
Guardia, M., Seara, T. M. and Teixeira, M. A.. Generic bifurcations of low codimension of planar Filippov systems. J. Differential Equations 250 (2011), 1967–2023.CrossRefGoogle Scholar
Gutierrez, C.. Smoothing continuous flows and the converse of Denjoy–Schwartz theorem. An. Acad. Brasil. Ci. 51(4) (1979), 581–589.Google Scholar
Jeffrey, M. R.. Nondeterminism in the limit of nonsmooth dynamics. Phys. Rev. Lett. 106 (2011),254103.CrossRefGoogle ScholarPubMed
m Meiss, J. D.. Differential Dynamical Systems. SIAM, Philadelphia, PA, 2007.CrossRefGoogle Scholar
Teixeira, M. A.. Perturbation Theory for Non-smooth Systems (Encyclopedia of Complexity and Systems Science). Ed. Meyers, R. A.. Springer, New York, 2009, pp. 6697–6709.Google Scholar