Three time scale singular perturbation problems and nonsmooth dynamical systems
Authors:
Pedro T. Cardin, Paulo R. da Silva and Marco A. Teixeira
Journal:
Quart. Appl. Math. 72 (2014), 673-687
MSC (2010):
Primary 34D15, 34N05, 34C20
DOI:
https://doi.org/10.1090/S0033-569X-2014-01360-X
Published electronically:
September 17, 2014
MathSciNet review:
3291821
Full-text PDF Free Access
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Abstract: In this paper we study three time scale singular perturbation problems \[ \varepsilon x’ = f(\mathbf {x},\varepsilon ,\delta ), \qquad y’ = g(\mathbf {x},\varepsilon ,\delta ), \qquad z’ = \delta h(\mathbf {x},\varepsilon ,\delta ), \] where $\mathbf {x} = (x,y,z) \in \mathbb {R}^n \times \mathbb {R}^m \times \mathbb {R}^p$, $\varepsilon$ and $\delta$ are two independent small parameters $(0<\varepsilon$, $\delta \ll 1$), and $f$, $g$, $h$ are $C^r$ functions, where $r$ is big enough for our purposes. We establish conditions for the existence of compact invariant sets (singular points, periodic and homoclinic orbits) when $\varepsilon , \delta > 0$. Our main strategy is to consider three time scales which generate three different limit problems. In addition, we prove that double regularization of nonsmooth dynamical systems with self-intersecting switching variety provides a class of three time scale singular perturbation problems.
References
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References
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- B. Deng and G. Hines, Food chain chaos due to Shilnikov’s orbit, Chaos 12 (2002), no. 3, 533–538. MR 1939450 (2003j:92014), DOI https://doi.org/10.1063/1.1482255
- N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations 31 (1979), no. 1, 53–98. MR 524817 (80m:58032), DOI https://doi.org/10.1016/0022-0396%2879%2990152-9
- A. F. Filippov, Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR 1028776 (90i:34002)
- G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol. 60 (2010), no. 3, 347–386. MR 2576546 (2011b:34151), DOI https://doi.org/10.1007/s00285-009-0266-7
- C. K. R. T. Jones, Geometric singular perturbation theory, Dynamical systems (Montecatini Terme, 1994) Lecture Notes in Math., vol. 1609, Springer, Berlin, 1995, pp. 44–118. MR 1374108 (97e:34105), DOI https://doi.org/10.1007/BFb0095239
- T. J. Kaper, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, (Baltimore, MD, 1998) Proc. Sympos. Appl. Math., vol. 56, Amer. Math. Soc., Providence, RI, 1999, pp. 85–131. MR 1718893 (2000h:34090)
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- R. M. May, Limit cycles in predator–prey communities. Science 177 (1972), 900–902.
- G. S. Medvedev and J. E. Cisternas, Multimodal regimes in a compartmental model of the dopamine neuron, Phys. D 194 (2004), no. 3-4, 333–356. MR 2075659, DOI https://doi.org/10.1016/j.physd.2004.02.006
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- S. Muratori and S. Rinaldi, Remarks on competitive coexistence, SIAM J. Appl. Math. 49 (1989), no. 5, 1462–1472. MR 1015073 (90k:92050), DOI https://doi.org/10.1137/0149088
- S. Rinaldi and S. Muratori, Slow–fast limit cycles in predator–prey models, Ecol. Model. 61 (1992), 287–308.
- M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator–prey interactions, Am. Nat. 97 (1963), 209–223.
- Y. Shimazu et al., Some problems in ecology oriented enviromentology, J. Earth Sci. Nagoya Univ. 20 (1972), 31–89.
- P. Szmolyan, Transversal heteroclinic and homoclinic orbits in singular perturbation problems, J. Differential Equations 92 (1991), no. 2, 252–281. MR 1120905 (92e:58185), DOI https://doi.org/10.1016/0022-0396%2891%2990049-F
- M. A. Teixeira, Perturbation Theory for Non-smooth Dynamical Systems, Encyclopedia of Complexity and Systems Science, Ed. G. Gaeta, Springer-Verlag, 2008.
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Additional Information
Pedro T. Cardin
Affiliation:
Departamento de Matemática – Faculdade de Engenharia de Ilha Solteira, UNESP – Univ Estadual Paulista, Rua Rio de Janeiro, 266, CEP 15385–000 Ilha Solteira, São Paulo, Brazil
Email:
pedrocardin@mat.feis.unesp.br
Paulo R. da Silva
Affiliation:
Departamento de Matemática – Instituto de Biociências, Letras e Ciências Exatas, UNESP – Univ Estadual Paulista, Rua Cristóvão Colombo, 2265, CEP 15054–000 S. J. Rio Preto, São Paulo, Brazil
Email:
prs@ibilce.unesp.br
Marco A. Teixeira
Affiliation:
IMECC–UNICAMP, CEP 13081–970, Campinas, São Paulo, Brazil
Email:
teixeira@ime.unicamp.br
Keywords:
Geometric theory,
singular perturbations,
three time scales,
nonsmooth dynamical systems
Received by editor(s):
October 12, 2012
Published electronically:
September 17, 2014
Article copyright:
© Copyright 2014
Brown University