Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Discrete and Continuous Dynamical Systems - Series B

Volumn 8, Issue 4, 2007, Pages 943-970

  Pokrovskii, Alexei ^a   Rasskazov, Oleg ^a   Visetti, Daniela ^a  

^a UNIVERSITY COLLEGE CORK   (Ireland)

Author keywords

Chaotic behaviour; Topological hyperbolicity; Transversal homoclinic orbit

Indexed keywords

EID: 37349100223     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2007.8.943     Document Type: Article

References (20)

1. M. Brokate, A. Pokrovskii, D. Rachinskii and O. Rasskazov, Differential equations with hysteresis via a canonical example, in "The Science of Hysteresis", Eds. Giorgio Bertotti, Isaak Mayergoyz, Academic Press, 1 (2006), 125-291.
2. K. Deimling, "Nonlinear Functional Analysis," Springer, Berlin, 1980.
3. J. P. Den Hartog, "Mechanical Vibrations," Dover, New York, 1985.
4. P. Diamond, P. Kloeden, M. Krasnosel'skii and A. Pokrovskii, Chaotic dynamics in nonsmooth perturbations of bishadowing system, Arab J. Math. Sci., 6 (2000), 41-74.
5. P. Diamond, P. Kloeden, V. Kozyakin and A. Pokrovskii, Semihyperbolic mappings, Journal of Nonlinear Science, 5 (1995), 419-431, Springer-Verlag, New York.
6. K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley and Sons, Chichester, 1990.
7. P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors, Physica, 9D (1983), 189-208.
8. B. Hasselblatt and A. E. Katok, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.
9. A. M. Krasnosel'skii and A. V. Pokrovskii, On large-amplitude subharmonics in a semilinear Duffing oscillator, (Russian), DAN, 391 (2003), 448-452.
10. A. M. Krasnosel'skii and A. V. Pokrovskii, Dissipativity of a nonresonance pendulum with ferromagnetic friction, (Russian), Avtomat. i Telemekh., (2006), 57-69; translation in Autom. Remote Control, 67 (2006), 221-232.
11. M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis," Springer, Berlin, 1984.
12. A. J. Lichtenberg and M. A. Lieberman, "Applied Mathematical Sciences Regular and Stochastic Motion," Springer-Verlag, New York, 1983.
13. W. S. Loud, Branching phenomena for periodic solutions of non-autonomous piecewise linear systems, Internat. J. Non-Linear Mech., 3 (1968), 273-293.
14. R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. London Math. Soc., (3) 79 (1999), 381-413.
15. A. V. Pokrovskii, Topological shadowing and split-hyperbolicity, J. Funct. Differ. Equ., 4 (1998), 335-360.
16. A. V. Pokrovskii, S. J. Szybka and J. G. McInerney, Topological degree in locating homoclinic structures for discrete dynamical systems, Izvestiya of RAEN, Series MMMIU, 5 (2001), 152-184.
17. R. Reissig, G. Sansone and R. Conti, "Qualitative Theorie Nichtlinearer Differentialgleihungen," Edizioni Cremonese, Rome, 1963.
18. X. Yang, Boundedness of solutions for semilinear Duffing equations, Appl. Math. Comput., 132 (2002), 423-436.
19. P. Zgliczyński, Fixed point index for iterations of maps, topological horseshoe and chaos, Topol. Methods Nonlinear Anal., 8 (1996), 169-177.
20. P. Zgliczyński, Computer assisted proof of the horseshoe dynamics in the Henon map, Random Comput. Dyn., 5 (1997), 1-17.