IJPAM: Volume 16, No. 2 (2004)

SYMBOLIC DYNAMICS
IN CHAOTIC WAVE VIBRATION

Sandra Vinagre$^1$, Ricardo Severino$^2$, J. Sousa Ramos$^3$
$^1$Department of Mathematics
University of Évora
Rua Romão Ramalho 59, Évora, 7000-671, PORTUGAL
e-mail: smv@uevora.pt
$^2$Department of Mathematics
University of Minho
Campus de Gualtar, Braga, 4710-057, PORTUGAL
e-mail: ricardo@math.uminho.pt
$^3$Department of Mathematics
I.S.T., Technical University of Lisbon
Avenida Rovisco Pais 1, Lisbon, 1049-001, PORTUGAL
e-mail: sramos@math.ist.utl.pt


Abstract.We consider a class of nonlinear boundary value problems for partial differential equations, whose solutions are, basically, characterized by the iteration of a nonlinear function. We apply methods of symbolic dynamics of discrete dynamical systems of the interval in order to compute the topological entropy associated to chaotic wave solutions. Then, from the variation of the entropy with the physical parameters we identify phase transitions. We also interpret the phase transition point in terms of the symbolic dynamics. Using these methods we formalize self-similar phenomena and study some of their properties.

Received: August 1, 2004

AMS Subject Classification: 37B10, 37E05, 35L20, 37B40, 37D45, 74H65

Key Words and Phrases: symbolic dynamics, chaotic vibration, boundary value problems, topological entropy, self-similarity, phase transition

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 16
Issue: 2