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High-frequency chaotic solutions for a slowly varying dynamical system

Published online by Cambridge University Press:  18 January 2006

PATRICIO FELMER ,
SALOMÉ MARTÍNEZ  and
KAZUNAGA TANAKA
PATRICIO FELMER
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (e-mail: [email protected])
SALOMÉ MARTÍNEZ
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (e-mail: [email protected])
KAZUNAGA TANAKA
Affiliation:
Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
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Abstract

In this article we study the asymptotic dynamics of highly oscillatory solutions for the unbalanced Allen–Cahn equation with a slowly varying coefficient. We describe the underlying structure of these solutions through a function we call the adiabatic profile, which accounts for the asymptotic area covered by the solutions in the phase space. In finite intervals, we construct solutions given any adiabatic profile. In the case of a periodic coefficient we show that the system has chaotic behavior by constructing high-frequency complex solutions which can be characterized by a bi-infinite sequence of real numbers in $[c_1,c_2]\cup\{ 0\}\ (0<c_1<c_2)$.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 26 , Issue 2 , April 2006 , pp. 379 - 407
Copyright
2006 Cambridge University Press

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