# American Institute of Mathematical Sciences

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November  2017, 16(6): 2337-2355. doi: 10.3934/cpaa.2017115

## Dynamics of a Class of ODEs via Wavelets

 1 Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668,13560-970, São Carlos, SP, Brazil 2 Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Tarfia s/n, 41012-Sevilla, Spain

Received  December 2016 Revised  March 2017 Published  July 2017

Fund Project: H.R. was partially supported by FAPESP grant 2015/19165-5. T.C. was partially supported by the projects MTM2015-63723-P (MINECO/ FEDER, EU) and P12-FQM-1492 (Junta de Andalucía). M.G. was partially supported by FAPESP grants 2013/07460-7,2016/08704-5, and 2016/21032-6, and by CNPq grants 305860/2013-5 and 310740/2016-9, Brazil.

The objective of this paper is to study a perturbed linear hyperbolic differential equation. The first part of this work is dedicated to study perturbation of the equilibrium (special solution) of a perturbed hyperbolic system. On the second part we analyze the stable and the unstable manifolds of a perturbed semilinear differential equation. We assume that the perturbed forcing function belongs to an $L_2$ class and that it is developed in a series of wavelets. Then we analyze the effect of this development on the special solution of the perturbed equation. Similar study is provided for the stable and unstable manifolds of this special solutions.

Citation: Hildebrando M. Rodrigues, Tomás Caraballo, Marcio Gameiro. Dynamics of a Class of ODEs via Wavelets. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2337-2355. doi: 10.3934/cpaa.2017115
##### References:

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##### References:
Scaling solution $x_j(t)$ for $j=2$ (red), and $j=-2$ (green)
Wavelet solution $y_j(t)$ for $j=1$ (red), and $j=-1$ (green).
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