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Dynamics of a Class of ODEs via Wavelets

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

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  • 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.

    Mathematics Subject Classification: 34G20, 35B15, 42C40, 42B25.


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  • Figure 1.  Scaling solution $x_j(t)$ for $j=2$ (red), and $ j=-2$ (green)

    Figure 2.  Wavelet solution $y_j(t)$ for $j=1$ (red), and $j=-1$ (green).

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