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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 and Applied Analysis, 2017, 16 (6) : 2337-2355. doi: 10.3934/cpaa.2017115
References:
[1]

G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, 1966.

[2]

T. Gnana BhaskarS. Hariharan and N. Nataraj, Heatlet approach to diffusion equation on unbounded domains, Applied Mathematics and Computation, 197 (2008), 891-903.  doi: 10.1016/j.amc.2007.08.060.

[3]

T. CaraballoR. Colucci and X. Han, Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing, Nonlinear Anal. Real World Appl., 31 (2016), 661-680.  doi: 10.1016/j.nonrwa.2016.03.007.

[4]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, SpringerBriefs in Mathematics, Springer International Publishing, 2016. doi: 10.1007/978-3-319-49247-6.

[5]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[6]

T. CaraballoG. Ł ukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[7]

W. A. Coppel, Almost periodic properties of ordinary differential equations, Anal. Mat. Pura Appl., 76 (1967), 27-49.  doi: 10.1007/BF02412227.

[8]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D-. C. Heath & Co. , Boston, 1965.

[9]

I. C. Daubechies and A. C. Gilbert, Harmonic Analysis, Wavelets and Applications, Proceedings of the IEEE. 84 (1996). doi: 10.1006/acha.1997.0234.

[10]

J. K. Hale, Ordinary Differential Equations, Second Edition, Krieger Publishing Co. , Huntington, New York, 1980.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes Math. , Vol. 840, Springer-Verlag, Berlin, 1981.

[12]

E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press LLC, 1996. doi: 10.1201/9781420049985.

[13]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stochastics & Dynamics, 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[14]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, volume 176 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[15]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis, 74 (2011), 2695-2719.  doi: 10.1016/j.na.2010.12.025.

[16]

S. Mallat, A Wavelet Tour of Signal Processing Academic Press, 1998.

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.

[18]

J. Shen and G. Strang, On wavelets fundamental solutions to the heat equation, J. of Differential Equations, 161 (2000), 403-421.  doi: 10.1006/jdeq.1999.3707.

[19]

B. Vidakovic and P. Mueller, Wavelets for Kids. A Tutorial Introduction Duke University. http://www2.isye.gatech.edu/~brani/wp/kidsA.pdf.

show all references

References:
[1]

G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, 1966.

[2]

T. Gnana BhaskarS. Hariharan and N. Nataraj, Heatlet approach to diffusion equation on unbounded domains, Applied Mathematics and Computation, 197 (2008), 891-903.  doi: 10.1016/j.amc.2007.08.060.

[3]

T. CaraballoR. Colucci and X. Han, Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing, Nonlinear Anal. Real World Appl., 31 (2016), 661-680.  doi: 10.1016/j.nonrwa.2016.03.007.

[4]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, SpringerBriefs in Mathematics, Springer International Publishing, 2016. doi: 10.1007/978-3-319-49247-6.

[5]

T. CaraballoX. Han and P. E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal., 47 (2015), 2178-2199.  doi: 10.1137/14099930X.

[6]

T. CaraballoG. Ł ukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal., 64 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[7]

W. A. Coppel, Almost periodic properties of ordinary differential equations, Anal. Mat. Pura Appl., 76 (1967), 27-49.  doi: 10.1007/BF02412227.

[8]

W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations, D-. C. Heath & Co. , Boston, 1965.

[9]

I. C. Daubechies and A. C. Gilbert, Harmonic Analysis, Wavelets and Applications, Proceedings of the IEEE. 84 (1996). doi: 10.1006/acha.1997.0234.

[10]

J. K. Hale, Ordinary Differential Equations, Second Edition, Krieger Publishing Co. , Huntington, New York, 1980.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer Lecture Notes Math. , Vol. 840, Springer-Verlag, Berlin, 1981.

[12]

E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press LLC, 1996. doi: 10.1201/9781420049985.

[13]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stochastics & Dynamics, 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.

[14]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, volume 176 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[15]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Analysis, 74 (2011), 2695-2719.  doi: 10.1016/j.na.2010.12.025.

[16]

S. Mallat, A Wavelet Tour of Signal Processing Academic Press, 1998.

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York Berlin Heidelberg Tokyo, 1983. doi: 10.1007/978-1-4612-5561-1.

[18]

J. Shen and G. Strang, On wavelets fundamental solutions to the heat equation, J. of Differential Equations, 161 (2000), 403-421.  doi: 10.1006/jdeq.1999.3707.

[19]

B. Vidakovic and P. Mueller, Wavelets for Kids. A Tutorial Introduction Duke University. http://www2.isye.gatech.edu/~brani/wp/kidsA.pdf.

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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