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May  2010, 9(3): 643-654. doi: 10.3934/cpaa.2010.9.643

Center manifolds for nonuniform trichotomies and arbitrary growth rates

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  June 2009 Revised  November 2009 Published  January 2010

We consider linear equations $v'=A(t)v$ in a Banach space that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates $e^{c\rho(t)}$ determined by a function $\rho(t)$. The usual exponential behavior with $\rho(t)=t$ is included as a very special case. For other functions the Lyapunov exponents may be infinite (either $+\infty$ or $-\infty$), but we can still distinguish between different asymptotic rates. Our main objective is to establish the existence of center manifolds for a large class of nonlinear perturbations $v'=A(t)v+f(t,v)$ assuming that the linear equation has the above general asymptotic behavior. We also allow the stable, unstable and central components of $v'=A(t)v$ to exhibit a nonuniform exponential behavior. We emphasize that our results are new even in the very particular case of perturbations of uniform exponential trichotomies with arbitrary growth rates.
Citation: Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure & Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643
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