# American Institute of Mathematical Sciences

January  2010, 26(1): 63-74. doi: 10.3934/dcds.2010.26.63

## Quadratic Lyapunov sequences 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  August 2008 Revised  August 2009 Published  October 2009

We characterize completely in terms of strict quadratic Lyapunov sequences when a linear nonautonomous dynamics admits a nonuniform exponential dichotomy. We allow asymptotic rates of the form $e^{c\rho(m)}$ determined by an arbitrary increasing sequence $\rho(m)$, which thus may correspond to infinite Lyapunov exponents, and not only the usual exponential behavior with $\rho(m)=m$. In particular, we obtain inverse theorems in this general setting, by constructing explicitly a strict quadratic Lyapunov sequence for each nonuniform exponential dichotomy. Furthermore, for a large class of perturbations and using only quadratic Lyapunov sequences, we show in a simple manner that if the linear dynamics admits a nonuniform exponential dichotomy, then the perturbed dynamics remains unstable.
Citation: Luis Barreira, Claudia Valls. Quadratic Lyapunov sequences and arbitrary growth rates. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 63-74. doi: 10.3934/dcds.2010.26.63
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