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*any*linear equation in block form in a finite-dimensional space, with three blocks having asymptotic rates $e^{c \rho(t)}$ respectively with $c$ negative, zero, and positive, admits a $\rho$-nonuniform exponential trichotomy. We also give explicit examples that cannot be made uniform and for which one cannot take $\rho(t)=t$ without making all Lyapunov exponents infinite. Furthermore, we obtain sharp bounds for the constants that determine the exponential trichotomy. These are expressed in terms of appropriate Lyapunov exponents that measure the growth rate with respect to the function $\rho$.

*nonuniformly partially hyperbolic dynamics*, corresponding to the existence of a nonuniform exponential trichotomy of the linear variational equation. We also consider the case of

*nonautonomous dynamics*.

$ \partial_t u = -\partial_x v - \alpha \partial_{x x x}v - \partial_x(u v), \quad \partial_t v = - \partial_x u - v \partial_x v,$

For $\alpha \leq 1$, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every $\alpha$, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form.

*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.

$\partial_t u = -\partial_x v - \alpha \partial_{x x x} v - \epsilon \partial_x(u v), \quad \partial_t v = - \partial_x u - \epsilon v \partial_x v,$

where $\epsilon$ is an small parameter and $\alpha \in (0,1)$. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for some values of $\alpha$, it contains solutions that are defined for large values of time and they are very close (of order $O(\epsilon)$) to a linear torus for long times (of order $O(\epsilon^{-1})$). The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

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