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In some recent papers, the so called $(H,\rho)$-induced dynamics of a system$\mathcal{S}$ whose time evolution is deduced adopting an operatorial approach,has been introduced. According to the formal mathematical apparatus of quantummechanics, $H$ denotes the Hamiltonian for $\mathcal{S}$, while $\rho$ is acertain rule applied periodically on $\mathcal{S}$. In this approach the ruleacts at specific times $k\tau$, with $k$ integer and $\tau$ fixed, by modifyingsome of the parameters entering $H$ according to the state variation of thesystem. As a result, a dynamics admitting an asymptotic equilibrium state canbe obtained. Here, we consider the limit for $\tau\rightarrow 0$, so that weintroduce a generalized model leading to asymptotic equilibria. Moreover, inthe case of a two-mode fermionic model, we are able to derive a relationlinking the parameters involved in the Hamiltonian to the asymptoticequilibrium states.

Generalized Hamiltonian for a two-mode fermionic model and asymptotic equilibria