Monotone Nonautonomous Dynamical Systems with a First Integral

In this chapter we study the problem of Bohr/Levitan almost periodicity, almost automorphy, almost recurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, and Poisson stability of solutions of monotone nonautonomous differential equations \(\displaystyle x'(t)=f(t,x(t)) \) having a strongly monotone first integral. We show that under some conditions every solution of Eq. (6.1.1) bounded on semiaxis has a limiting regime which has the same character of recurrence in time t as the right hand side f of Eq. (6.1.1).