Periodic and Unbounded Solutions of Periodic Systems

This paper deals with various cases of resonance, which is a fundamental concept of science and engineering. Specifically, we study the connections between periodic and unbounded solutions for several classes of equations and systems. In particular, we extend the classical Massera’s theorem, dealing with periodic systems of the type \(\begin{aligned} x'=A(t)x+f(t) , \end{aligned}\) $$\begin{array}{c}\hfill x^{\prime }=A(t)x+f(t),\end{array}$$ and clarify that this theorem deals with a case of resonance. Then we provide instability results for the corresponding semilinear systems, with the linear part at resonance. We also use the solution curves developed previously by the author to establish the instability results for pendulum-like equations, and for first-order periodic equations.