Stability of periodic or anti-periodic solutions of certain differential systems (Q1780140)

Summary: We consider the nonlinear differential system in \(\mathbb{R}^2\) \[ \begin{gathered} u'(t)+ ku(t)(u(t)^2+ v(t)^2)-\lambda u(t)= h_1(t),\\ v'(t)+ kv(t)(u(t)^2+ v(t)^2)-\lambda v(t)= h_2(t).\end{gathered} \] In order to study the stability of anti-periodic solutions of this system, a result of Lyapounov in the stability theory of autonomous nonlinear ordinary differential equations (ODEs) is enlarged to differential systems of the form \(u'= F(t, u)\) in the Hilbert space framework with finite dimension. On the other hand, a result of R. Bellman in the instability theory of autonomous nonlinear ODEs is enlarged to differential systems where \(L\), the linearized operator for \(F\), is a nonautonomous periodic operator.