On the boundedness and periodicity of the solutions to a certain vector differential equation of third-order (Q1305369)

The third-order system of nonlinear differential equations \[ \dddot X+F(X,\dot X)\ddot X+B\dot X+H(X)=P(t,X,\dot X,\ddot X)\tag{1} \] is considered, with \(X\in\mathbb{R}^n\), \(F,H\) and \(P\) are continuous, \(B\) is a real constant symmetric \(n\times n\)-matrix and the dots denote differentiation with respect to \(t\). Sufficient conditions are established for the ultimate boundedness of solutions and for the existence of periodic solutions to (1).