Three-dimensional time-varying nonlinear systems containing a Hamilton system (Q631697)

For some specific function \(H(x,y)\), all orbits of the system \[ x'= \frac{\partial}{\partial y}H(x,y),\qquad y'= -\frac{\partial }{\partial x}H(x,y) \] near the origin are isolated closed curves surrounding the origin. The author obtains sufficient conditions on the functions \(f(t),g(t)\) and \(h(t)\) that guarantee that the zero solution of the following system is uniformly stable and asymptotically stable \[ x'=\frac{\partial }{\partial y}H(x,y), \quad y'= -\frac{\partial }{\partial x}H(x,y)+f(t),\quad z'= -g(t)\frac{\partial}{\partial y}H(x,y)-h(t)z. \] Several examples are given to illustrate the author's results.