Global attraction to the origin in a parametrically driven nonlinear oscillator (Q1826672)

Sufficient conditions are given for the coefficients in the equation for a parametrically driven nonlinear oscillator so that the origin is a global attractor and that the trajectories converge monotonically to the origin. The equation considered is \(\dot x = y\), \(\dot y = -\gamma - f(t)x^{2m-1}\), \((x,y)\in \mathbb R^2\), \(t\geq 0\), where \(f(t)\) is bounded from above \(0\leq f\leq f^+\). It is proved that the origin is a global attractor if \(\gamma >-\min(f'(t)/2f(t))\) and letting \(u(t)= x(t)^{-2m}\), \(v(t) =\dot u(t)\), the trajectories \(x(t)\), \(y(t)\) for which \(u(t),v(t)\) enter the region \(u>0\), \(v>0\), \(u>4f^+/\gamma^2\), \(v/u < m(\gamma +\sqrt{\gamma^2-4f^+/u)}\) converge monotonically to zero. Further behavior of the trajectories is discussed based on numerical simulations.