Antidamping ($\ddot{x}+2\lambda \dot{x}+{\omega }_0^{2}x=0,\lambda <0$)

antidamping is a velocity-proportional term that supplies rather than removes mechanical energy. For the displayed damped harmonic oscillator with negative damping coefficient, $d[({\dot{x}}^2+{\omega }_0^2{x}^2)/2]/dt=-2\lambda {\dot{x}}^2\ge 0$. causality of a driven solution and its long-time stability are different properties.