A dual-oscillator approach to complex-stiffness damping based on fourth-order dynamics

It is shown that the classical damped harmonic oscillator belongs to family of fourth-order Pais-Uhlenbeck oscillators. follows solutions equation make action stationary. Two systematic approaches are given for deriving from equation, and it may be possible use these methods identify stationary principles other dissipative systems which do not conform Hamilton's principle. also every $x$, there exists a two-parameter dual oscillators $y$ satisfying equation. The any its duals can interpreted as system two coupled with atypical spring stiffnesses (not necessarily positive real-valued). For overdamped systems, resulting should physically achievable have engineering applications. Finally, new physical interpretation optimal damping ratio $\zeta=1/\sqrt{2}$ in control theory.