Symmetry property of the Lyapunov spectra of a class of dissipative dynamical systems with viscous damping

It is shown that the Lyapunov spectra of a class of dissipative dynamical systems with viscous damping are symmetric with respect to a constant determined by the dissipation of the system. A property of the associated linearized flow similar to the symplectic property is established. Furthermore, numerical computations of the characteristic exponents of 15 coupled damped Duffing oscillators driven by a van der Pol oscillator are presented.