The evolution of covariant Lyapunov modes in QOD hard disc systems

We describe the time evolution of the covariant Lyapunov modes for a quasi-one-dimensional hard disc system from knowledge of the Gram–Schmidt Lyapunov modes. For the zero modes the time evolution is exact, but for the transverse and longitudinal momentum modes of all orders the time evolution is an accurate approximation. This description, based on the dynamics alone, allows the calculation of angle distributions and means for the conjugate covariant modes. From a complete knowledge of the functional forms of the Gram–Schmidt modes, the localizations of both the Gram–Schmidt and covariant Lyapunov modes can be obtained. These localizations agree to high accuracy with those obtained by direct numerical calculation