Computing Lyapunov Exponents for Time-Delay Systems

The hall mark property of a chaotic attractor, namely sensitive dependence on initial condition, has been associated by the Lyapunov exponents to characterize the degree of exponential divergence/convergence of trajectories arising from nearby initial conditions. At first, we will describe briefly the concept of Lyapunov exponent and the procedure for computing Lyapunov exponents of the flow of a dynamical system described by n-dimensional ordinary differential equations (ODEs), which is then extended to scalar delay differential equations (DDEs), which are essentially an infinite-dimensional systems. An important step in computing Lyapunov exponents of DDEs is that it is necessary to approximate the continuous evolution of an infinitedimensional system by a finite-dimensional (appreciably large) iterated mapping. Then the Lyapunov exponents of the finite-dimensional map can be calculated by computing simultaneously the reference trajectories from the original map and the trajectories from their linearized equations of motion. Alternatively, it can also be calculated by computing the evolution of infinitesimal volume element formed by a set of infinitesimal separation vectors corresponding to the trajectories starting from nearby initial conditions.