Positive Lyapunov exponent from time series of strange nonchaotic system

Abstract

Because of the complex properties of high-dimensional nonlinear systems, e.g. neural networks and cardiac tissue, it has become a standard practice to study the reconstructed attractor from measured time series. Here, the authors show that the time-series methods for estimating Lyapunov exponents can give a positive exponent when they are applied to the time series of strange nonchaotic systems. Strange nonchaotic systems are characterized by an unstable phase space that generates repeatedly expanding dynamics. If some variables of a strange nonchaotic system are Independent of the others, the expanding dynamics can occur in different time intervals for different variables. It is then possible to find at any time a variable undergoing expanding dynamics and generating a disordered trajectory. In this case, if the observable signal is a sum of these variables, the time series is ill-conditioned. Thus with the time series method, the obtained maximum Lyapunov exponent can be positive. As two examples, a two-neuron system driven by quasiperiodic forces and an unidirectionally coupled 100-logistic-map lattice driven by quasiperiodic forces are discussed numerically. The authors' research suggests that there are some limitations for the use of time-series methods for complex systems.