Extreme sensitive dependence on parameters and initial conditions in spatio-temporal chaotic dynamical systems

Abstract

We investigate the sensitive dependence of asymptotic attractors on both initial conditions and parameters in spatio-temporal chaotic dynamical systems. Our models of spatio-temporal systems are globally coupled two-dimensional maps and locally coupled ordinary differential equations. It is found that extreme sensitive dependence occurs commonly in both phase space and parameter space of these systems. That is, for an initial condition and/or a parameter value that leads to chaotic attractors, there are initial conditions and/or parameter values arbitrarily nearby that lead to nonchaotic attractors. This indicates the occurrence of an extreme type of fractal structure in both phase space and parameter space. A scaling exponent used to characterize extreme sensitive dependence on initial conditions and parameters is determined to be near zero in both phase space and parameter space. Accordingly, there is a significant probability of error in numerical computations intended to determine asymptotic attractors, regardless of the precision with which initial conditions or parameters are specified. Consequently, fundamental statistical properties of asymptotic attractors cannot be computed reliably for particular parameter values and initial conditions.