Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability.

Severe obstruction to shadowing of computer-generated trajectories can occur in nonhyperbolic chaotic systems with unstable dimension variability. That is, when the dimension of the unstable eigenspace changes along a trajectory in the invariant set, no true trajectory of reasonable length can be found to exist near any numerically generated trajectory. An important quantity characterizing the shadowability of numerical trajectories is the shadowing time, which measures for how long a trajectory remains valid. This time depends sensitively on initial condition. Here we show that the probability distribution of the shadowing time contains two distinct scaling behaviors: an algebraic scaling for short times and an exponential scaling for long times. The exponential behavior depends on system details but the small-time algebraic behavior appears to be universal. We describe the computational procedure for computing the shadowing time and give a physical analysis for the observed scaling behaviors.