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

Younghae Do, Ying-Cheng Lai

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Number of pages1
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume69
Issue number1
DOIs
StatePublished - Jan 1 2004

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Shadowing
Chaotic System
Unstable
statistics
trajectories
Trajectory
Statistics
scaling
Scaling Behavior
Scaling
Eigenspace
Invariant Set
Obstruction
Initial conditions
Probability Distribution
Valid
Distinct
Computing

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics

Cite this

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abstract = "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.",
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