### 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 language | English |
---|---|

Article number | 016213 |

Number of pages | 10 |

Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |

Volume | 69 |

Issue number | 1 Part 2 |

DOIs | |

Publication status | Published - 30 Jan 2004 |

### Keywords

- on-off intermittency
- Lyapunov exponents
- dynamic systems
- trajectories
- orbits
- sets

### Cite this

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*,

*69*(1 Part 2), [016213]. https://doi.org/10.1103/PhysRevE.69.016213

**Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability.** / Do, Younghae; Lai, Ying-Cheng.

Research output: Contribution to journal › Article

*Physical Review. E, Statistical, Nonlinear and Soft Matter Physics*, vol. 69, no. 1 Part 2, 016213. https://doi.org/10.1103/PhysRevE.69.016213

}

TY - JOUR

T1 - Statistics of shadowing time in nonhyperbolic chaotic systems with unstable dimension variability

AU - Do, Younghae

AU - Lai, Ying-Cheng

PY - 2004/1/30

Y1 - 2004/1/30

N2 - 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.

AB - 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.

KW - on-off intermittency

KW - Lyapunov exponents

KW - dynamic systems

KW - trajectories

KW - orbits

KW - sets

U2 - 10.1103/PhysRevE.69.016213

DO - 10.1103/PhysRevE.69.016213

M3 - Article

VL - 69

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 1 Part 2

M1 - 016213

ER -