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Transient Behavior of Stochastic Dynamics

  • Author / Creator
    Liu, Shuo
  • This thesis concerns dynamical systems subject to small noise perturbations. Our purpose is to obtain a deep understanding of how small noise perturbations influence the original unperturbed dynamical system, especially over long but finite time intervals. We consider two special systems, the random linear recurrence equation which is a discrete system, and the underdamped Langevin equation which is a continuous system.

    According to the classical perturbation theory, small perturbations do not impact much of the dynamics over a short time. When the time tends to infinity, in many situations, the flow can eventually merge with one of the steady states of the perturbed system. As far as we know, small perturbations gradually influence the dynamics in an accumulative way over time, resulting in the so-called transient behavior which describes the process of such a gradual change of dynamics from the unperturbed system to the steady state of the corresponding perturbed system.

    We demonstrate this change by utilizing the cut-off phenomenon in the random linear recurrence system. We find the time window when the solution evolves from a deterministic value to the limiting distribution. The underdamped Langevin equation is essentially a slow-fast system admitting three time scales. The unperturbed Hamiltonian system plays a dominant role in the short time scale. In the intermediate time scale, the fast variables have already tended to their stationary distributions and the slow variables transition from their initial deterministic values to the marginal distribution of the slow variables with respect to the system’s stationary distribution. Finally, in the long time scale, all variables are slaved by the stationary distribution.

  • Subjects / Keywords
  • Graduation date
    Fall 2022
  • Type of Item
    Thesis
  • Degree
    Doctor of Philosophy
  • DOI
    https://doi.org/10.7939/r3-9yze-4477
  • License
    This thesis is made available by the University of Alberta Library with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.