Physical systems evolving on time-dependent domains

• Author / Creator

  Ghadiri Motlagh, Mahdi

• Despite the ubiquity of physical systems evolving on time-dependent spatial domains ranging from
  crystal growth, formation of patterns and shapes in biology and living organisms – animals skin
  patterns, tentacle patterns on Hydra, whorled leaves, teeth primordia in the alligator – to quantum
  particles traveling in a time-evolving potential, fluid motion, fluid-structure interaction, and galaxies
  agglomeration in the expanding Universe, to name a few, understanding their regular and chaotic
  dynamical properties is still in a quite rudimentary state. The underlying theme of this dissertation is
  to explore the key differences in the dynamics – both regular and chaotic – between extended systems
  on time-fixed and time-dependent spatial domains, studied here with the synergy of experimental
  and theoretical approaches and numerical simulations.
  In the quest to understand dynamics of distributed systems on time-dependent spatial domains,
  in chapter 2, we study experimentally the response to domain deformations by Faraday wave patterns
  – standing waves formed on the free surface of a liquid layer due to its vertical vibration – chosen
  as a paradigm owing to their historical use in testing new theories and ideas. In our experimental
  setup of a vibrating water container with controlled positions of lateral walls and liquid layer depth,
  the characteristics of the patterns are measured using the Fourier transform profilometry technique,
  which allows us to reconstruct an accurate time history of the pattern three-dimensional landscape
  and reveal how it reacts to the domain dynamics on various length- and time-scales.
  Analysis of Faraday waves on growing, shrinking, and oscillating domains leads to a number of intriguing results. First, the observation of a transverse instability – namely, when a twodimensional pattern experiences an instability in the direction orthogonal to the direction of the
  domain deformation – provides a new facet to the stability picture compared to one-dimensional systems in which the longitudinal (Eckhaus) instability accounts for pattern transformation on time-dependent domains. Second, the domain evolution rate is found to be a key factor dictating the patterns observed on the path between the initial and final domain aspect ratios. Its effects range from
  allowing the formation of complex sequences of patterns to impeding the appearance of any new
  pattern on the path. Third, the shrinkage-growth process turns out to be generally irreversible on a
  horizontally evolving domain, but becomes reversible in the case of a time-dependent liquid layer
  depth, i.e. when the dilution and convective effects are absent. These experimentally observed
  enigmatic effects of the domain size variations in time are complemented here with appropriate
  theoretical insights elucidating the nature of the phenomena and disentangling the dynamics of
  two-dimensional pattern evolution, which proves to be more intricate compared to one-dimensional
  systems.
  In chapter 3, we present the experimental discovery of a novel mechanism to control chaos by
  time-variation of the spatial domain size. Moreover, depending upon the rate of the latter the chaotic
  state may be prevented altogether. As a testbed to traverse the edge of chaos by varying the domain
  size, we have chosen the Faraday waves phenomenon, which is a paradigmatic example in pattern forming systems due to its simplicity and richness, in particular known to exhibit temporal chaos.
  The experimental findings are disentangled with theoretical insights and numerical modeling, which
  also demonstrates the ability to control spatio-temporal chaos. These findings may shed some light
  on biological systems and life, which require ‘a healthy dose of chaos’ for proper operation (Korolja et al., 2019) and hence often balance on the edge of chaos. The latter concept has also been
  applied in many other areas (Waldrop, 1993): in economy, creative destruction represents the driving force within a market economy; in social science, the dynamic interaction between individuals
  and macro-levels such as laws, religions, and governments imposing too much order and limiting
  individual development in the name of conformity, ultimately leading to stasis; in human cognition
  and creativity (Schwartz, 2014), the states at the edge of chaos can be seen to be maximally novel
  while still connected to ones in the ordered regime – the hallmark of innovative thinking.

• Subjects / Keywords

  • Time dependent domains
  • Pattern formation
  • Faraday waves
  • Chaos

• Graduation date

  Spring 2021

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-sps4-r190

• License

  This thesis is made available by the University of Alberta Libraries 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.