Physical systems evolving on time-dependent domains

Abstract

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.