UK Nonlinear News, November 2000
Forced symmetry-breaking, or the introduction of small perturbations that reduce the symmetry of a system, can be responsible for complex dynamics in a system that would otherwise behave in a regular manner. This observation is of particular interest because in real physical systems symmetries are rarely exact and symmetry-breaking imperfections must be assumed to be present. This dissertation shows that forced symmetry-breaking can lead to periodic or irregular bursts of very large dynamic range. The results are applied to relevant hydrodynamical systems.
First, a model of large aspect-ratio binary fluid convection which considers the competition between two nearly degenerate modes of opposite parity is studied. The model corresponds to a Hopf bifurcation with broken D4 symmetry, where the ``interchange'' symmetry between the two modes is weakly broken because of the large but finite aspect-ratio. For open parameter regimes, it is shown that periodic or irregular bursting with very large dynamic range may occur close to threshold. The bursts are found to be associated with global bifurcations involving fixed points and periodic orbits ``at infinity''. Global connections involving finite amplitude states are also present. The intricate sequence of bifurcations that take place is described in several cases, and the robustness of the results to small higher order terms in the amplitude equations is demonstrated.
Second, the effect of resonant temporal forcing on a system undergoing a Hopf bifurcation with D4 symmetry is studied. The forcing breaks the continuous normal form symmetry of the amplitude equations. For the example which is considered, a type of gluing bifurcation (called a ``supergluing bifurcation'') is found and described. It is also shown that the attracting quasiperiodic solution which exists in the absence of forcing ``wrinkles'' into chaos as the amplitude of the forcing is increased, with the overall behavior governed by the approach of the attractor to solutions at infinity. Windows in which stable periodic solutions exist are also found and are associated with the traversal in parameter space through Arnol'd tongues. In addition, it is shown that bursts related to those found in the model of large aspect-ratio binary fluid convection may arise due to the forcing. Here the onset of bursting can be via an interior crisis or Type I intermittency.
Finally, other mechanisms that lead to behavior that has been called bursting are reviewed and compared to the new mechanism described in this dissertation.
The thesis is available from http://www.math.princeton.edu/~jmoehlis
Source: Jeff Moehlis
The reaction-diffusion (Turing) mechanism is one of the simplest and most elegant theories for biological pattern formation. The recent experimental realisation of Turing patterns in chemical systems has fostered renewed interest in reaction-diffusion theory, however, its relevance to many biological problems has been questioned because of the perceived failure of the mechanism to generate patterns reliably. A recent paper suggesting the involvement of reaction-diffusion in fish skin patterns has implicated domain growth as an important mechanism controlling pattern selection. In this thesis we present a systematic study of the effects of domain growth on reaction-diffusion patterns, and discuss the implications for reliable pattern generation.
Starting from the postulate that tissue growth rates are locally determined, we derive general evolution equations for reaction-diffusion on growing domains as a problem in kinematics. We argue that the biologically plausible scenario is to consider domain growth on a longer timescale than pattern formation. Then it is found that the solution goes through a sequence of recognisable (quasi-steady) patterns. Using symmetry arguments relating different pattern modes we show that for uniform domain growth the solution evolves by frequency-doubling, the regular splitting or insertion of peaks in the pattern. For pattern formation in two spatial dimensions domain growth is found to select rectangular lattices, rather than the hexagonal planform that is preferred on the fixed domain. For nonuniform growth the local tissue expansion rate varies across the domain and splitting or insertion may be restricted to regions of the domain where the growth is sufficiently fast.
The behaviour of solutions can be studied asymptotically and peak splitting and insertion are shown to occur according to the form of the reaction nullclines. We highlight a novel behaviour, frequency-tripling, where both mechanisms operate simultaneously, which is realised when quadratic terms are absent from the reaction kinetics. Any particular pattern in a sequence remains established until the domain is sufficiently large that a transition to a higher pattern mode occurs. This presents a degree of scale invariance. The pattern which persists finally is not strongly dependent on the final domain size, and hence domain growth can provide a mechanism for reliable pattern selection.
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