© 2014 IOP Publishing Ltd & London Mathematical Society. We explore the bifurcation structure of a modified Cahn-Hilliard equation that describes a system that may undergo a first-order phase transition and is kept permanently out of equilibrium by a lateral driving. This forms a simple model, e.g., for the deposition of stripe patterns of different phases of surfactant molecules through Langmuir-Blodgett transfer. Employing continuation techniques the bifurcation structure is numerically investigated using the non-dimensional transfer velocity as the main control parameter. It is found that the snaking structure of steady front states is intertwined with a large number of branches of time-periodic solutions that emerge from Hopf or period-doubling bifurcations and end in global bifurcations (sniper and homoclinic). Overall the bifurcation diagram has a harp-like appearance. This is complemented by a two-parameter study in non-dimensional transfer velocity and domain size (as a measure of the distance to the phase transition threshold) that elucidates through which local and global codimension 2 bifurcations the entire harp-like structure emerges.
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: The authors are grateful to the Newton Institute in Cambridge, UK, for its hospitality during their stay at the programme 'Mathematical Modelling and Analysis of Complex Fluids and Active Media in Evolving Domains' where part of this work was done. MHK acknowledges the support by the Human Frontier Science Program (Grant RGP0052/2009-C). This publication is based in part on work supported by Award No KUK-C1-013-04, made by the King Abdullah University of Science and Technology (KAUST), and LabEX ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL*.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.