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Published June 3, 2019 | Published
Journal Article Open

Breathers and other time-periodic solutions in an array of cantilevers decorated with magnets


In this article, the existence, stability and bifurcation structure of time-periodic solutions (including ones that also have the property of spatial localization, i.e., breathers) are studied in an array of cantilevers that have magnetic tips. The repelling magnetic tips are responsible for the intersite nonlinearity of the system, whereas the cantilevers are responsible for the onsite (potentially nonlinear) force. The relevant model is of the mixed Fermi-Pasta-Ulam-Tsingou and Klein-Gordon type with both damping and driving. In the case of base excitation, we provide experimental results to validate the model. In particular, we identify regions of bistability in the model and in the experiment, which agree with minimal tuning of the system parameters. We carry out additional numerical explorations in order to contrast the base excitation problem with the boundary excitation problem and the problem with a single mass defect. We find that the base excitation problem is more stable than the boundary excitation problem and that breathers are possible in the defect system. The effect of an onsite nonlinearity is also considered, where it is shown that bistability is possible for both softening and hardening cubic nonlinearities.

Additional Information

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0). Received: 11 December 2018, Accepted: 19 April 2019, Published: 03 June 2019. This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1615037 (CC) and DMS-1809074 (PGK). CC would like to thank Carina Spiro and Jacob Hart of Bowdoin College and Chenzhang Zhou of the École Polytechnique for technical assistance. EGC would like to thank the Department of Mathematics at Bowdoin College for the kind hospitality where the initial stages of this work were carried out. PGK gratefully acknowledges support from the US-AFOSR via FA9550-17-1-0114. The authors declare no conflict of interest.

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