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Bifurcation to Instability Through the Lens of the Maslov Index

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Abstract

The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems (Cornwell and Jones in SIAM J Appl Dyn Syst 17(1):754–787, 2018). In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These standing waves are known to become unstable as parameters vary. Our goal is to see how this established stability criterion manifests itself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism for instability. We further suggest how this mechanism might be used to reveal new instabilities in singularly perturbed models.

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Notes

  1. The reader must decide whether tracking the unstable manifold should be done with [25] or without [5] the use of differential forms.

  2. For a proof that \(E^{s/u}(\lambda ,\xi )\) are Lagrangian subspaces for fixed \(\lambda \ge 0\) and \(\xi \in {\mathbb {R}}\), see [13, Theorem 2.3].

  3. This is true specifically for the second corner, which is the transition from the first slow segment to the fast back. Although multiple heteroclinic connections exist at the other two corners as well, neither path taken would result in a conjugate point.

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Acknowledgements

C.J. acknowledges support from the US Office of Naval Research under Grant Number N00014-18-1-2204.

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Authors and Affiliations

  1. Johns Hopkins University Applied Physics Lab, 11100 Johns Hopkins Road, Laurel, MD, 20723-6099, USA

    Paul Cornwell

  2. Department of Mathematics, UNC Chapel Hill, Phillips Hall CB #3250, Chapel Hill, NC, 27516, USA

    Christopher K. R. T. Jones & Claire Kiers

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  1. Paul Cornwell
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Correspondence to Paul Cornwell.

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Cornwell, P., Jones, C.K.R.T. & Kiers, C. Bifurcation to Instability Through the Lens of the Maslov Index. J Dyn Diff Equat 36 (Suppl 1), 127–148 (2024). https://doi.org/10.1007/s10884-021-10017-1

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