## Uncovering the patterns behind patterns, Elizabeth Makrides

*Figure 1: Examples of bifurcation diagrams corresponding to widely disparate physical systems with similar underlying mathematical structure. Figures are reproduced from published works as follows: (a) plane Couette flow [8]; (b) celluar buckling [2]; (c) crime hotspots [3]; (d) optical cavity solitons [6]; (e) vegetation patterns [7]. *

**Description**: In this project we will explore localized patterns on the plane. Such patterns appear in a wide variety of physical contexts, which include -- but are not limited to! -- fluid flows, crime hot spots, buckling problems, vegetation growth and optical systems. Bifurcation diagrams for widely disparate systems have proved to be remarkably similar; see Figure 1.

Various types of patterns which are periodic in one direction and localized in the other, including those termed "rolls", "spots and stripes" and "squares" have been investigated (see, for example, [1], and refer to Figure 2 for visualizations of particular patterns). We would like to use this time and computing resources to better understand the relationships between these patterns and, in particular, their connections in parameter space.

As time and interest allow, we may also explore the formation of large localized hexagon patches. Computing resources were specifically identified as a limiting factor in a 2008 study [4]. However, since this time the underlying package capabilities have progressed substantially, and revisiting this problem with new computing resources may enable substantial progress.

*Figure 2: Examples of patterns observed in the Swift-Hohenberg Equation on an infinite cylinder. See [1] and [5]. *

**Prerequisites**: While there are no formal prerequistes for participation in this project, some familiarity with differential equations and dynamical systems theory would be helpful. In particular, a basic understanding of bifurcation theory underlies most of the work. I would be happy to review with any interested students lacking this background; Strogatz [9] also provides an accessible and useful introduction. Programming for this project will be conducted in C/C++. Current software is written in MATLAB, so our first task will be converting this software. Depending on our progress we may also use Auto07p, a Fortran based program for continuation, but no knowledge of Fortran is expected.

**References**

[1] D. Avitabile, D. J. B. Lloyd, J. Burke, E. Knobloch and B. Sandstede. To snake or not to snake in the planar Swift-Hohenberg equation. SIAM J. Appl. Dyn. Syst. 9 (2010) 704–733.

[2] G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J. Budd and G. J. Lord. Cellular buckling in long structures. Nonlin. Dynam. 21 (2000) 3–29.

[3] D. J. B. Lloyd and H. O’Farrell. On localised hotspots of an urban crime model. Physica D 253 (2013) 23–39.

[4] D. J. B. Lloyd, B. Sandstede, D. Avitabile and A. R. Champneys. Localized hexagon patterns of the planar Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 7 (2008) 1049–1100.

[5] E. Makrides and B. Sandstede. Predicting the bifurcation structure of localized snaking patterns. Physica D 269 (2014).

[6] J. M. McSloy, W. J. Firth, G. K. Harkness and G.-L. Oppo. Computationally determined existence and stability of transverse structures II. Multipeaked cavity solitons. Phys. Rev. E 66 (2002) 046606.

[7] E. Meron. Pattern-formation approach to modelling spatially extended ecosystems. Ecological Modelling 234 (2012) 70–82.

[8] T. M. Schneider, J. F. Gibson and J. Burke. Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104 (2010) 104501.

[9] S. H. Strogatz. Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering. Studies in Nonlinearity, Westview Press, 2001.