{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:24:48Z","timestamp":1787239488253,"version":"build-2736575974"},"reference-count":0,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2023,2]]},"abstract":"In this issue the Education section presents two contributions. The first paper is \u201cChaos Game Representation\u201d (CGR), written by Eunice Y. S. Chan and Robert M. Corless. The chaos game is an algorithm which allows us to produce pictures like fractal structures associated with one-dimensional sequences of integers and, in this way, to \u201cvisualize\u201d such sequences. In particular, these pictures can help to generate conjectures and give an intuition of the \u201cdistance\u201d between sequences. CGR is based on an algorithm from chaotic dynamics called the \u201cchaos game,\u201d popularized by Barnsley. It involves a polygon, a dividing rate, and randomly chosen sequences of vertices of the polygon leading to various fractal-like patterns depending on the game board and the dividing rate. For a triangular game board and some choices of dividing rate, one gets the Sierpinski triangle. The appearance of patterns is explained by the overlapping of the attractors of CGR. This overlapping can be controlled by an appropriate choice of the dividing rate.<\/jats:p>\n The authors also demonstrate, in an experimental way, that the number of vertices in the game board may influence the final patterns. The article provides a nice description of how CGR has been applied to DNA and protein (amino acids) sequencing and visualization. Finally, the CGR is also discussed when the random choice of vertices is replaced by digits of $\\pi$, Fibonacci sequence, and prime numbers, getting various patterns in each case.<\/jats:p>\n The construction of CGR given in this paper can be introduced in a \u201cdynamical systems\u201d or \u201cmodeling\u201d courses classroom.<\/jats:p>\n This teaching module also aims to stimulate a discussion about randomness and its meaning.<\/jats:p>\n The second paper, \u201cSurprises in a Classic Boundary-Layer Problem,\u201d is presented by William A. Clark, Mario W. Gomes, Arnaldo Rodriguez-Gonzalez, Leo C. Stein, and Steven H. Strogatz. It provides a detailed analysis of a nonlinear boundary-value problem<\/jats:p>\n $$ (1.1) \\varepsilon y \u201d = yy' - y , \\quad y(0) = 1 ,\\quad y(1) = -1 $$<\/jats:p>\n and exhibits a number of previously overlooked properties of its solutions. Equation (1.1) is considered in the classic textbook by Mark Holmes [ Introduction to Perturbation Methods, Springer, New York, 1995], where it is argued that it has a unique solution, however after imposing some convexity\/concavity assumptions on solutions near the boundary. In the absence of such an assumption it turns out that (1.1) actually has three solutions whenever $\\varepsilon >0$ is sufficiently small.<\/jats:p>\n In particular, the authors show the existence of a pitchfork bifurcation parameter $\\varepsilon_c=0.2159869288903\\dots$ such that for all $0 < \\varepsilon < \\varepsilon_c $ there are three solutions of the above equation, while for $\\varepsilon \\geq \\varepsilon_c $ the solution is unique. The bifurcation parameter $\\varepsilon_c $ is obtained by an explicit elegant calculation. Then the value of $y'(0)$ for each solution is discussed whenever $\\varepsilon >0$ is sufficiently small. For two of them it is transcendentally close to 1, and explicit estimates of this closeness are explained. Then the occurring pitchfork bifurcation is discussed in more detail. The study relies on converting the second order ODE into a system of two first order ODEs and then investigating the phase portrait to understand qualitative properties of solutions. In particular, solutions are conservative and some of them enjoy symmetry-like properties. Finally, using the obtained estimates, the shooting method is applied to solve (1.1) numerically. The detailed proofs and calculations make this paper almost self-contained and provide a methodology of a complete analysis of (1.1). This problem could be included in courses on perturbation methods, applied dynamical systems, or numerical analysis.<\/jats:p>","DOI":"10.1137\/23n975648","type":"journal-article","created":{"date-parts":[[2023,2,9]],"date-time":"2023-02-09T12:38:38Z","timestamp":1675946318000},"page":"259-260","source":"Crossref","is-referenced-by-count":0,"title":["Education"],"prefix":"10.1137","volume":"65","author":[{"given":"H\u00e9l\u00e8ne","family":"Frankowska","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,2,9]]},"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23N975648","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:34:32Z","timestamp":1787236472000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23N975648"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["10.1137\/23N975648"],"URL":"https:\/\/doi.org\/10.1137\/23n975648","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2]]}}}