ABSTRACT
Dynamical systems are used to model a variety of time-dependent systems. Visualizations of dynamical systems can display a large amount of information in a single image, but generating these images requires both mathematical and programming expertise. dsmodels is a domain-specific language (DSL) for visualizing two-dimensional dynamical systems. dsmodels speeds up the process of visualizing a dynamical system by providing primitives to capture models, encapsulate features, and depict the overall behavior of the system. dsmodels can also simulate dynamical systems to compute attractors and their basis of attraction, allowing for rapid prototyping or informal analysis. We present dsmodels using a case study of population models.
- H. N. Agiza. 1999. On the Analysis of Stability, Bifurcation, Chaos and Chaos Control of Kopel Map. Chaos, Solitons & Fractals 10, 11 (Nov 1999), 1909--1916.Google Scholar
- E.G. Altmann. 2017. Introduction to dynamical systems using billiards. (2017). https://www.pks.mpg.de/nonlinear-dynamics-and-time-series-analysis/visualization-of-dynamical-systems/ (Accessed on 02/07/2018).Google Scholar
- L. Assas, S. Elaydi, E. Kwessi, G. Livadiotis, and D. Ribble. 2014. Hierarchical competition models with Allee effects. Journal of Biological Dynamics 9, sup1 (jun 2014), 32--44.Google Scholar
- G. I. Bischi, L. Stefanini, and L. Gardini. 1998. Synchronization, intermittency and critical curves in a duopoly game. Mathematics and Computers in Simulation 44, 6 (1998), 559--585. Google ScholarDigital Library
- C. Brewer. 2017. ColorBrewer: Color Advice for Maps. (2017). http://colorbrewer2.org/ (Accessed on 02/06/2018).Google Scholar
- S. Elaydi. 2008. Discrete Chaos with Applications in Science and Engineering. Chapman & Hall, London, England.Google Scholar
- M. Ezekiel. 1938. The cobweb theorem. The Quarterly Journal of Economics 52, 2 (1938), 255--280.Google ScholarCross Ref
- M. Fowler. 2011. Domain-specific languages. Addison-Wesley, Upper Saddle River, NJ. Google ScholarDigital Library
- G. Grothendieck, L. Kates, and T. Petzoldt. 2016. proto: Prototype Object--Based Programming. CRAN. https://CRAN.R-project.org/package=proto R package version 1.0.0.Google Scholar
- E. Gunawan. 2013. PyDYN. (2013). https://github.com/rwl/PyDyn (Accessed on 02/07/2018).Google Scholar
- Wolfram Research, Inc. 2017. Mathematica, Version 11.2. (2017). Champaign, IL, 2017.Google Scholar
- D. Kahle and H. Wickham. 2013. ggmap: Spatial Visualization with ggplot2. The R Journal 5, 1 (2013), 144--161. http://journal.r-project.org/archive/2013-1/kahlewickham.pdfGoogle ScholarCross Ref
- H. Kitano. 2001. Foundations of Systems Biology. The MIT Press, Massachusetts, USA.Google Scholar
- G. Livadiotis, L. Assas, S. Elaydi, E. Kwessi, and D. Ribble. 2014. Competition models with Allee effects. Journal of Difference Equations and Applications 20, 8 (May 2014), 1127--1151.Google ScholarCross Ref
- M. Martelli. 1999. Introduction to Discrete Dynamical Systems and Chaos. John Wiley & Sons, Inc., Texas, USA.Google Scholar
- MATLAB. 2017. version 1.7.0_60 (R2017a). The MathWorks Inc., Massachusetts, USA.Google Scholar
- R. M. May. 1976. Simple mathematical models with very complicated dynamics. Nature 261, 5560 (1976), 459.Google Scholar
- M. Mernik, J. Heering, and A. M. Sloane. 2005. When and How to Develop Domain-Specific Languages. ACM Comput. Surv. 37, 4 (Dec 2005), 316--344. Google ScholarDigital Library
- H.O. Peitgen and P.H. Richter. 2013. The beauty of fractals: images of complex dynamical systems. Springer Science & Business Media, Berlin, Germany.Google Scholar
- H. Poincaré. 1899. Les méthodes nouvelles de la mécanique céleste: Invariants intégraux. Solutions périodiques du deuxième genre. Solutions doublement asymptotiques. Gauthier-Villars et fils, Paris, France.Google Scholar
- R Core Team. 2016. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/Google Scholar
- C. N. Stein and S. Fogarty. 2017. dsmodels: A Language to Facilitate Simulation and Visualization of Two-Dimensional Dynamical Systems. Trinity University. https://CRAN.R-project.org/package=dsmodels R package version 1.1.0.Google Scholar
- C. N. Stein and S. Fogarty. 2017. dsmodels API. (2017). http://www.cs.trinity.edu/-sfogarty/dsmodels/index.html (Accessed 07/22/2017).Google Scholar
- Grace Development Team. 2007. Grace. (2007). http://plasma-gate.weizmann.ac.il/Grace/ (Accessed on 02/07/2018).Google Scholar
- H. Wickham. 2009. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, New York, USA. http://ggplot2.org Google ScholarDigital Library
- H. Wickham, P. Danenberg, and M. Eugster. 2017. roxygen2: In-Line Documentation for R. CRAN. https://CRAN.R-project.org/package=roxygen2 R package version 6.0.1.Google Scholar
- S. Wiggins. 2003. Introduction to applied nonlinear dynamical systems and chaos. Vol. 2. Springer Science & Business Media, Berlin, Germany.Google Scholar
- X. Q. Zhao. 2017. Dynamical Systems in Population Biology. Springer Science & Business Media, Berlin, Germany.Google Scholar
Index Terms
- dsmodels: A Little Language for Dynamical Systems
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