Table des matières
Integration of Ordinary Differential Equations
- Ordinary Differential Equations (ODE, ODEs)
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- « most widely known member of the Runge–Kutta family is generally referred to as “RK4”, “classical Runge–Kutta method” or simply as “the Runge–Kutta method »
Applications
- chemical kinetics
- population dynamics, Logistic function (and equation)
- Pierre François Verhulst model and non-linearity
- bifurcation, period doubling and routes to chaos
- Oscillating chemical reactions : Belousov-Zhabotinsky, Brusselator, Oregonator
- Lotka–Volterra equations (predator–prey equations)
- molecular dynamics, Verlet integration
Lorenz system
« The Lorenz system is a system of ordinary differential equations (the Lorenz equations, note it is not Lorentz) first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight. »
- Numerical solutions (and codes) :
- http://jakevdp.github.io/blog/2013/02/16/animating-the-lorentz-system-in-3d/ (2012, Jake Vanderplas)
- with Ipython notebook
- http://nbviewer.ipython.org/gist/dpsanders/d417c1ffbb76f13f678c, including a tutorial on ODEs
- https://ipywidgets.readthedocs.io/en/stable/examples/Lorenz%20Differential%20Equations.html, in the Jupyter Widgets documentation (2016…)
- On active state
Python libraries
- on Stackoverflow :
References
- Numerical recipes, The Art of Scientific Computing 3rd Edition, William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery, 2007, isbn: 9780521880688
- predator–prey models :
- see seminal and recent references here : Evolutionarily stable strategies in stable and periodically fluctuating populations: The Rosenzweig–MacArthur predator–prey model Katrin Grunert, Helge Holden, Espen R. Jakobsen, Nils Chr. Stenseth, Proceedings of the National Academy of Sciences Jan 2021, 118 (4) e2017463118; DOI: 10.1073/pnas.2017463118
- Automatic Code Generation with SymPy → applications to ODE and chemical kinetics
- Runge-Kutta Numerical Integration of Ordinary Differential Equations in Python Andrew Joseph Davies, Medium, 23/01/2022