Chaos Theory and the Logistic Map

Logistic map bifurcation diagram showing the period-doubling path to chaosUsing Python to visualize chaos, fractals, and self-similarity to better understand the limits of knowledge and prediction. Download/cite the article here and try pynamical yourself.

Chaos theory is a branch of mathematics that deals with nonlinear dynamical systems. A system is just a set of interacting components that form a larger whole. Nonlinear means that due to feedback or multiplicative effects between the components, the whole becomes something greater than just adding up the individual parts. Lastly, dynamical means the system changes over time based on its current state. In the following piece (adapted from this article), I break down some of this jargon, visualize interesting characteristics of chaos, and discuss its implications for knowledge and prediction.

Chaotic systems are a simple sub-type of nonlinear dynamical systems. They may contain very few interacting parts and these may follow very simple rules, but these systems all have a very sensitive dependence on their initial conditions. Despite their deterministic simplicity, over time these systems can produce totally unpredictable and wildly divergent (aka, chaotic) behavior. Edward Lorenz, the father of chaos theory, described chaos as “when the present determines the future, but the approximate present does not approximately determine the future.”

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Using geopandas on Windows

projected-shapefile-gps-coordinatesThis guide was updated in June 2016 to reflect changes to the dependencies and the ability to install with Python wheels.

I recently went through the exercise of installing geopandas on Windows and getting it to run. Having learned several valuable lessons, I thought I’d share them with the world in case anyone else is trying to get this toolkit working in a Windows environment (also see this GitHub gist I put together).

It seems that pip installing geopandas works fine on Linux and Mac. However, several of its dependencies have C extensions that cause compilation failures with pip on Windows. This guide gets around that issue. For preliminaries, I have this working on Windows 7, 8, and 10. My Python environments are Anaconda, 64-bit, with both Python 2.7 and 3.5. I’m running geopandas version 0.2 with GDAL 2.0.2, Fiona 1.7.0, pyproj 1.9.5.1, and shapely 1.5.16.

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