The Mathlet Toolkit: How to Create Your Own Interactive/Dynamic Java Graphing Applets
Below I provide instructions for writing, compiling, and then posting java applets for mathematical graphing (mathlets). My students (Victor Skutovsky, Tim Andrews, and Andrew August) and I have created tools (the Mathlet Toolkit), and some sample mathlets, that should enable anyone (with java experience or not) to create and post their own mathlets. I will show you how to download the Mathlet Toolkit, and to download and use Eclipse (one of the standard development environments for java) to compile the mathlets. For detailed instructions click on Getting Started.
Next, I show you how to modify the code of an existing mathlet to create your own mathlet using the Mathlet Toolkit. Click on Creating Mathlets.
Here is a pdf of the Powerpoint I used in my talk at the ICTCM meeting in Orlando in March 2012. Some of the information here is repeated in other places on this website, but some is not (in particular, the slides on the basic operation of the mathlets after they are created). Creating and Incorporating Dynamic Applets for Differential Equations the Easy Way
Sample Mathlets
Below are the sample mathlets that come with the file MathletToolkit.zip (not including Template, which is a generic mathlet used to create new mathlets). The “Getting Started” link above has the download for the file MathletToolkit.zip, which contains all code for the mathlets below.
Document versions: Some of the mathlets have a “document version” which turns the mathlet into a live document by adding instructions, background information, and so on.
Differential Equations and Iterated Maps:
First order DE grapher: graphs solution curves to a single first-order differential equation. FirstOrder
DE systems grapher: graphs solution curves to a system of first-order differential equations, or second order equation if put in first-order form. Systems The document version contains text that explains how to write a second order equation as a system. SystemsDocument
Comparing numerical and exact solutions to a DE: graphs both the exact solution to a simple DE (Newton's law of cooling) and a numerical approximation (using any one of three numerical methods). DEandSolution The document version explains how to use a parameter y0 in an input box so that the exact and numerical solutions can be compared for different parameter values. DEandSolutionDocument
The harvested logistic DE and bifurcation: graphs solution curves for the logistic differential equation (with harvesting) and its constant solutions in one window, and plots the right-hand side of the differential equation as a function in another window in order to better visualize why and when an important bifurcation occurs. LogisticDE The document version adds text and equations to turn the applet into a mini lesson. LogisticDEDocument
Numerical methods as dynamical systems: three applets that are related. The first applet graphs solution curves of the logistic differential equation three times with three different numerical methods to show how the dynamics of fourth-order Runge-Kutta (Rk4), second-order Runge-Kutta (Rk2), and Euler's method are related when applied to the logistic differential equation, as the growth parameter is increased. The second applet graphs two iterated maps to focus on Rk2 versus Euler by writing out the iterated maps explicitly for each method. These iterated maps are then used in the third applet which graphs and overlays two bifurcation diagrams to verify what is discovered in the previous two applets, and reveals new relationships. Tim Andrews, one the of the students who helped with the creation of the Mathlet Toolkit, gave several talks on these relationships. EulerRk2 EulerRk2Maps EulerRk2Bifurcation A document version of the first one gives some background. EulerRk2Document
Three views of a damped pendulum: graphs solution curves of the differential equation governing a damped, rigid pendulum (by first writing the second-order equation as a first-order system), and a function which has in interesting relationship to the DE in the case of large damping. Pendulum The document version explains the large damping phenomena. PendulumDocument
Bifurcations of a pendulum and the trace determinant plane: graphs solution curves for a damped, rigid pendulum in the phase plane and simultaneously graphs the points in the trace determinant plane that give the stability of the fixed points of the pendulum equation. PendTraceDet The document version explains how to use the applet to interpret bifurcations. PendTraceDetDocument
The Lorenz equations: graphs the system of 3 differential equations known as the Lorenz equations. Also illustrates rotations in three dimensions. Lorenz
The Henon map: graphs the well known Henon two dimensional iterated map. As the a parameter is increased one observes period doubling leading to a strange attractor. Henon The document version explains how to observe period doubling and a strange attractor. HenonDocument
Functions:
A function and its derivative: graphs a function and its derivative on the same coordinate system. Also, vertical lines are drawn where the function has a maximum or minimum and whee the derivative is zero in order to drive home the relationship. FunctionDerivative
A spiral using parametric functions: graphs a simple three-dimensional spiral using a three-dimensional parametric function (can also be rotated). Spiral3D
Iterated Function Systems:
Fractals: the last two illustrate iterated functions systems, which can be used to create fractals. The first is the Serpinski triangle, with some parameters which can be interactively adjusted, and the second is the Barnsley fern (no parameters). Serpinski Fern