In this course, we will use a variety of numerical solvers and other software tools. You do not need to purchase any of them as they will all be available either
on the web
(so you can use them at home) or
in the Honors College Computer Lab
(MHC 111). Here is a list of some software tools and where to find them, thanks to Bob Devaney at Boston University.
The following tools require no special software -- only a Java-enabled web browser (like Netscape Navigator or Internet Explorer) and access to the World Wide Web. Therefore, they can be used on a home computer, outside our computer lab.
Single ODE solvers and plotters: they can plot the slope field for a single differential equation of the form dy/dt = f(t,y) and plot its solution curves.
Java Applet beta versions of
John Polking's ODE tools.
Be patient with the download and press the big button labeled DFIELD. Several windows will pop up; you may close some of them. It may happen that the applet doesn't work (as its author says, it's a work in progress).
Several TI-83 programs for slope field and Euler's method plotting.
Too long to post here; it's faster to come to my office (MHC 162) and download them yourself...
Interactive Differential Equations (IDE)
suite by
Addison Wesley Longman.
A great visualization suite of tools with pre-set equations. Unfortunately, you cannot enter your own equations in the demo version.
In our class discussions so far, I have used the following IDE tools:
Saddle-node
(a source-sink pair appears or disappears)
Trans-critical
(a source and a sink pass each other and change types)
Pitchfork
(same-type equilibrium pair appears or disappears and another equilibrium changes type)
ODE system solvers and plotters: they can plot the direction field for a system of differential equations of the form dx/dt = f(x,y), dy/dt = g(x,y) and plot its solution curves (trajectories). These tools can be used also for single ODEs, since the first-order ODE dy/dx = f(x,y) is equivalent to the first-order system dx/dt = 1, dy/dt = f(x,y).
Phase portrait plotter and ODE system solver by
Richard Mansfield at Penn State.
Two applet windows pop up immediately and won't close until you leave the page.
Discrete dynamical system plotters:
they can plot the discrete orbit of a dynamical system for different values of a given parameter. The applets were developed at Boston University.
Web Diagram.
Plots web diagrams for a pre-set list of dynamical systems; you can adjust the parameter.
Linear Web.
Plots web diagrams and time series for the linear dynamical system; you can adjust the slope and intercept.
Orbit (Bifurcation) Diagram.
Plots the "steady-state" behavior (stable periodic points or chaos) of a dynamical system (selected from a list) for a range of parameter values. (The bifurcation diagram for the Logistics is also known as the "Feigenbaum Road to Chaos".)
Histogram plotter.
Plots time series or a histogram of a dynamical system (selected from a list). Try cx(1-x) for c=4 and x(0)=0.2.
A simple example
of plotting slope fields and solutions of a single ODE. It requires the special (and familiar) commercial software Maple. Therefore, it can be used only in our computer lab. To see it in action, just
follow the link above and, when prompted, choose to "Open the page from it's current location" -- Maple will start automatically.
An introduction
to plotting direction fields and solutions of a system of ODEs.