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Analyzing a system with two parameters: use a stability diagram to show where bifurcations occur and to identify regions of bistability

A stability diagram allows us to see where saddle node bifurcations occur in a two parameter space. This allows us to identify how many stable fixed points exist in different parts of parameter space

Plot multiple time series curves for approximate solutions of a differential equation in Mathematica

Use Mathematica to make a phase line by using graphics commands.

Mathematica commands to plot the vector field associated with a first order differential equation.

Bifurcation diagram example.

Consider the period of an oscillation as the system approaches a bifurcation where the oscillation will disappear. How does the period change with the parameter value?

An oscillation example with a bifurcation.

Modeling oscillation with a 1d dynamical system.

Bead on a spinning hoop nondimensionalization example from Strogatz 3.5.

This project was created with Explain Everything™ Interactive Whiteboard for iPad.

nondimensionalizing, step by step

Identifying the dimension of each parameter or variable in a differential equation will ultimately allow us to simplify such equations into a nondimensional form.

Subcritical pitchfork bifurcation diagram (contrasted with the supercritical pitchfork diagram). Note that the pitchfork can open to either the right or the left. It is the stability of the branches that sets sub vs supercritical.

supercritical and subcritical pitchfork

phase portrait example

Lorenz logistic sine Rossler

Change in a function with time

C06 intro to 2d

phase difference

oscillator example

oscillator in 1d

Dimension in differential equations

Flux of a vector field

double integral example

Orbit diagram of logistic map

Orbit diagrams for maps with one peak

Tangent bifurcations leading to periodic orbits

Structure of orbit diagram