ABSTRACT
This chapter introduces two different families that illustrate many of the fundamental concepts of bifurcation theory. A bifurcation diagram is a plot that summarizes how the dynamics of a system change as the parameter changes. Ideally, it should tell us about all periodic points, both attracting and repelling, and any other complicated behavior that may occur. The chapter discusses how to numerically generate bifurcation diagrams that completely describe all of the attracting dynamics. Probably, the most common bifurcation is the tangent or saddle-node bifurcation. A tangent bifurcation occurs when an attracting fixed point and a repelling fixed point merge at some parameter value. The tangent bifurcation theorem describes what often happens when the derivative at the fixed point is equal to 1.
