ABSTRACT
This chapter is concerned with some important aspects of bifurcations in simple one-dimensional systems. In its most general form, bifurcation theory is concerned with equilibrium solutions of nonlinear systems. In the present context the equilibrium solutions of interest are fixed points and periodic orbits, stable or unstable. Even this can be very technical for general non-linear discrete systems. One should use the Bifurcation Diagrams window of Chaosfor Java to experiment with these bifurcation and final state diagrams for oneself, to check that period doubling does indeed continue in an infinite cascade, and to examine the structure of some unstable periodic orbits. One of the common features of final state diagrams is the fact that a chaotic attractor may change its size discontinuously, or even appear or disappear suddenly, at a critical value of a parameter. Such an occurrence is an example of a crisis.
