A central problem in nonlinear dynamics is that of discovering how the qualitative dynamical properties of orbits change and evolve as a dynamical system is continuously changed. More specifically, consider a dynamical system which depends on a single scalar parameter. We ask, what happens to the orbits of the system if we examine them at different values of the parameter? We have already met this question and substantially answered it for the case of the logistic map, xn=1 = rxn(1 − xn). In particular, we found in Chapter 2 that as the parameter r is increased there is a period doubling cascade, terminating in an accumulation of an infinite number of period doublings, followed by a parameter domain in which chaos and periodic ‘windows’ are finely intermixed. Another example of a context in which we have addressed this question is our discussion in Chapter 6 of Arnold tongues and the transition from quasiperiodicity to chaos. Still another aspect of this question is the types of generic bifurcations of periodic orbits which can occur as a parameter is varied. In this regard recall our discussions of the generic bifurcations of periodic orbits of one-dimensional maps (Section 2.3) and of the Hopf bifurcation (Chapter 6).

In this chapter we shall be interested in transitions of the system behavior with variation of a parameter such that the transitions involve chaotic orbits.