Dynamics provides the concepts and tools to describe and understand complicated long-term behavior in systems that evolve in time. An excellent way to acquire and appreciate these concepts and tools is in a gradual progression from simple to complex behavior during which examples, concepts, and tools all evolve toward greater sophistication.

Accordingly, the course begins with the most simple dynamical behavior (a universal steady state), which turns out to be a fundamental tool as well. (This latter theme is developed in detail in Chapter 9.) Chapter 2 gently progresses to also include systems with several steady states. Chapter 3 studies linear maps as slightly more complicated yet quite tractable systems that double as a central tool later.

Complexity first appears in Chapter 4 (and moves to higher dimension in Chapter 5). We encounter orbits whose long-term behavior involves close but not exact returns to the initial state (recurrence), as well as distribution throughout the space of states. However, this potential complexity is tempered by the regularity of individual asymptotics and uniformity of behavior across orbits. This is also borne out by statistical analysis, which is introduced here. Mechanical systems (Chapter 6) provide natural examples where some complexity (recurrence) is forced while the intrinsic structure often limits complexity to the level of the previous two chapters.

The highest level of complexity is reached in Chapter 7, where complicated recurrence is highly irregular for individual orbits and the asymptotic behavior of orbits is thoroughly heterogeneous and inextricably intertwined.