Chaotic Behavior of Interval Maps and Total Variations of iterates

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f : I → I is chaotic, then its iterates f will display heightened oscillatory behavior or profiles as n → ∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of f on subintervals. There are four distinctive cases of the growth of total variations of f as n → ∞: (i) the total variations of f on I remain bounded; (ii) they grow unbounded, but not exponentially with respect to n; (iii) they grow with an exponential rate with respect to n; (iv) they grow unbounded on every subinterval of I . We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of f are more extreme, the more complex is the behavior of the system.