Chaotic vibration of the wave equation studied through the unbounded growth of the total variation.

摘要

The study of chaotic dynamical systems is an active research field. Of particular interest is chaos in partial differential equations. In this dissertation, we first look at interval maps. Our first result shows a relationship between the total variations and the chaotic dynamic property interval map. If the interval map has sensitive dependence on initial data, then the total variations of the nth iterate fn on each subinterval will grow unboundedly as n → ∞. The converse theorem is also true, if, in addition, f itself has only finitely many extreme points. Such interval maps will have infinitely many periodic points of prime periods 2k, k = 1,2,…. This result suggests that we can use the property of unbounded growth of total variations to study some chaotic dynamic systems, since sensitive dependence and infinitely many periodic points are some of the most important characteristics of chaos. In the second part, we use total variations to study a chaotic infinite dimensional system. We study a certain wave equation with a nonlinear boundary condition and prove that under some conditions that system displays chaotic behavior in the total variation sense previously discussed.